List of Syntax, Axioms (ax-) and
Definitions (df-)
Ref | Expression (see link for any distinct variable requirements)
|
wn 3 | wff ¬ 𝜑 |
wi 4 | wff (𝜑 → 𝜓) |
ax-mp 5 | ⊢ 𝜑
& ⊢ (𝜑 → 𝜓) ⇒ ⊢ 𝜓 |
ax-1 6 | ⊢ (𝜑 → (𝜓 → 𝜑)) |
ax-2 7 | ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))) |
ax-3 8 | ⊢ ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑)) |
wb 195 | wff (𝜑 ↔ 𝜓) |
df-bi 196 | ⊢ ¬ (((𝜑 ↔ 𝜓) → ¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑))) → ¬ (¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)) → (𝜑 ↔ 𝜓))) |
wo 382 | wff (𝜑 ∨ 𝜓) |
wa 383 | wff (𝜑 ∧ 𝜓) |
df-or 384 | ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓)) |
df-an 385 | ⊢ ((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜓)) |
wif 1006 | wff if-(𝜑, 𝜓, 𝜒) |
df-ifp 1007 | ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒))) |
w3o 1030 | wff (𝜑 ∨ 𝜓 ∨ 𝜒) |
w3a 1031 | wff (𝜑 ∧ 𝜓 ∧ 𝜒) |
df-3or 1032 | ⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((𝜑 ∨ 𝜓) ∨ 𝜒)) |
df-3an 1033 | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒)) |
wnan 1439 | wff (𝜑 ⊼ 𝜓) |
df-nan 1440 | ⊢ ((𝜑 ⊼ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓)) |
wxo 1456 | wff (𝜑 ⊻ 𝜓) |
df-xor 1457 | ⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓)) |
wal 1473 | wff ∀𝑥𝜑 |
cv 1474 | class 𝑥 |
wceq 1475 | wff 𝐴 = 𝐵 |
wtru 1476 | wff ⊤ |
df-tru 1478 | ⊢ (⊤ ↔ (∀𝑥 𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥)) |
wfal 1480 | wff ⊥ |
df-fal 1481 | ⊢ (⊥ ↔ ¬
⊤) |
whad 1523 | wff hadd(𝜑, 𝜓, 𝜒) |
df-had 1524 | ⊢ (hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ⊻ 𝜓) ⊻ 𝜒)) |
wcad 1536 | wff cadd(𝜑, 𝜓, 𝜒) |
df-cad 1537 | ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓)))) |
wex 1695 | wff ∃𝑥𝜑 |
df-ex 1696 | ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑) |
wnf 1699 | wff Ⅎ𝑥𝜑 |
wnfOLD 1700 | wff Ⅎ𝑥𝜑 |
df-nf 1701 | ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) |
df-nfOLD 1712 | ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑)) |
ax-gen 1713 | ⊢ 𝜑 ⇒ ⊢ ∀𝑥𝜑 |
ax-4 1728 | ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓)) |
ax-5 1827 | ⊢ (𝜑 → ∀𝑥𝜑) |
wsb 1867 | wff [𝑦 / 𝑥]𝜑 |
df-sb 1868 | ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) |
ax-6 1875 | ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦 |
ax-7 1922 | ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧)) |
wcel 1977 | wff 𝐴 ∈ 𝐵 |
ax-8 1979 | ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧)) |
ax-9 1986 | ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦)) |
ax-10 2006 | ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) |
ax-11 2021 | ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) |
ax-12 2034 | ⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
ax-13 2234 | ⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) |
weu 2458 | wff ∃!𝑥𝜑 |
wmo 2459 | wff ∃*𝑥𝜑 |
df-eu 2462 | ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) |
df-mo 2463 | ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) |
ax-ext 2590 | ⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦) |
cab 2596 | class {𝑥 ∣ 𝜑} |
df-clab 2597 | ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ [𝑥 / 𝑦]𝜑) |
df-cleq 2603 | ⊢ (∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧) → 𝑦 = 𝑧) ⇒ ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) |
df-clel 2606 | ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵)) |
wnfc 2738 | wff Ⅎ𝑥𝐴 |
df-nfc 2740 | ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴) |
wne 2780 | wff 𝐴 ≠ 𝐵 |
wnel 2781 | wff 𝐴 ∉ 𝐵 |
df-ne 2782 | ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) |
df-nel 2783 | ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵) |
wral 2896 | wff ∀𝑥 ∈ 𝐴 𝜑 |
wrex 2897 | wff ∃𝑥 ∈ 𝐴 𝜑 |
wreu 2898 | wff ∃!𝑥 ∈ 𝐴 𝜑 |
wrmo 2899 | wff ∃*𝑥 ∈ 𝐴 𝜑 |
crab 2900 | class {𝑥 ∈ 𝐴 ∣ 𝜑} |
df-ral 2901 | ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) |
df-rex 2902 | ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) |
df-reu 2903 | ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) |
df-rmo 2904 | ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) |
df-rab 2905 | ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
cvv 3173 | class V |
df-v 3175 | ⊢ V = {𝑥 ∣ 𝑥 = 𝑥} |
wcdeq 3385 | wff CondEq(𝑥 = 𝑦 → 𝜑) |
df-cdeq 3386 | ⊢ (CondEq(𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑)) |
wsbc 3402 | wff [𝐴 / 𝑥]𝜑 |
df-sbc 3403 | ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) |
csb 3499 | class ⦋𝐴 / 𝑥⦌𝐵 |
df-csb 3500 | ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} |
cdif 3537 | class (𝐴 ∖ 𝐵) |
cun 3538 | class (𝐴 ∪ 𝐵) |
cin 3539 | class (𝐴 ∩ 𝐵) |
wss 3540 | wff 𝐴 ⊆ 𝐵 |
wpss 3541 | wff 𝐴 ⊊ 𝐵 |
df-dif 3543 | ⊢ (𝐴 ∖ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)} |
df-un 3545 | ⊢ (𝐴 ∪ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)} |
df-in 3547 | ⊢ (𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} |
df-ss 3554 | ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴) |
df-pss 3556 | ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵)) |
csymdif 3805 | class (𝐴 △ 𝐵) |
df-symdif 3806 | ⊢ (𝐴 △ 𝐵) = ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐴)) |
c0 3874 | class ∅ |
df-nul 3875 | ⊢ ∅ = (V ∖ V) |
cif 4036 | class if(𝜑, 𝐴, 𝐵) |
df-if 4037 | ⊢ if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ ¬ 𝜑))} |
cpw 4108 | class 𝒫 𝐴 |
df-pw 4110 | ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴} |
csn 4125 | class {𝐴} |
df-sn 4126 | ⊢ {𝐴} = {𝑥 ∣ 𝑥 = 𝐴} |
cpr 4127 | class {𝐴, 𝐵} |
df-pr 4128 | ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) |
ctp 4129 | class {𝐴, 𝐵, 𝐶} |
df-tp 4130 | ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) |
cop 4131 | class 〈𝐴, 𝐵〉 |
df-op 4132 | ⊢ 〈𝐴, 𝐵〉 = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} |
cotp 4133 | class 〈𝐴, 𝐵, 𝐶〉 |
df-ot 4134 | ⊢ 〈𝐴, 𝐵, 𝐶〉 = 〈〈𝐴, 𝐵〉, 𝐶〉 |
cuni 4372 | class ∪
𝐴 |
df-uni 4373 | ⊢ ∪ 𝐴 = {𝑥 ∣ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)} |
cint 4410 | class ∩
𝐴 |
df-int 4411 | ⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)} |
ciun 4455 | class ∪ 𝑥 ∈ 𝐴 𝐵 |
ciin 4456 | class ∩ 𝑥 ∈ 𝐴 𝐵 |
df-iun 4457 | ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} |
df-iin 4458 | ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} |
wdisj 4553 | wff Disj 𝑥 ∈ 𝐴 𝐵 |
df-disj 4554 | ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) |
wbr 4583 | wff 𝐴𝑅𝐵 |
df-br 4584 | ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) |
copab 4642 | class {〈𝑥, 𝑦〉 ∣ 𝜑} |
cmpt 4643 | class (𝑥 ∈ 𝐴 ↦ 𝐵) |
df-opab 4644 | ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} |
df-mpt 4645 | ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
wtr 4680 | wff Tr 𝐴 |
df-tr 4681 | ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴) |
ax-rep 4699 | ⊢ (∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))) |
ax-sep 4709 | ⊢ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)) |
ax-nul 4717 | ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 |
ax-pow 4769 | ⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) |
ax-pr 4833 | ⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) |
cep 4947 | class E |
cid 4948 | class I |
df-eprel 4949 | ⊢ E = {〈𝑥, 𝑦〉 ∣ 𝑥 ∈ 𝑦} |
df-id 4953 | ⊢ I = {〈𝑥, 𝑦〉 ∣ 𝑥 = 𝑦} |
wpo 4957 | wff 𝑅 Po 𝐴 |
wor 4958 | wff 𝑅 Or 𝐴 |
df-po 4959 | ⊢ (𝑅 Po 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) |
df-so 4960 | ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))) |
wfr 4994 | wff 𝑅 Fr 𝐴 |
wse 4995 | wff 𝑅 Se 𝐴 |
wwe 4996 | wff 𝑅 We 𝐴 |
df-fr 4997 | ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦)) |
df-se 4998 | ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) |
df-we 4999 | ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴)) |
cxp 5036 | class (𝐴 × 𝐵) |
ccnv 5037 | class ◡𝐴 |
cdm 5038 | class dom 𝐴 |
crn 5039 | class ran 𝐴 |
cres 5040 | class (𝐴 ↾ 𝐵) |
cima 5041 | class (𝐴 “ 𝐵) |
ccom 5042 | class (𝐴 ∘ 𝐵) |
wrel 5043 | wff Rel 𝐴 |
df-xp 5044 | ⊢ (𝐴 × 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)} |
df-rel 5045 | ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V)) |
df-cnv 5046 | ⊢ ◡𝐴 = {〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} |
df-co 5047 | ⊢ (𝐴 ∘ 𝐵) = {〈𝑥, 𝑦〉 ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)} |
df-dm 5048 | ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} |
df-rn 5049 | ⊢ ran 𝐴 = dom ◡𝐴 |
df-res 5050 | ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V)) |
df-ima 5051 | ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) |
cpred 5596 | class Pred(𝑅, 𝐴, 𝑋) |
df-pred 5597 | ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋})) |
word 5639 | wff Ord 𝐴 |
con0 5640 | class On |
wlim 5641 | wff Lim 𝐴 |
csuc 5642 | class suc 𝐴 |
df-ord 5643 | ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴)) |
df-on 5644 | ⊢ On = {𝑥 ∣ Ord 𝑥} |
df-lim 5645 | ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ 𝐴 = ∪ 𝐴)) |
df-suc 5646 | ⊢ suc 𝐴 = (𝐴 ∪ {𝐴}) |
cio 5766 | class (℩𝑥𝜑) |
df-iota 5768 | ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} |
wfun 5798 | wff Fun 𝐴 |
wfn 5799 | wff 𝐴 Fn 𝐵 |
wf 5800 | wff 𝐹:𝐴⟶𝐵 |
wf1 5801 | wff 𝐹:𝐴–1-1→𝐵 |
wfo 5802 | wff 𝐹:𝐴–onto→𝐵 |
wf1o 5803 | wff 𝐹:𝐴–1-1-onto→𝐵 |
cfv 5804 | class (𝐹‘𝐴) |
wiso 5805 | wff 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) |
df-fun 5806 | ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I )) |
df-fn 5807 | ⊢ (𝐴 Fn 𝐵 ↔ (Fun 𝐴 ∧ dom 𝐴 = 𝐵)) |
df-f 5808 | ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) |
df-f1 5809 | ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹)) |
df-fo 5810 | ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) |
df-f1o 5811 | ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵)) |
df-fv 5812 | ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥) |
df-isom 5813 | ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) |
crio 6510 | class (℩𝑥 ∈ 𝐴 𝜑) |
df-riota 6511 | ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) |
co 6549 | class (𝐴𝐹𝐵) |
coprab 6550 | class {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
cmpt2 6551 | class (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
df-ov 6552 | ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) |
df-oprab 6553 | ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = 〈〈𝑥, 𝑦〉, 𝑧〉 ∧ 𝜑)} |
df-mpt2 6554 | ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} |
cof 6793 | class ∘𝑓
𝑅 |
cofr 6794 | class ∘𝑟
𝑅 |
df-of 6795 | ⊢ ∘𝑓
𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))) |
df-ofr 6796 | ⊢ ∘𝑟 𝑅 = {〈𝑓, 𝑔〉 ∣ ∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓‘𝑥)𝑅(𝑔‘𝑥)} |
crpss 6834 | class
[⊊] |
df-rpss 6835 | ⊢ [⊊] = {〈𝑥, 𝑦〉 ∣ 𝑥 ⊊ 𝑦} |
ax-un 6847 | ⊢ ∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) |
com 6957 | class ω |
df-om 6958 | ⊢ ω = {𝑥 ∈ On ∣ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)} |
c1st 7057 | class 1st |
c2nd 7058 | class 2nd |
df-1st 7059 | ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥}) |
df-2nd 7060 | ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥}) |
csupp 7182 | class supp |
df-supp 7183 | ⊢ supp = (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}}) |
ctpos 7238 | class tpos 𝐹 |
df-tpos 7239 | ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})) |
ccur 7278 | class curry 𝐴 |
cunc 7279 | class uncurry 𝐴 |
df-cur 7280 | ⊢ curry 𝐹 = (𝑥 ∈ dom dom 𝐹 ↦ {〈𝑦, 𝑧〉 ∣ 〈𝑥, 𝑦〉𝐹𝑧}) |
df-unc 7281 | ⊢ uncurry 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑦(𝐹‘𝑥)𝑧} |
cund 7285 | class Undef |
df-undef 7286 | ⊢ Undef = (𝑠 ∈ V ↦ 𝒫 ∪ 𝑠) |
cwrecs 7293 | class wrecs(𝑅, 𝐴, 𝐹) |
df-wrecs 7294 | ⊢ wrecs(𝑅, 𝐴, 𝐹) = ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} |
wsmo 7329 | wff Smo 𝐴 |
df-smo 7330 | ⊢ (Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∀𝑦 ∈ dom 𝐴(𝑥 ∈ 𝑦 → (𝐴‘𝑥) ∈ (𝐴‘𝑦)))) |
crecs 7354 | class recs(𝐹) |
df-recs 7355 | ⊢ recs(𝐹) = wrecs( E , On, 𝐹) |
crdg 7392 | class rec(𝐹, 𝐼) |
df-rdg 7393 | ⊢ rec(𝐹, 𝐼) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))) |
cseqom 7429 | class seq𝜔(𝐹, 𝐼) |
df-seqom 7430 | ⊢ seq𝜔(𝐹, 𝐼) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ 〈suc 𝑖, (𝑖𝐹𝑣)〉), 〈∅, ( I ‘𝐼)〉) “
ω) |
c1o 7440 | class
1𝑜 |
c2o 7441 | class
2𝑜 |
c3o 7442 | class
3𝑜 |
c4o 7443 | class
4𝑜 |
coa 7444 | class
+𝑜 |
comu 7445 | class
·𝑜 |
coe 7446 | class
↑𝑜 |
df-1o 7447 | ⊢ 1𝑜 = suc
∅ |
df-2o 7448 | ⊢ 2𝑜 = suc
1𝑜 |
df-3o 7449 | ⊢ 3𝑜 = suc
2𝑜 |
df-4o 7450 | ⊢ 4𝑜 = suc
3𝑜 |
df-oadd 7451 | ⊢ +𝑜 = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ suc 𝑧), 𝑥)‘𝑦)) |
df-omul 7452 | ⊢ ·𝑜 = (𝑥 ∈ On, 𝑦 ∈ On ↦ (rec((𝑧 ∈ V ↦ (𝑧 +𝑜 𝑥)), ∅)‘𝑦)) |
df-oexp 7453 | ⊢ ↑𝑜 = (𝑥 ∈ On, 𝑦 ∈ On ↦ if(𝑥 = ∅, (1𝑜 ∖
𝑦), (rec((𝑧 ∈ V ↦ (𝑧 ·𝑜
𝑥)),
1𝑜)‘𝑦))) |
wer 7626 | wff 𝑅 Er 𝐴 |
cec 7627 | class [𝐴]𝑅 |
cqs 7628 | class (𝐴 / 𝑅) |
df-er 7629 | ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅)) |
df-ec 7631 | ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴}) |
df-qs 7635 | ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} |
cmap 7744 | class
↑𝑚 |
cpm 7745 | class
↑pm |
df-map 7746 | ⊢ ↑𝑚 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∣ 𝑓:𝑦⟶𝑥}) |
df-pm 7747 | ⊢ ↑pm =
(𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓}) |
cixp 7794 | class X𝑥 ∈ 𝐴 𝐵 |
df-ixp 7795 | ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)} |
cen 7838 | class ≈ |
cdom 7839 | class ≼ |
csdm 7840 | class ≺ |
cfn 7841 | class Fin |
df-en 7842 | ⊢ ≈ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} |
df-dom 7843 | ⊢ ≼ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦} |
df-sdom 7844 | ⊢ ≺ = ( ≼ ∖ ≈
) |
df-fin 7845 | ⊢ Fin = {𝑥 ∣ ∃𝑦 ∈ ω 𝑥 ≈ 𝑦} |
cfsupp 8158 | class finSupp |
df-fsupp 8159 | ⊢ finSupp = {〈𝑟, 𝑧〉 ∣ (Fun 𝑟 ∧ (𝑟 supp 𝑧) ∈ Fin)} |
cfi 8199 | class fi |
df-fi 8200 | ⊢ fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}) |
csup 8229 | class sup(𝐴, 𝐵, 𝑅) |
cinf 8230 | class inf(𝐴, 𝐵, 𝑅) |
df-sup 8231 | ⊢ sup(𝐴, 𝐵, 𝑅) = ∪ {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝑅𝑧))} |
df-inf 8232 | ⊢ inf(𝐴, 𝐵, 𝑅) = sup(𝐴, 𝐵, ◡𝑅) |
coi 8297 | class OrdIso(𝑅, 𝐴) |
df-oi 8298 | ⊢ OrdIso(𝑅, 𝐴) = if((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴), (recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) ↾ {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (recs((ℎ ∈ V ↦ (℩𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣))) “ 𝑥)𝑧𝑅𝑡}), ∅) |
char 8344 | class har |
cwdom 8345 | class
≼* |
df-har 8346 | ⊢ har = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦 ≼ 𝑥}) |
df-wdom 8347 | ⊢ ≼* = {〈𝑥, 𝑦〉 ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑥)} |
ax-reg 8380 | ⊢ (∃𝑦 𝑦 ∈ 𝑥 → ∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥))) |
ax-inf 8418 | ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦))) |
ax-inf2 8421 | ⊢ ∃𝑥(∃𝑦(𝑦 ∈ 𝑥 ∧ ∀𝑧 ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑧 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))))) |
ccnf 8441 | class CNF |
df-cnf 8442 | ⊢ CNF = (𝑥 ∈ On, 𝑦 ∈ On ↦ (𝑓 ∈ {𝑔 ∈ (𝑥 ↑𝑚 𝑦) ∣ 𝑔 finSupp ∅} ↦
⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝑥 ↑𝑜 (ℎ‘𝑘)) ·𝑜 (𝑓‘(ℎ‘𝑘))) +𝑜 𝑧)), ∅)‘dom ℎ))) |
ctc 8495 | class TC |
df-tc 8496 | ⊢ TC = (𝑥 ∈ V ↦ ∩ {𝑦
∣ (𝑥 ⊆ 𝑦 ∧ Tr 𝑦)}) |
cr1 8508 | class
𝑅1 |
crnk 8509 | class rank |
df-r1 8510 | ⊢ 𝑅1 =
rec((𝑥 ∈ V ↦
𝒫 𝑥),
∅) |
df-rank 8511 | ⊢ rank = (𝑥 ∈ V ↦ ∩ {𝑦
∈ On ∣ 𝑥 ∈
(𝑅1‘suc 𝑦)}) |
ccrd 8644 | class card |
cale 8645 | class ℵ |
ccf 8646 | class cf |
wacn 8647 | class AC 𝐴 |
df-card 8648 | ⊢ card = (𝑥 ∈ V ↦ ∩ {𝑦
∈ On ∣ 𝑦 ≈
𝑥}) |
df-aleph 8649 | ⊢ ℵ = rec(har, ω) |
df-cf 8650 | ⊢ cf = (𝑥 ∈ On ↦ ∩ {𝑦
∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑣 ∈ 𝑥 ∃𝑢 ∈ 𝑧 𝑣 ⊆ 𝑢))}) |
df-acn 8651 | ⊢ AC 𝐴 = {𝑥 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ((𝒫 𝑥 ∖ {∅})
↑𝑚 𝐴)∃𝑔∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝑓‘𝑦))} |
wac 8821 | wff
CHOICE |
df-ac 8822 | ⊢ (CHOICE ↔
∀𝑥∃𝑓(𝑓 ⊆ 𝑥 ∧ 𝑓 Fn dom 𝑥)) |
ccda 8872 | class
+𝑐 |
df-cda 8873 | ⊢ +𝑐 = (𝑥 ∈ V, 𝑦 ∈ V ↦ ((𝑥 × {∅}) ∪ (𝑦 ×
{1𝑜}))) |
cfin1a 8983 | class FinIa |
cfin2 8984 | class
FinII |
cfin4 8985 | class
FinIV |
cfin3 8986 | class
FinIII |
cfin5 8987 | class FinV |
cfin6 8988 | class
FinVI |
cfin7 8989 | class
FinVII |
df-fin1a 8990 | ⊢ FinIa = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ∈ Fin ∨ (𝑥 ∖ 𝑦) ∈ Fin)} |
df-fin2 8991 | ⊢ FinII = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝒫 𝑥((𝑦 ≠ ∅ ∧ [⊊] Or
𝑦) → ∪ 𝑦
∈ 𝑦)} |
df-fin4 8992 | ⊢ FinIV = {𝑥 ∣ ¬ ∃𝑦(𝑦 ⊊ 𝑥 ∧ 𝑦 ≈ 𝑥)} |
df-fin3 8993 | ⊢ FinIII = {𝑥 ∣ 𝒫 𝑥 ∈ FinIV} |
df-fin5 8994 | ⊢ FinV = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 ≺ (𝑥 +𝑐 𝑥))} |
df-fin6 8995 | ⊢ FinVI = {𝑥 ∣ (𝑥 ≺ 2𝑜 ∨ 𝑥 ≺ (𝑥 × 𝑥))} |
df-fin7 8996 | ⊢ FinVII = {𝑥 ∣ ¬ ∃𝑦 ∈ (On ∖ ω)𝑥 ≈ 𝑦} |
ax-cc 9140 | ⊢ (𝑥 ≈ ω → ∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) |
ax-dc 9151 | ⊢ ((∃𝑦∃𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)) |
ax-ac 9164 | ⊢ ∃𝑦∀𝑧∀𝑤((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → ∃𝑣∀𝑢(∃𝑡((𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡) ∧ (𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦)) ↔ 𝑢 = 𝑣)) |
ax-ac2 9168 | ⊢ ∃𝑦∀𝑧∃𝑣∀𝑢((𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑦 → ((𝑣 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧 ∈ 𝑣))) ∨ (¬ 𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑥 → ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣))))) |
cgch 9321 | class GCH |
df-gch 9322 | ⊢ GCH = (Fin ∪ {𝑥 ∣ ∀𝑦 ¬ (𝑥 ≺ 𝑦 ∧ 𝑦 ≺ 𝒫 𝑥)}) |
cwina 9383 | class
Inaccw |
cina 9384 | class Inacc |
df-wina 9385 | ⊢ Inaccw = {𝑥 ∣ (𝑥 ≠ ∅ ∧ (cf‘𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑥 𝑦 ≺ 𝑧)} |
df-ina 9386 | ⊢ Inacc = {𝑥 ∣ (𝑥 ≠ ∅ ∧ (cf‘𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑥 𝒫 𝑦 ≺ 𝑥)} |
cwun 9401 | class WUni |
cwunm 9402 | class wUniCl |
df-wun 9403 | ⊢ WUni = {𝑢 ∣ (Tr 𝑢 ∧ 𝑢 ≠ ∅ ∧ ∀𝑥 ∈ 𝑢 (∪ 𝑥 ∈ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢))} |
df-wunc 9404 | ⊢ wUniCl = (𝑥 ∈ V ↦ ∩ {𝑢
∈ WUni ∣ 𝑥
⊆ 𝑢}) |
ctsk 9449 | class Tarski |
df-tsk 9450 | ⊢ Tarski = {𝑦 ∣ (∀𝑧 ∈ 𝑦 (𝒫 𝑧 ⊆ 𝑦 ∧ ∃𝑤 ∈ 𝑦 𝒫 𝑧 ⊆ 𝑤) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))} |
cgru 9491 | class Univ |
df-gru 9492 | ⊢ Univ = {𝑢 ∣ (Tr 𝑢 ∧ ∀𝑥 ∈ 𝑢 (𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢 ↑𝑚 𝑥)∪
ran 𝑦 ∈ 𝑢))} |
ax-groth 9524 | ⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 (∀𝑤(𝑤 ⊆ 𝑧 → 𝑤 ∈ 𝑦) ∧ ∃𝑤 ∈ 𝑦 ∀𝑣(𝑣 ⊆ 𝑧 → 𝑣 ∈ 𝑤)) ∧ ∀𝑧(𝑧 ⊆ 𝑦 → (𝑧 ≈ 𝑦 ∨ 𝑧 ∈ 𝑦))) |
ctskm 9538 | class tarskiMap |
df-tskm 9539 | ⊢ tarskiMap = (𝑥 ∈ V ↦ ∩ {𝑦
∈ Tarski ∣ 𝑥
∈ 𝑦}) |
cnpi 9545 | class N |
cpli 9546 | class
+N |
cmi 9547 | class
·N |
clti 9548 | class
<N |
cplpq 9549 | class
+pQ |
cmpq 9550 | class
·pQ |
cltpq 9551 | class
<pQ |
ceq 9552 | class
~Q |
cnq 9553 | class Q |
c1q 9554 | class
1Q |
cerq 9555 | class
[Q] |
cplq 9556 | class
+Q |
cmq 9557 | class
·Q |
crq 9558 | class
*Q |
cltq 9559 | class
<Q |
cnp 9560 | class P |
c1p 9561 | class
1P |
cpp 9562 | class
+P |
cmp 9563 | class
·P |
cltp 9564 | class
<P |
cer 9565 | class
~R |
cnr 9566 | class R |
c0r 9567 | class
0R |
c1r 9568 | class
1R |
cm1r 9569 | class
-1R |
cplr 9570 | class
+R |
cmr 9571 | class
·R |
cltr 9572 | class
<R |
df-ni 9573 | ⊢ N = (ω
∖ {∅}) |
df-pli 9574 | ⊢ +N = (
+𝑜 ↾ (N ×
N)) |
df-mi 9575 | ⊢
·N = ( ·𝑜 ↾
(N × N)) |
df-lti 9576 | ⊢ <N = ( E ∩
(N × N)) |
df-plpq 9609 | ⊢ +pQ = (𝑥 ∈ (N ×
N), 𝑦 ∈
(N × N) ↦ 〈(((1st
‘𝑥)
·N (2nd ‘𝑦)) +N
((1st ‘𝑦)
·N (2nd ‘𝑥))), ((2nd ‘𝑥)
·N (2nd ‘𝑦))〉) |
df-mpq 9610 | ⊢ ·pQ = (𝑥 ∈ (N ×
N), 𝑦 ∈
(N × N) ↦ 〈((1st
‘𝑥)
·N (1st ‘𝑦)), ((2nd ‘𝑥)
·N (2nd ‘𝑦))〉) |
df-ltpq 9611 | ⊢ <pQ =
{〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (N ×
N) ∧ 𝑦
∈ (N × N)) ∧ ((1st
‘𝑥)
·N (2nd ‘𝑦)) <N
((1st ‘𝑦)
·N (2nd ‘𝑥)))} |
df-enq 9612 | ⊢ ~Q = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (N ×
N) ∧ 𝑦
∈ (N × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 〈𝑧, 𝑤〉 ∧ 𝑦 = 〈𝑣, 𝑢〉) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)))} |
df-nq 9613 | ⊢ Q = {𝑥 ∈ (N ×
N) ∣ ∀𝑦 ∈ (N ×
N)(𝑥
~Q 𝑦 → ¬ (2nd ‘𝑦) <N
(2nd ‘𝑥))} |
df-erq 9614 | ⊢ [Q] = (
~Q ∩ ((N × N)
× Q)) |
df-plq 9615 | ⊢ +Q =
(([Q] ∘ +pQ ) ↾
(Q × Q)) |
df-mq 9616 | ⊢
·Q = (([Q] ∘
·pQ ) ↾ (Q ×
Q)) |
df-1nq 9617 | ⊢ 1Q =
〈1𝑜, 1𝑜〉 |
df-rq 9618 | ⊢ *Q =
(◡ ·Q
“ {1Q}) |
df-ltnq 9619 | ⊢ <Q = (
<pQ ∩ (Q ×
Q)) |
df-np 9682 | ⊢ P = {𝑥 ∣ ((∅ ⊊
𝑥 ∧ 𝑥 ⊊ Q) ∧
∀𝑦 ∈ 𝑥 (∀𝑧(𝑧 <Q 𝑦 → 𝑧 ∈ 𝑥) ∧ ∃𝑧 ∈ 𝑥 𝑦 <Q 𝑧))} |
df-1p 9683 | ⊢ 1P =
{𝑥 ∣ 𝑥 <Q
1Q} |
df-plp 9684 | ⊢ +P = (𝑥 ∈ P, 𝑦 ∈ P ↦
{𝑤 ∣ ∃𝑣 ∈ 𝑥 ∃𝑢 ∈ 𝑦 𝑤 = (𝑣 +Q 𝑢)}) |
df-mp 9685 | ⊢
·P = (𝑥 ∈ P, 𝑦 ∈ P ↦ {𝑤 ∣ ∃𝑣 ∈ 𝑥 ∃𝑢 ∈ 𝑦 𝑤 = (𝑣 ·Q 𝑢)}) |
df-ltp 9686 | ⊢ <P = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ P ∧ 𝑦 ∈ P) ∧
𝑥 ⊊ 𝑦)} |
df-enr 9756 | ⊢ ~R = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (P ×
P) ∧ 𝑦
∈ (P × P)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 〈𝑧, 𝑤〉 ∧ 𝑦 = 〈𝑣, 𝑢〉) ∧ (𝑧 +P 𝑢) = (𝑤 +P 𝑣)))} |
df-nr 9757 | ⊢ R =
((P × P) / ~R
) |
df-plr 9758 | ⊢ +R =
{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧
∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [〈𝑤, 𝑣〉] ~R ∧
𝑦 = [〈𝑢, 𝑓〉] ~R ) ∧
𝑧 = [〈(𝑤 +P
𝑢), (𝑣 +P 𝑓)〉]
~R ))} |
df-mr 9759 | ⊢
·R = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧
∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [〈𝑤, 𝑣〉] ~R ∧
𝑦 = [〈𝑢, 𝑓〉] ~R ) ∧
𝑧 = [〈((𝑤
·P 𝑢) +P (𝑣
·P 𝑓)), ((𝑤 ·P 𝑓) +P
(𝑣
·P 𝑢))〉] ~R
))} |
df-ltr 9760 | ⊢ <R =
{〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ R ∧
𝑦 ∈ R)
∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = [〈𝑧, 𝑤〉] ~R ∧
𝑦 = [〈𝑣, 𝑢〉] ~R ) ∧
(𝑧
+P 𝑢)<P (𝑤 +P
𝑣)))} |
df-0r 9761 | ⊢ 0R =
[〈1P, 1P〉]
~R |
df-1r 9762 | ⊢ 1R =
[〈(1P +P
1P), 1P〉]
~R |
df-m1r 9763 | ⊢ -1R =
[〈1P, (1P
+P 1P)〉]
~R |
cc 9813 | class ℂ |
cr 9814 | class ℝ |
cc0 9815 | class 0 |
c1 9816 | class 1 |
ci 9817 | class i |
caddc 9818 | class + |
cltrr 9819 | class
<ℝ |
cmul 9820 | class · |
df-c 9821 | ⊢ ℂ = (R
× R) |
df-0 9822 | ⊢ 0 =
〈0R,
0R〉 |
df-1 9823 | ⊢ 1 =
〈1R,
0R〉 |
df-i 9824 | ⊢ i =
〈0R,
1R〉 |
df-r 9825 | ⊢ ℝ = (R
× {0R}) |
df-add 9826 | ⊢ + = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 〈(𝑤 +R 𝑢), (𝑣 +R 𝑓)〉))} |
df-mul 9827 | ⊢ · = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 〈((𝑤 ·R 𝑢) +R
(-1R ·R (𝑣
·R 𝑓))), ((𝑣 ·R 𝑢) +R
(𝑤
·R 𝑓))〉))} |
df-lt 9828 | ⊢ <ℝ =
{〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧
∃𝑧∃𝑤((𝑥 = 〈𝑧, 0R〉 ∧
𝑦 = 〈𝑤,
0R〉) ∧ 𝑧 <R 𝑤))} |
ax-cnex 9871 | ⊢ ℂ ∈ V |
ax-resscn 9872 | ⊢ ℝ ⊆ ℂ |
ax-1cn 9873 | ⊢ 1 ∈ ℂ |
ax-icn 9874 | ⊢ i ∈ ℂ |
ax-addcl 9875 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ) |
ax-addrcl 9876 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ) |
ax-mulcl 9877 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ) |
ax-mulrcl 9878 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ) |
ax-mulcom 9879 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴)) |
ax-addass 9880 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))) |
ax-mulass 9881 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))) |
ax-distr 9882 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))) |
ax-i2m1 9883 | ⊢ ((i · i) + 1) = 0 |
ax-1ne0 9884 | ⊢ 1 ≠ 0 |
ax-1rid 9885 | ⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴) |
ax-rnegex 9886 | ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) |
ax-rrecex 9887 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1) |
ax-cnre 9888 | ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝐴 = (𝑥 + (i · 𝑦))) |
ax-pre-lttri 9889 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 <ℝ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <ℝ 𝐴))) |
ax-pre-lttrn 9890 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ 𝐶) → 𝐴 <ℝ 𝐶)) |
ax-pre-ltadd 9891 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 → (𝐶 + 𝐴) <ℝ (𝐶 + 𝐵))) |
ax-pre-mulgt0 9892 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0
<ℝ 𝐴
∧ 0 <ℝ 𝐵) → 0 <ℝ (𝐴 · 𝐵))) |
ax-pre-sup 9893 | ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 <ℝ 𝑥) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 <ℝ 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 <ℝ 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <ℝ 𝑧))) |
ax-addf 9894 | ⊢ + :(ℂ ×
ℂ)⟶ℂ |
ax-mulf 9895 | ⊢ · :(ℂ ×
ℂ)⟶ℂ |
cpnf 9950 | class +∞ |
cmnf 9951 | class -∞ |
cxr 9952 | class
ℝ* |
clt 9953 | class < |
cle 9954 | class ≤ |
df-pnf 9955 | ⊢ +∞ = 𝒫 ∪ ℂ |
df-mnf 9956 | ⊢ -∞ = 𝒫
+∞ |
df-xr 9957 | ⊢ ℝ* = (ℝ
∪ {+∞, -∞}) |
df-ltxr 9958 | ⊢ < = ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) ×
{+∞}) ∪ ({-∞} × ℝ))) |
df-le 9959 | ⊢ ≤ = ((ℝ*
× ℝ*) ∖ ◡
< ) |
cmin 10145 | class − |
cneg 10146 | class -𝐴 |
df-sub 10147 | ⊢ − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)) |
df-neg 10148 | ⊢ -𝐴 = (0 − 𝐴) |
cdiv 10563 | class / |
df-div 10564 | ⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦
(℩𝑧 ∈
ℂ (𝑦 · 𝑧) = 𝑥)) |
cn 10897 | class ℕ |
df-nn 10898 | ⊢ ℕ = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 1) “
ω) |
c2 10947 | class 2 |
c3 10948 | class 3 |
c4 10949 | class 4 |
c5 10950 | class 5 |
c6 10951 | class 6 |
c7 10952 | class 7 |
c8 10953 | class 8 |
c9 10954 | class 9 |
c10 10955 | class 10 |
df-2 10956 | ⊢ 2 = (1 + 1) |
df-3 10957 | ⊢ 3 = (2 + 1) |
df-4 10958 | ⊢ 4 = (3 + 1) |
df-5 10959 | ⊢ 5 = (4 + 1) |
df-6 10960 | ⊢ 6 = (5 + 1) |
df-7 10961 | ⊢ 7 = (6 + 1) |
df-8 10962 | ⊢ 8 = (7 + 1) |
df-9 10963 | ⊢ 9 = (8 + 1) |
df-10OLD 10964 | ⊢ 10 = (9 + 1) |
cn0 11169 | class
ℕ0 |
df-n0 11170 | ⊢ ℕ0 = (ℕ
∪ {0}) |
cxnn0 11240 | class
ℕ0* |
df-xnn0 11241 | ⊢ ℕ0* =
(ℕ0 ∪ {+∞}) |
cz 11254 | class ℤ |
df-z 11255 | ⊢ ℤ = {𝑛 ∈ ℝ ∣ (𝑛 = 0 ∨ 𝑛 ∈ ℕ ∨ -𝑛 ∈ ℕ)} |
cdc 11369 | class ;𝐴𝐵 |
df-dec 11370 | ⊢ ;𝐴𝐵 = (((9 + 1) · 𝐴) + 𝐵) |
cuz 11563 | class
ℤ≥ |
df-uz 11564 | ⊢ ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘}) |
cq 11664 | class ℚ |
df-q 11665 | ⊢ ℚ = ( / “ (ℤ
× ℕ)) |
crp 11708 | class
ℝ+ |
df-rp 11709 | ⊢ ℝ+ = {𝑥 ∈ ℝ ∣ 0 <
𝑥} |
cxne 11819 | class -𝑒𝐴 |
cxad 11820 | class
+𝑒 |
cxmu 11821 | class
·e |
df-xneg 11822 | ⊢ -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) |
df-xadd 11823 | ⊢ +𝑒 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ*
↦ if(𝑥 = +∞,
if(𝑦 = -∞, 0,
+∞), if(𝑥 = -∞,
if(𝑦 = +∞, 0,
-∞), if(𝑦 = +∞,
+∞, if(𝑦 = -∞,
-∞, (𝑥 + 𝑦)))))) |
df-xmul 11824 | ⊢ ·e = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ*
↦ if((𝑥 = 0 ∨
𝑦 = 0), 0, if((((0 <
𝑦 ∧ 𝑥 = +∞) ∨ (𝑦 < 0 ∧ 𝑥 = -∞)) ∨ ((0 < 𝑥 ∧ 𝑦 = +∞) ∨ (𝑥 < 0 ∧ 𝑦 = -∞))), +∞, if((((0 < 𝑦 ∧ 𝑥 = -∞) ∨ (𝑦 < 0 ∧ 𝑥 = +∞)) ∨ ((0 < 𝑥 ∧ 𝑦 = -∞) ∨ (𝑥 < 0 ∧ 𝑦 = +∞))), -∞, (𝑥 · 𝑦))))) |
cioo 12046 | class (,) |
cioc 12047 | class (,] |
cico 12048 | class [,) |
cicc 12049 | class [,] |
df-ioo 12050 | ⊢ (,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ*
↦ {𝑧 ∈
ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)}) |
df-ioc 12051 | ⊢ (,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ*
↦ {𝑧 ∈
ℝ* ∣ (𝑥 < 𝑧 ∧ 𝑧 ≤ 𝑦)}) |
df-ico 12052 | ⊢ [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ*
↦ {𝑧 ∈
ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) |
df-icc 12053 | ⊢ [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ*
↦ {𝑧 ∈
ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)}) |
cfz 12197 | class ... |
df-fz 12198 | ⊢ ... = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ (𝑚 ≤ 𝑘 ∧ 𝑘 ≤ 𝑛)}) |
cfzo 12334 | class ..^ |
df-fzo 12335 | ⊢ ..^ = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ (𝑚...(𝑛 − 1))) |
cfl 12453 | class ⌊ |
cceil 12454 | class ⌈ |
df-fl 12455 | ⊢ ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))) |
df-ceil 12456 | ⊢ ⌈ = (𝑥 ∈ ℝ ↦
-(⌊‘-𝑥)) |
cmo 12530 | class mod |
df-mod 12531 | ⊢ mod = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ+ ↦ (𝑥 − (𝑦 · (⌊‘(𝑥 / 𝑦))))) |
cseq 12663 | class seq𝑀( + , 𝐹) |
df-seq 12664 | ⊢ seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) “ ω) |
cexp 12722 | class ↑ |
df-exp 12723 | ⊢ ↑ = (𝑥 ∈ ℂ, 𝑦 ∈ ℤ ↦ if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ ×
{𝑥}))‘-𝑦))))) |
cfa 12922 | class ! |
df-fac 12923 | ⊢ ! = ({〈0, 1〉} ∪ seq1( ·
, I )) |
cbc 12951 | class C |
df-bc 12952 | ⊢ C = (𝑛 ∈ ℕ0, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0)) |
chash 12979 | class # |
df-hash 12980 | ⊢ # = (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ∘ card)
∪ ((V ∖ Fin) × {+∞})) |
cword 13146 | class Word 𝑆 |
clsw 13147 | class lastS |
cconcat 13148 | class ++ |
cs1 13149 | class 〈“𝐴”〉 |
csubstr 13150 | class substr |
csplice 13151 | class splice |
creverse 13152 | class reverse |
creps 13153 | class repeatS |
df-word 13154 | ⊢ Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆} |
df-lsw 13155 | ⊢ lastS = (𝑤 ∈ V ↦ (𝑤‘((#‘𝑤) − 1))) |
df-concat 13156 | ⊢ ++ = (𝑠 ∈ V, 𝑡 ∈ V ↦ (𝑥 ∈ (0..^((#‘𝑠) + (#‘𝑡))) ↦ if(𝑥 ∈ (0..^(#‘𝑠)), (𝑠‘𝑥), (𝑡‘(𝑥 − (#‘𝑠)))))) |
df-s1 13157 | ⊢ 〈“𝐴”〉 = {〈0, ( I ‘𝐴)〉} |
df-substr 13158 | ⊢ substr = (𝑠 ∈ V, 𝑏 ∈ (ℤ × ℤ) ↦
if(((1st ‘𝑏)..^(2nd ‘𝑏)) ⊆ dom 𝑠, (𝑥 ∈ (0..^((2nd ‘𝑏) − (1st
‘𝑏))) ↦ (𝑠‘(𝑥 + (1st ‘𝑏)))), ∅)) |
df-splice 13159 | ⊢ splice = (𝑠 ∈ V, 𝑏 ∈ V ↦ (((𝑠 substr 〈0, (1st
‘(1st ‘𝑏))〉) ++ (2nd ‘𝑏)) ++ (𝑠 substr 〈(2nd
‘(1st ‘𝑏)), (#‘𝑠)〉))) |
df-reverse 13160 | ⊢ reverse = (𝑠 ∈ V ↦ (𝑥 ∈ (0..^(#‘𝑠)) ↦ (𝑠‘(((#‘𝑠) − 1) − 𝑥)))) |
df-reps 13161 | ⊢ repeatS = (𝑠 ∈ V, 𝑛 ∈ ℕ0 ↦ (𝑥 ∈ (0..^𝑛) ↦ 𝑠)) |
ccsh 13385 | class cyclShift |
df-csh 13386 | ⊢ cyclShift = (𝑤 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)}, 𝑛 ∈ ℤ ↦ if(𝑤 = ∅, ∅, ((𝑤 substr 〈(𝑛 mod (#‘𝑤)), (#‘𝑤)〉) ++ (𝑤 substr 〈0, (𝑛 mod (#‘𝑤))〉)))) |
cs2 13437 | class 〈“𝐴𝐵”〉 |
cs3 13438 | class 〈“𝐴𝐵𝐶”〉 |
cs4 13439 | class 〈“𝐴𝐵𝐶𝐷”〉 |
cs5 13440 | class 〈“𝐴𝐵𝐶𝐷𝐸”〉 |
cs6 13441 | class 〈“𝐴𝐵𝐶𝐷𝐸𝐹”〉 |
cs7 13442 | class 〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”〉 |
cs8 13443 | class 〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”〉 |
df-s2 13444 | ⊢ 〈“𝐴𝐵”〉 = (〈“𝐴”〉 ++
〈“𝐵”〉) |
df-s3 13445 | ⊢ 〈“𝐴𝐵𝐶”〉 = (〈“𝐴𝐵”〉 ++ 〈“𝐶”〉) |
df-s4 13446 | ⊢ 〈“𝐴𝐵𝐶𝐷”〉 = (〈“𝐴𝐵𝐶”〉 ++ 〈“𝐷”〉) |
df-s5 13447 | ⊢ 〈“𝐴𝐵𝐶𝐷𝐸”〉 = (〈“𝐴𝐵𝐶𝐷”〉 ++ 〈“𝐸”〉) |
df-s6 13448 | ⊢ 〈“𝐴𝐵𝐶𝐷𝐸𝐹”〉 = (〈“𝐴𝐵𝐶𝐷𝐸”〉 ++ 〈“𝐹”〉) |
df-s7 13449 | ⊢ 〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”〉 = (〈“𝐴𝐵𝐶𝐷𝐸𝐹”〉 ++ 〈“𝐺”〉) |
df-s8 13450 | ⊢ 〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”〉 = (〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”〉 ++ 〈“𝐻”〉) |
ctcl 13572 | class t+ |
crtcl 13573 | class t* |
df-trcl 13574 | ⊢ t+ = (𝑥 ∈ V ↦ ∩ {𝑧
∣ (𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}) |
df-rtrcl 13575 | ⊢ t* = (𝑥 ∈ V ↦ ∩ {𝑧
∣ (( I ↾ (dom 𝑥
∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}) |
crelexp 13608 | class
↑𝑟 |
df-relexp 13609 | ⊢ ↑𝑟 = (𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛))) |
crtrcl 13643 | class t*rec |
df-rtrclrec 13644 | ⊢ t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) |
cshi 13654 | class shift |
df-shft 13655 | ⊢ shift = (𝑓 ∈ V, 𝑥 ∈ ℂ ↦ {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ ℂ ∧ (𝑦 − 𝑥)𝑓𝑧)}) |
csgn 13674 | class sgn |
df-sgn 13675 | ⊢ sgn = (𝑥 ∈ ℝ* ↦ if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1))) |
ccj 13684 | class ∗ |
cre 13685 | class ℜ |
cim 13686 | class ℑ |
df-cj 13687 | ⊢ ∗ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑥 + 𝑦) ∈ ℝ ∧ (i · (𝑥 − 𝑦)) ∈ ℝ))) |
df-re 13688 | ⊢ ℜ = (𝑥 ∈ ℂ ↦ ((𝑥 + (∗‘𝑥)) / 2)) |
df-im 13689 | ⊢ ℑ = (𝑥 ∈ ℂ ↦ (ℜ‘(𝑥 / i))) |
csqrt 13821 | class √ |
cabs 13822 | class abs |
df-sqrt 13823 | ⊢ √ = (𝑥 ∈ ℂ ↦ (℩𝑦 ∈ ℂ ((𝑦↑2) = 𝑥 ∧ 0 ≤ (ℜ‘𝑦) ∧ (i · 𝑦) ∉
ℝ+))) |
df-abs 13824 | ⊢ abs = (𝑥 ∈ ℂ ↦ (√‘(𝑥 · (∗‘𝑥)))) |
clsp 14049 | class lim sup |
df-limsup 14050 | ⊢ lim sup = (𝑥 ∈ V ↦ inf(ran (𝑘 ∈ ℝ ↦ sup(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*),
ℝ*, < )), ℝ*, < )) |
cli 14063 | class ⇝ |
crli 14064 | class
⇝𝑟 |
co1 14065 | class 𝑂(1) |
clo1 14066 | class
≤𝑂(1) |
df-clim 14067 | ⊢ ⇝ = {〈𝑓, 𝑦〉 ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈ ℤ
∀𝑘 ∈
(ℤ≥‘𝑗)((𝑓‘𝑘) ∈ ℂ ∧ (abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥))} |
df-rlim 14068 | ⊢ ⇝𝑟 = {〈𝑓, 𝑥〉 ∣ ((𝑓 ∈ (ℂ ↑pm
ℝ) ∧ 𝑥 ∈
ℂ) ∧ ∀𝑦
∈ ℝ+ ∃𝑧 ∈ ℝ ∀𝑤 ∈ dom 𝑓(𝑧 ≤ 𝑤 → (abs‘((𝑓‘𝑤) − 𝑥)) < 𝑦))} |
df-o1 14069 | ⊢ 𝑂(1) = {𝑓 ∈ (ℂ
↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚} |
df-lo1 14070 | ⊢ ≤𝑂(1) = {𝑓 ∈ (ℝ ↑pm
ℝ) ∣ ∃𝑥
∈ ℝ ∃𝑚
∈ ℝ ∀𝑦
∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ 𝑚} |
csu 14264 | class Σ𝑘 ∈ 𝐴 𝐵 |
df-sum 14265 | ⊢ Σ𝑘 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))) |
cprod 14474 | class ∏𝑘 ∈ 𝐴 𝐵 |
df-prod 14475 | ⊢ ∏𝑘 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈
(ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))) |
cfallfac 14574 | class FallFac |
crisefac 14575 | class RiseFac |
df-risefac 14576 | ⊢ RiseFac = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ0 ↦
∏𝑘 ∈ (0...(𝑛 − 1))(𝑥 + 𝑘)) |
df-fallfac 14577 | ⊢ FallFac = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ0 ↦
∏𝑘 ∈ (0...(𝑛 − 1))(𝑥 − 𝑘)) |
cbp 14616 | class BernPoly |
df-bpoly 14617 | ⊢ BernPoly = (𝑚 ∈ ℕ0, 𝑥 ∈ ℂ ↦ (wrecs(
< , ℕ0, (𝑔 ∈ V ↦ ⦋(#‘dom
𝑔) / 𝑛⦌((𝑥↑𝑛) − Σ𝑘 ∈ dom 𝑔((𝑛C𝑘) · ((𝑔‘𝑘) / ((𝑛 − 𝑘) + 1))))))‘𝑚)) |
ce 14631 | class exp |
ceu 14632 | class e |
csin 14633 | class sin |
ccos 14634 | class cos |
ctan 14635 | class tan |
cpi 14636 | class π |
df-ef 14637 | ⊢ exp = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ ℕ0
((𝑥↑𝑘) / (!‘𝑘))) |
df-e 14638 | ⊢ e =
(exp‘1) |
df-sin 14639 | ⊢ sin = (𝑥 ∈ ℂ ↦ (((exp‘(i
· 𝑥)) −
(exp‘(-i · 𝑥))) / (2 · i))) |
df-cos 14640 | ⊢ cos = (𝑥 ∈ ℂ ↦ (((exp‘(i
· 𝑥)) +
(exp‘(-i · 𝑥))) / 2)) |
df-tan 14641 | ⊢ tan = (𝑥 ∈ (◡cos “ (ℂ ∖ {0})) ↦
((sin‘𝑥) /
(cos‘𝑥))) |
df-pi 14642 | ⊢ π = inf((ℝ+
∩ (◡sin “ {0})), ℝ,
< ) |
cdvds 14821 | class ∥ |
df-dvds 14822 | ⊢ ∥ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ∃𝑛 ∈ ℤ (𝑛 · 𝑥) = 𝑦)} |
cbits 14979 | class bits |
csad 14980 | class sadd |
csmu 14981 | class smul |
df-bits 14982 | ⊢ bits = (𝑛 ∈ ℤ ↦ {𝑚 ∈ ℕ0 ∣ ¬ 2
∥ (⌊‘(𝑛 /
(2↑𝑚)))}) |
df-sad 15011 | ⊢ sadd = (𝑥 ∈ 𝒫 ℕ0, 𝑦 ∈ 𝒫
ℕ0 ↦ {𝑘 ∈ ℕ0 ∣
hadd(𝑘 ∈ 𝑥, 𝑘 ∈ 𝑦, ∅ ∈ (seq0((𝑐 ∈ 2𝑜, 𝑚 ∈ ℕ0
↦ if(cadd(𝑚 ∈
𝑥, 𝑚 ∈ 𝑦, ∅ ∈ 𝑐), 1𝑜, ∅)), (𝑛 ∈ ℕ0
↦ if(𝑛 = 0, ∅,
(𝑛 −
1))))‘𝑘))}) |
df-smu 15036 | ⊢ smul = (𝑥 ∈ 𝒫 ℕ0, 𝑦 ∈ 𝒫
ℕ0 ↦ {𝑘 ∈ ℕ0 ∣ 𝑘 ∈ (seq0((𝑝 ∈ 𝒫
ℕ0, 𝑚
∈ ℕ0 ↦ (𝑝 sadd {𝑛 ∈ ℕ0 ∣ (𝑚 ∈ 𝑥 ∧ (𝑛 − 𝑚) ∈ 𝑦)})), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))‘(𝑘 + 1))}) |
cgcd 15054 | class gcd |
df-gcd 15055 | ⊢ gcd = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∧ 𝑦 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑥 ∧ 𝑛 ∥ 𝑦)}, ℝ, < ))) |
clcm 15139 | class lcm |
clcmf 15140 | class lcm |
df-lcm 15141 | ⊢ lcm = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < ))) |
df-lcmf 15142 | ⊢ lcm = (𝑧 ∈ 𝒫 ℤ ↦ if(0 ∈
𝑧, 0, inf({𝑛 ∈ ℕ ∣
∀𝑚 ∈ 𝑧 𝑚 ∥ 𝑛}, ℝ, < ))) |
cprime 15223 | class ℙ |
df-prm 15224 | ⊢ ℙ = {𝑝 ∈ ℕ ∣ {𝑛 ∈ ℕ ∣ 𝑛 ∥ 𝑝} ≈
2𝑜} |
cnumer 15279 | class numer |
cdenom 15280 | class denom |
df-numer 15281 | ⊢ numer = (𝑦 ∈ ℚ ↦ (1st
‘(℩𝑥
∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝑦 = ((1st ‘𝑥) / (2nd ‘𝑥)))))) |
df-denom 15282 | ⊢ denom = (𝑦 ∈ ℚ ↦ (2nd
‘(℩𝑥
∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝑦 = ((1st ‘𝑥) / (2nd ‘𝑥)))))) |
codz 15306 | class
odℤ |
cphi 15307 | class ϕ |
df-odz 15308 | ⊢ odℤ = (𝑛 ∈ ℕ ↦ (𝑥 ∈ {𝑥 ∈ ℤ ∣ (𝑥 gcd 𝑛) = 1} ↦ inf({𝑚 ∈ ℕ ∣ 𝑛 ∥ ((𝑥↑𝑚) − 1)}, ℝ, <
))) |
df-phi 15309 | ⊢ ϕ = (𝑛 ∈ ℕ ↦ (#‘{𝑥 ∈ (1...𝑛) ∣ (𝑥 gcd 𝑛) = 1})) |
cpc 15379 | class pCnt |
df-pc 15380 | ⊢ pCnt = (𝑝 ∈ ℙ, 𝑟 ∈ ℚ ↦ if(𝑟 = 0, +∞, (℩𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑟 = (𝑥 / 𝑦) ∧ 𝑧 = (sup({𝑛 ∈ ℕ0 ∣ (𝑝↑𝑛) ∥ 𝑥}, ℝ, < ) − sup({𝑛 ∈ ℕ0
∣ (𝑝↑𝑛) ∥ 𝑦}, ℝ, < )))))) |
cgz 15471 | class ℤ[i] |
df-gz 15472 | ⊢ ℤ[i] = {𝑥 ∈ ℂ ∣ ((ℜ‘𝑥) ∈ ℤ ∧
(ℑ‘𝑥) ∈
ℤ)} |
cvdwa 15507 | class AP |
cvdwm 15508 | class MonoAP |
cvdwp 15509 | class PolyAP |
df-vdwap 15510 | ⊢ AP = (𝑘 ∈ ℕ0 ↦ (𝑎 ∈ ℕ, 𝑑 ∈ ℕ ↦ ran
(𝑚 ∈ (0...(𝑘 − 1)) ↦ (𝑎 + (𝑚 · 𝑑))))) |
df-vdwmc 15511 | ⊢ MonoAP = {〈𝑘, 𝑓〉 ∣ ∃𝑐(ran (AP‘𝑘) ∩ 𝒫 (◡𝑓 “ {𝑐})) ≠ ∅} |
df-vdwpc 15512 | ⊢ PolyAP = {〈〈𝑚, 𝑘〉, 𝑓〉 ∣ ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑𝑚
(1...𝑚))(∀𝑖 ∈ (1...𝑚)((𝑎 + (𝑑‘𝑖))(AP‘𝑘)(𝑑‘𝑖)) ⊆ (◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...𝑚) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖))))) = 𝑚)} |
cram 15541 | class Ramsey |
df-ram 15543 | ⊢ Ramsey = (𝑚 ∈ ℕ0, 𝑟 ∈ V ↦ inf({𝑛 ∈ ℕ0
∣ ∀𝑠(𝑛 ≤ (#‘𝑠) → ∀𝑓 ∈ (dom 𝑟 ↑𝑚 {𝑦 ∈ 𝒫 𝑠 ∣ (#‘𝑦) = 𝑚})∃𝑐 ∈ dom 𝑟∃𝑥 ∈ 𝒫 𝑠((𝑟‘𝑐) ≤ (#‘𝑥) ∧ ∀𝑦 ∈ 𝒫 𝑥((#‘𝑦) = 𝑚 → (𝑓‘𝑦) = 𝑐)))}, ℝ*, <
)) |
cprmo 15573 | class #p |
df-prmo 15574 | ⊢ #p = (𝑛 ∈ ℕ0 ↦
∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1)) |
cstr 15691 | class Struct |
cnx 15692 | class ndx |
csts 15693 | class sSet |
cslot 15694 | class Slot 𝐴 |
cbs 15695 | class Base |
cress 15696 | class
↾s |
df-struct 15697 | ⊢ Struct = {〈𝑓, 𝑥〉 ∣ (𝑥 ∈ ( ≤ ∩ (ℕ ×
ℕ)) ∧ Fun (𝑓
∖ {∅}) ∧ dom 𝑓 ⊆ (...‘𝑥))} |
df-ndx 15698 | ⊢ ndx = ( I ↾ ℕ) |
df-slot 15699 | ⊢ Slot 𝐴 = (𝑥 ∈ V ↦ (𝑥‘𝐴)) |
df-base 15700 | ⊢ Base = Slot 1 |
df-sets 15701 | ⊢ sSet = (𝑠 ∈ V, 𝑒 ∈ V ↦ ((𝑠 ↾ (V ∖ dom {𝑒})) ∪ {𝑒})) |
df-ress 15702 | ⊢ ↾s = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑥, 𝑤, (𝑤 sSet 〈(Base‘ndx), (𝑥 ∩ (Base‘𝑤))〉))) |
cplusg 15768 | class +g |
cmulr 15769 | class .r |
cstv 15770 | class
*𝑟 |
csca 15771 | class Scalar |
cvsca 15772 | class
·𝑠 |
cip 15773 | class
·𝑖 |
cts 15774 | class TopSet |
cple 15775 | class le |
coc 15776 | class oc |
cds 15777 | class dist |
cunif 15778 | class UnifSet |
chom 15779 | class Hom |
cco 15780 | class comp |
df-plusg 15781 | ⊢ +g = Slot 2 |
df-mulr 15782 | ⊢ .r = Slot 3 |
df-starv 15783 | ⊢ *𝑟 = Slot
4 |
df-sca 15784 | ⊢ Scalar = Slot 5 |
df-vsca 15785 | ⊢ ·𝑠 = Slot
6 |
df-ip 15786 | ⊢
·𝑖 = Slot 8 |
df-tset 15787 | ⊢ TopSet = Slot 9 |
df-ple 15788 | ⊢ le = Slot ;10 |
df-ocomp 15790 | ⊢ oc = Slot ;11 |
df-ds 15791 | ⊢ dist = Slot ;12 |
df-unif 15792 | ⊢ UnifSet = Slot ;13 |
df-hom 15793 | ⊢ Hom = Slot ;14 |
df-cco 15794 | ⊢ comp = Slot ;15 |
crest 15904 | class
↾t |
ctopn 15905 | class TopOpen |
df-rest 15906 | ⊢ ↾t = (𝑗 ∈ V, 𝑥 ∈ V ↦ ran (𝑦 ∈ 𝑗 ↦ (𝑦 ∩ 𝑥))) |
df-topn 15907 | ⊢ TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾t
(Base‘𝑤))) |
ctg 15921 | class topGen |
cpt 15922 | class
∏t |
c0g 15923 | class 0g |
cgsu 15924 | class
Σg |
df-0g 15925 | ⊢ 0g = (𝑔 ∈ V ↦ (℩𝑒(𝑒 ∈ (Base‘𝑔) ∧ ∀𝑥 ∈ (Base‘𝑔)((𝑒(+g‘𝑔)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑔)𝑒) = 𝑥)))) |
df-gsum 15926 | ⊢ Σg = (𝑤 ∈ V, 𝑓 ∈ V ↦ ⦋{𝑥 ∈ (Base‘𝑤) ∣ ∀𝑦 ∈ (Base‘𝑤)((𝑥(+g‘𝑤)𝑦) = 𝑦 ∧ (𝑦(+g‘𝑤)𝑥) = 𝑦)} / 𝑜⦌if(ran 𝑓 ⊆ 𝑜, (0g‘𝑤), if(dom 𝑓 ∈ ran ..., (℩𝑥∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛))), (℩𝑥∃𝑔[(◡𝑓 “ (V ∖ 𝑜)) / 𝑦](𝑔:(1...(#‘𝑦))–1-1-onto→𝑦 ∧ 𝑥 = (seq1((+g‘𝑤), (𝑓 ∘ 𝑔))‘(#‘𝑦))))))) |
df-topgen 15927 | ⊢ topGen = (𝑥 ∈ V ↦ {𝑦 ∣ 𝑦 ⊆ ∪ (𝑥 ∩ 𝒫 𝑦)}) |
df-pt 15928 | ⊢ ∏t = (𝑓 ∈ V ↦
(topGen‘{𝑥 ∣
∃𝑔((𝑔 Fn dom 𝑓 ∧ ∀𝑦 ∈ dom 𝑓(𝑔‘𝑦) ∈ (𝑓‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝑓 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝑓‘𝑦)) ∧ 𝑥 = X𝑦 ∈ dom 𝑓(𝑔‘𝑦))})) |
cprds 15929 | class Xs |
cpws 15930 | class
↑s |
df-prds 15931 | ⊢ Xs = (𝑠 ∈ V, 𝑟 ∈ V ↦ ⦋X𝑥 ∈
dom 𝑟(Base‘(𝑟‘𝑥)) / 𝑣⦌⦋(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) / ℎ⦌(({〈(Base‘ndx),
𝑣〉,
〈(+g‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥))))〉, 〈(.r‘ndx),
(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥))))〉} ∪ {〈(Scalar‘ndx),
𝑠〉, 〈(
·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠
‘(𝑟‘𝑥))(𝑔‘𝑥))))〉,
〈(·𝑖‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥)))))〉}) ∪ ({〈(TopSet‘ndx),
(∏t‘(TopOpen ∘ 𝑟))〉, 〈(le‘ndx), {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥))}〉, 〈(dist‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, <
))〉} ∪ {〈(Hom ‘ndx), ℎ〉, 〈(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ (𝑐ℎ(2nd ‘𝑎)), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(〈((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)〉(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))〉}))) |
df-pws 15933 | ⊢ ↑s = (𝑟 ∈ V, 𝑖 ∈ V ↦ ((Scalar‘𝑟)Xs(𝑖 × {𝑟}))) |
cordt 15982 | class ordTop |
cxrs 15983 | class
ℝ*𝑠 |
df-ordt 15984 | ⊢ ordTop = (𝑟 ∈ V ↦
(topGen‘(fi‘({dom 𝑟} ∪ ran ((𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑦𝑟𝑥}) ∪ (𝑥 ∈ dom 𝑟 ↦ {𝑦 ∈ dom 𝑟 ∣ ¬ 𝑥𝑟𝑦})))))) |
df-xrs 15985 | ⊢ ℝ*𝑠 =
({〈(Base‘ndx), ℝ*〉,
〈(+g‘ndx), +𝑒 〉,
〈(.r‘ndx), ·e 〉} ∪
{〈(TopSet‘ndx), (ordTop‘ ≤ )〉, 〈(le‘ndx),
≤ 〉, 〈(dist‘ndx), (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ*
↦ if(𝑥 ≤ 𝑦, (𝑦 +𝑒
-𝑒𝑥),
(𝑥 +𝑒
-𝑒𝑦)))〉}) |
cqtop 15986 | class qTop |
cimas 15987 | class
“s |
cqus 15988 | class
/s |
cxps 15989 | class
×s |
df-qtop 15990 | ⊢ qTop = (𝑗 ∈ V, 𝑓 ∈ V ↦ {𝑠 ∈ 𝒫 (𝑓 “ ∪ 𝑗) ∣ ((◡𝑓 “ 𝑠) ∩ ∪ 𝑗) ∈ 𝑗}) |
df-imas 15991 | ⊢ “s = (𝑓 ∈ V, 𝑟 ∈ V ↦
⦋(Base‘𝑟) / 𝑣⦌(({〈(Base‘ndx), ran
𝑓〉,
〈(+g‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {〈〈(𝑓‘𝑝), (𝑓‘𝑞)〉, (𝑓‘(𝑝(+g‘𝑟)𝑞))〉}〉,
〈(.r‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {〈〈(𝑓‘𝑝), (𝑓‘𝑞)〉, (𝑓‘(𝑝(.r‘𝑟)𝑞))〉}〉} ∪
{〈(Scalar‘ndx), (Scalar‘𝑟)〉, 〈(
·𝑠 ‘ndx), ∪ 𝑞 ∈ 𝑣 (𝑝 ∈ (Base‘(Scalar‘𝑟)), 𝑥 ∈ {(𝑓‘𝑞)} ↦ (𝑓‘(𝑝( ·𝑠
‘𝑟)𝑞)))〉,
〈(·𝑖‘ndx), ∪ 𝑝 ∈ 𝑣 ∪ 𝑞 ∈ 𝑣 {〈〈(𝑓‘𝑝), (𝑓‘𝑞)〉, (𝑝(·𝑖‘𝑟)𝑞)〉}〉}) ∪
{〈(TopSet‘ndx), ((TopOpen‘𝑟) qTop 𝑓)〉, 〈(le‘ndx), ((𝑓 ∘ (le‘𝑟)) ∘ ◡𝑓)〉, 〈(dist‘ndx), (𝑥 ∈ ran 𝑓, 𝑦 ∈ ran 𝑓 ↦ inf(∪
𝑛 ∈ ℕ ran (𝑔 ∈ {ℎ ∈ ((𝑣 × 𝑣) ↑𝑚 (1...𝑛)) ∣ ((𝑓‘(1st ‘(ℎ‘1))) = 𝑥 ∧ (𝑓‘(2nd ‘(ℎ‘𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝑓‘(2nd ‘(ℎ‘𝑖))) = (𝑓‘(1st ‘(ℎ‘(𝑖 + 1)))))} ↦
(ℝ*𝑠 Σg
((dist‘𝑟) ∘ 𝑔))), ℝ*, <
))〉})) |
df-qus 15992 | ⊢ /s = (𝑟 ∈ V, 𝑒 ∈ V ↦ ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟)) |
df-xps 15993 | ⊢ ×s = (𝑟 ∈ V, 𝑠 ∈ V ↦ (◡(𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ ◡({𝑥} +𝑐 {𝑦})) “s
((Scalar‘𝑟)Xs◡({𝑟} +𝑐 {𝑠})))) |
cmre 16065 | class Moore |
cmrc 16066 | class mrCls |
cmri 16067 | class mrInd |
cacs 16068 | class ACS |
df-mre 16069 | ⊢ Moore = (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠
∈ 𝑐))}) |
df-mrc 16070 | ⊢ mrCls = (𝑐 ∈ ∪ ran
Moore ↦ (𝑥 ∈
𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠})) |
df-mri 16071 | ⊢ mrInd = (𝑐 ∈ ∪ ran
Moore ↦ {𝑠 ∈
𝒫 ∪ 𝑐 ∣ ∀𝑥 ∈ 𝑠 ¬ 𝑥 ∈ ((mrCls‘𝑐)‘(𝑠 ∖ {𝑥}))}) |
df-acs 16072 | ⊢ ACS = (𝑥 ∈ V ↦ {𝑐 ∈ (Moore‘𝑥) ∣ ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))}) |
ccat 16148 | class Cat |
ccid 16149 | class Id |
chomf 16150 | class
Homf |
ccomf 16151 | class
compf |
df-cat 16152 | ⊢ Cat = {𝑐 ∣ [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ][(comp‘𝑐) / 𝑜]∀𝑥 ∈ 𝑏 (∃𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(〈𝑦, 𝑥〉𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(〈𝑥, 𝑥〉𝑜𝑦)𝑔) = 𝑓) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)((𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓) ∈ (𝑥ℎ𝑧) ∧ ∀𝑤 ∈ 𝑏 ∀𝑘 ∈ (𝑧ℎ𝑤)((𝑘(〈𝑦, 𝑧〉𝑜𝑤)𝑔)(〈𝑥, 𝑦〉𝑜𝑤)𝑓) = (𝑘(〈𝑥, 𝑧〉𝑜𝑤)(𝑔(〈𝑥, 𝑦〉𝑜𝑧)𝑓))))} |
df-cid 16153 | ⊢ Id = (𝑐 ∈ Cat ↦
⦋(Base‘𝑐) / 𝑏⦌⦋(Hom
‘𝑐) / ℎ⦌⦋(comp‘𝑐) / 𝑜⦌(𝑥 ∈ 𝑏 ↦ (℩𝑔 ∈ (𝑥ℎ𝑥)∀𝑦 ∈ 𝑏 (∀𝑓 ∈ (𝑦ℎ𝑥)(𝑔(〈𝑦, 𝑥〉𝑜𝑥)𝑓) = 𝑓 ∧ ∀𝑓 ∈ (𝑥ℎ𝑦)(𝑓(〈𝑥, 𝑥〉𝑜𝑦)𝑔) = 𝑓)))) |
df-homf 16154 | ⊢ Homf = (𝑐 ∈ V ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦))) |
df-comf 16155 | ⊢ compf = (𝑐 ∈ V ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)), 𝑦 ∈ (Base‘𝑐) ↦ (𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝑐)𝑦), 𝑓 ∈ ((Hom ‘𝑐)‘𝑥) ↦ (𝑔(𝑥(comp‘𝑐)𝑦)𝑓)))) |
coppc 16194 | class oppCat |
df-oppc 16195 | ⊢ oppCat = (𝑓 ∈ V ↦ ((𝑓 sSet 〈(Hom ‘ndx), tpos (Hom
‘𝑓)〉) sSet
〈(comp‘ndx), (𝑢
∈ ((Base‘𝑓)
× (Base‘𝑓)),
𝑧 ∈ (Base‘𝑓) ↦ tpos (〈𝑧, (2nd ‘𝑢)〉(comp‘𝑓)(1st ‘𝑢)))〉)) |
cmon 16211 | class Mono |
cepi 16212 | class Epi |
df-mon 16213 | ⊢ Mono = (𝑐 ∈ Cat ↦
⦋(Base‘𝑐) / 𝑏⦌⦋(Hom
‘𝑐) / ℎ⦌(𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ {𝑓 ∈ (𝑥ℎ𝑦) ∣ ∀𝑧 ∈ 𝑏 Fun ◡(𝑔 ∈ (𝑧ℎ𝑥) ↦ (𝑓(〈𝑧, 𝑥〉(comp‘𝑐)𝑦)𝑔))})) |
df-epi 16214 | ⊢ Epi = (𝑐 ∈ Cat ↦ tpos
(Mono‘(oppCat‘𝑐))) |
csect 16227 | class Sect |
cinv 16228 | class Inv |
ciso 16229 | class Iso |
df-sect 16230 | ⊢ Sect = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ {〈𝑓, 𝑔〉 ∣ [(Hom ‘𝑐) / ℎ]((𝑓 ∈ (𝑥ℎ𝑦) ∧ 𝑔 ∈ (𝑦ℎ𝑥)) ∧ (𝑔(〈𝑥, 𝑦〉(comp‘𝑐)𝑥)𝑓) = ((Id‘𝑐)‘𝑥))})) |
df-inv 16231 | ⊢ Inv = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ ◡(𝑦(Sect‘𝑐)𝑥)))) |
df-iso 16232 | ⊢ Iso = (𝑐 ∈ Cat ↦ ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝑐))) |
ccic 16278 | class
≃𝑐 |
df-cic 16279 | ⊢ ≃𝑐 = (𝑐 ∈ Cat ↦
((Iso‘𝑐) supp
∅)) |
cssc 16290 | class
⊆cat |
cresc 16291 | class
↾cat |
csubc 16292 | class Subcat |
df-ssc 16293 | ⊢ ⊆cat = {〈ℎ, 𝑗〉 ∣ ∃𝑡(𝑗 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡ℎ ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝑗‘𝑥))} |
df-resc 16294 | ⊢ ↾cat = (𝑐 ∈ V, ℎ ∈ V ↦ ((𝑐 ↾s dom dom ℎ) sSet 〈(Hom ‘ndx),
ℎ〉)) |
df-subc 16295 | ⊢ Subcat = (𝑐 ∈ Cat ↦ {ℎ ∣ (ℎ ⊆cat (Homf
‘𝑐) ∧ [dom
dom ℎ / 𝑠]∀𝑥 ∈ 𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥ℎ𝑥) ∧ ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝑐)𝑧)𝑓) ∈ (𝑥ℎ𝑧)))}) |
cfunc 16337 | class Func |
cidfu 16338 | class
idfunc |
ccofu 16339 | class
∘func |
cresf 16340 | class
↾f |
df-func 16341 | ⊢ Func = (𝑡 ∈ Cat, 𝑢 ∈ Cat ↦ {〈𝑓, 𝑔〉 ∣ [(Base‘𝑡) / 𝑏](𝑓:𝑏⟶(Base‘𝑢) ∧ 𝑔 ∈ X𝑧 ∈ (𝑏 × 𝑏)(((𝑓‘(1st ‘𝑧))(Hom ‘𝑢)(𝑓‘(2nd ‘𝑧))) ↑𝑚
((Hom ‘𝑡)‘𝑧)) ∧ ∀𝑥 ∈ 𝑏 (((𝑥𝑔𝑥)‘((Id‘𝑡)‘𝑥)) = ((Id‘𝑢)‘(𝑓‘𝑥)) ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑚 ∈ (𝑥(Hom ‘𝑡)𝑦)∀𝑛 ∈ (𝑦(Hom ‘𝑡)𝑧)((𝑥𝑔𝑧)‘(𝑛(〈𝑥, 𝑦〉(comp‘𝑡)𝑧)𝑚)) = (((𝑦𝑔𝑧)‘𝑛)(〈(𝑓‘𝑥), (𝑓‘𝑦)〉(comp‘𝑢)(𝑓‘𝑧))((𝑥𝑔𝑦)‘𝑚))))}) |
df-idfu 16342 | ⊢ idfunc = (𝑡 ∈ Cat ↦
⦋(Base‘𝑡) / 𝑏⦌〈( I ↾ 𝑏), (𝑧 ∈ (𝑏 × 𝑏) ↦ ( I ↾ ((Hom ‘𝑡)‘𝑧)))〉) |
df-cofu 16343 | ⊢ ∘func = (𝑔 ∈ V, 𝑓 ∈ V ↦ 〈((1st
‘𝑔) ∘
(1st ‘𝑓)),
(𝑥 ∈ dom dom
(2nd ‘𝑓),
𝑦 ∈ dom dom
(2nd ‘𝑓)
↦ ((((1st ‘𝑓)‘𝑥)(2nd ‘𝑔)((1st ‘𝑓)‘𝑦)) ∘ (𝑥(2nd ‘𝑓)𝑦)))〉) |
df-resf 16344 | ⊢ ↾f = (𝑓 ∈ V, ℎ ∈ V ↦ 〈((1st
‘𝑓) ↾ dom dom
ℎ), (𝑥 ∈ dom ℎ ↦ (((2nd ‘𝑓)‘𝑥) ↾ (ℎ‘𝑥)))〉) |
cful 16385 | class Full |
cfth 16386 | class Faith |
df-full 16387 | ⊢ Full = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ {〈𝑓, 𝑔〉 ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)(Hom ‘𝑑)(𝑓‘𝑦)))}) |
df-fth 16388 | ⊢ Faith = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ {〈𝑓, 𝑔〉 ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun ◡(𝑥𝑔𝑦))}) |
cnat 16424 | class Nat |
cfuc 16425 | class FuncCat |
df-nat 16426 | ⊢ Nat = (𝑡 ∈ Cat, 𝑢 ∈ Cat ↦ (𝑓 ∈ (𝑡 Func 𝑢), 𝑔 ∈ (𝑡 Func 𝑢) ↦ ⦋(1st
‘𝑓) / 𝑟⦌⦋(1st
‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈
(Base‘𝑡)((𝑟‘𝑥)(Hom ‘𝑢)(𝑠‘𝑥)) ∣ ∀𝑥 ∈ (Base‘𝑡)∀𝑦 ∈ (Base‘𝑡)∀ℎ ∈ (𝑥(Hom ‘𝑡)𝑦)((𝑎‘𝑦)(〈(𝑟‘𝑥), (𝑟‘𝑦)〉(comp‘𝑢)(𝑠‘𝑦))((𝑥(2nd ‘𝑓)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝑔)𝑦)‘ℎ)(〈(𝑟‘𝑥), (𝑠‘𝑥)〉(comp‘𝑢)(𝑠‘𝑦))(𝑎‘𝑥))})) |
df-fuc 16427 | ⊢ FuncCat = (𝑡 ∈ Cat, 𝑢 ∈ Cat ↦ {〈(Base‘ndx),
(𝑡 Func 𝑢)〉, 〈(Hom ‘ndx), (𝑡 Nat 𝑢)〉, 〈(comp‘ndx), (𝑣 ∈ ((𝑡 Func 𝑢) × (𝑡 Func 𝑢)), ℎ ∈ (𝑡 Func 𝑢) ↦ ⦋(1st
‘𝑣) / 𝑓⦌⦋(2nd
‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑡 Nat 𝑢)ℎ), 𝑎 ∈ (𝑓(𝑡 Nat 𝑢)𝑔) ↦ (𝑥 ∈ (Base‘𝑡) ↦ ((𝑏‘𝑥)(〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉(comp‘𝑢)((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))〉}) |
cinito 16461 | class InitO |
ctermo 16462 | class TermO |
czeroo 16463 | class ZeroO |
df-inito 16464 | ⊢ InitO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝑐)𝑏)}) |
df-termo 16465 | ⊢ TermO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝑐)𝑎)}) |
df-zeroo 16466 | ⊢ ZeroO = (𝑐 ∈ Cat ↦ ((InitO‘𝑐) ∩ (TermO‘𝑐))) |
cdoma 16493 | class
doma |
ccoda 16494 | class
coda |
carw 16495 | class Arrow |
choma 16496 | class
Homa |
df-doma 16497 | ⊢ doma = (1st
∘ 1st ) |
df-coda 16498 | ⊢ coda = (2nd
∘ 1st ) |
df-homa 16499 | ⊢ Homa = (𝑐 ∈ Cat ↦ (𝑥 ∈ ((Base‘𝑐) × (Base‘𝑐)) ↦ ({𝑥} × ((Hom ‘𝑐)‘𝑥)))) |
df-arw 16500 | ⊢ Arrow = (𝑐 ∈ Cat ↦ ∪ ran (Homa‘𝑐)) |
cida 16526 | class
Ida |
ccoa 16527 | class
compa |
df-ida 16528 | ⊢ Ida = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐) ↦ 〈𝑥, 𝑥, ((Id‘𝑐)‘𝑥)〉)) |
df-coa 16529 | ⊢ compa = (𝑐 ∈ Cat ↦ (𝑔 ∈ (Arrow‘𝑐), 𝑓 ∈ {ℎ ∈ (Arrow‘𝑐) ∣ (coda‘ℎ) =
(doma‘𝑔)} ↦
〈(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(〈(doma‘𝑓),
(doma‘𝑔)〉(comp‘𝑐)(coda‘𝑔))(2nd ‘𝑓))〉)) |
csetc 16548 | class SetCat |
df-setc 16549 | ⊢ SetCat = (𝑢 ∈ V ↦ {〈(Base‘ndx),
𝑢〉, 〈(Hom
‘ndx), (𝑥 ∈
𝑢, 𝑦 ∈ 𝑢 ↦ (𝑦 ↑𝑚 𝑥))〉,
〈(comp‘ndx), (𝑣
∈ (𝑢 × 𝑢), 𝑧 ∈ 𝑢 ↦ (𝑔 ∈ (𝑧 ↑𝑚 (2nd
‘𝑣)), 𝑓 ∈ ((2nd
‘𝑣)
↑𝑚 (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))〉}) |
ccatc 16567 | class CatCat |
df-catc 16568 | ⊢ CatCat = (𝑢 ∈ V ↦ ⦋(𝑢 ∩ Cat) / 𝑏⦌{〈(Base‘ndx), 𝑏〉, 〈(Hom ‘ndx),
(𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 Func 𝑦))〉, 〈(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))〉}) |
cestrc 16585 | class ExtStrCat |
df-estrc 16586 | ⊢ ExtStrCat = (𝑢 ∈ V ↦ {〈(Base‘ndx),
𝑢〉, 〈(Hom
‘ndx), (𝑥 ∈
𝑢, 𝑦 ∈ 𝑢 ↦ ((Base‘𝑦) ↑𝑚
(Base‘𝑥)))〉,
〈(comp‘ndx), (𝑣
∈ (𝑢 × 𝑢), 𝑧 ∈ 𝑢 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚
(Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd
‘𝑣))
↑𝑚 (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓)))〉}) |
cxpc 16631 | class
×c |
c1stf 16632 | class
1stF |
c2ndf 16633 | class
2ndF |
cprf 16634 | class
〈,〉F |
df-xpc 16635 | ⊢ ×c = (𝑟 ∈ V, 𝑠 ∈ V ↦
⦋((Base‘𝑟) × (Base‘𝑠)) / 𝑏⦌⦋(𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1st ‘𝑢)(Hom ‘𝑟)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑠)(2nd ‘𝑣)))) / ℎ⦌{〈(Base‘ndx), 𝑏〉, 〈(Hom ‘ndx),
ℎ〉,
〈(comp‘ndx), (𝑥
∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ 〈((1st ‘𝑔)(〈(1st
‘(1st ‘𝑥)), (1st ‘(2nd
‘𝑥))〉(comp‘𝑟)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(〈(2nd
‘(1st ‘𝑥)), (2nd ‘(2nd
‘𝑥))〉(comp‘𝑠)(2nd ‘𝑦))(2nd ‘𝑓))〉))〉}) |
df-1stf 16636 | ⊢ 1stF = (𝑟 ∈ Cat, 𝑠 ∈ Cat ↦
⦋((Base‘𝑟) × (Base‘𝑠)) / 𝑏⦌〈(1st ↾
𝑏), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (1st ↾ (𝑥(Hom ‘(𝑟 ×c 𝑠))𝑦)))〉) |
df-2ndf 16637 | ⊢ 2ndF = (𝑟 ∈ Cat, 𝑠 ∈ Cat ↦
⦋((Base‘𝑟) × (Base‘𝑠)) / 𝑏⦌〈(2nd ↾
𝑏), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (2nd ↾ (𝑥(Hom ‘(𝑟 ×c 𝑠))𝑦)))〉) |
df-prf 16638 | ⊢ 〈,〉F = (𝑓 ∈ V, 𝑔 ∈ V ↦ ⦋dom
(1st ‘𝑓) /
𝑏⦌〈(𝑥 ∈ 𝑏 ↦ 〈((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)〉), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (ℎ ∈ dom (𝑥(2nd ‘𝑓)𝑦) ↦ 〈((𝑥(2nd ‘𝑓)𝑦)‘ℎ), ((𝑥(2nd ‘𝑔)𝑦)‘ℎ)〉))〉) |
cevlf 16672 | class
evalF |
ccurf 16673 | class
curryF |
cuncf 16674 | class
uncurryF |
cdiag 16675 | class
Δfunc |
df-evlf 16676 | ⊢ evalF = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ 〈(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (Base‘𝑐) ↦ ((1st ‘𝑓)‘𝑥)), (𝑥 ∈ ((𝑐 Func 𝑑) × (Base‘𝑐)), 𝑦 ∈ ((𝑐 Func 𝑑) × (Base‘𝑐)) ↦ ⦋(1st
‘𝑥) / 𝑚⦌⦋(1st
‘𝑦) / 𝑛⦌(𝑎 ∈ (𝑚(𝑐 Nat 𝑑)𝑛), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝑐)(2nd ‘𝑦)) ↦ ((𝑎‘(2nd ‘𝑦))(〈((1st
‘𝑚)‘(2nd ‘𝑥)), ((1st
‘𝑚)‘(2nd ‘𝑦))〉(comp‘𝑑)((1st ‘𝑛)‘(2nd
‘𝑦)))(((2nd ‘𝑥)(2nd ‘𝑚)(2nd ‘𝑦))‘𝑔))))〉) |
df-curf 16677 | ⊢ curryF = (𝑒 ∈ V, 𝑓 ∈ V ↦
⦋(1st ‘𝑒) / 𝑐⦌⦋(2nd
‘𝑒) / 𝑑⦌〈(𝑥 ∈ (Base‘𝑐) ↦ 〈(𝑦 ∈ (Base‘𝑑) ↦ (𝑥(1st ‘𝑓)𝑦)), (𝑦 ∈ (Base‘𝑑), 𝑧 ∈ (Base‘𝑑) ↦ (𝑔 ∈ (𝑦(Hom ‘𝑑)𝑧) ↦ (((Id‘𝑐)‘𝑥)(〈𝑥, 𝑦〉(2nd ‘𝑓)〈𝑥, 𝑧〉)𝑔)))〉), (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑔 ∈ (𝑥(Hom ‘𝑐)𝑦) ↦ (𝑧 ∈ (Base‘𝑑) ↦ (𝑔(〈𝑥, 𝑧〉(2nd ‘𝑓)〈𝑦, 𝑧〉)((Id‘𝑑)‘𝑧)))))〉) |
df-uncf 16678 | ⊢ uncurryF = (𝑐 ∈ V, 𝑓 ∈ V ↦ (((𝑐‘1) evalF (𝑐‘2))
∘func ((𝑓 ∘func ((𝑐‘0)
1stF (𝑐‘1))) 〈,〉F
((𝑐‘0)
2ndF (𝑐‘1))))) |
df-diag 16679 | ⊢ Δfunc = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ (〈𝑐, 𝑑〉 curryF (𝑐
1stF 𝑑))) |
chof 16711 | class
HomF |
cyon 16712 | class Yon |
df-hof 16713 | ⊢ HomF = (𝑐 ∈ Cat ↦
〈(Homf ‘𝑐), ⦋(Base‘𝑐) / 𝑏⦌(𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ (𝑏 × 𝑏) ↦ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝑐)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝑐)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝑐)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝑐)(2nd ‘𝑦))ℎ)(〈(1st ‘𝑦), (1st ‘𝑥)〉(comp‘𝑐)(2nd ‘𝑦))𝑓))))〉) |
df-yon 16714 | ⊢ Yon = (𝑐 ∈ Cat ↦ (〈𝑐, (oppCat‘𝑐)〉 curryF
(HomF‘(oppCat‘𝑐)))) |
cpreset 16749 | class Preset |
cdrs 16750 | class Dirset |
df-preset 16751 | ⊢ Preset = {𝑓 ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧))} |
df-drs 16752 | ⊢ Dirset = {𝑓 ∈ Preset ∣
[(Base‘𝑓) /
𝑏][(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧))} |
cpo 16763 | class Poset |
cplt 16764 | class lt |
club 16765 | class lub |
cglb 16766 | class glb |
cjn 16767 | class join |
cmee 16768 | class meet |
df-poset 16769 | ⊢ Poset = {𝑓 ∣ ∃𝑏∃𝑟(𝑏 = (Base‘𝑓) ∧ 𝑟 = (le‘𝑓) ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦 ∧ 𝑦𝑟𝑧) → 𝑥𝑟𝑧)))} |
df-plt 16781 | ⊢ lt = (𝑝 ∈ V ↦ ((le‘𝑝) ∖ I )) |
df-lub 16797 | ⊢ lub = (𝑝 ∈ V ↦ ((𝑠 ∈ 𝒫 (Base‘𝑝) ↦ (℩𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑦(le‘𝑝)𝑥 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑦(le‘𝑝)𝑧 → 𝑥(le‘𝑝)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑦(le‘𝑝)𝑥 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑦(le‘𝑝)𝑧 → 𝑥(le‘𝑝)𝑧))})) |
df-glb 16798 | ⊢ glb = (𝑝 ∈ V ↦ ((𝑠 ∈ 𝒫 (Base‘𝑝) ↦ (℩𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑥(le‘𝑝)𝑦 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑧(le‘𝑝)𝑦 → 𝑧(le‘𝑝)𝑥)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑥(le‘𝑝)𝑦 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑧(le‘𝑝)𝑦 → 𝑧(le‘𝑝)𝑥))})) |
df-join 16799 | ⊢ join = (𝑝 ∈ V ↦ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ {𝑥, 𝑦} (lub‘𝑝)𝑧}) |
df-meet 16800 | ⊢ meet = (𝑝 ∈ V ↦ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ {𝑥, 𝑦} (glb‘𝑝)𝑧}) |
ctos 16856 | class Toset |
df-toset 16857 | ⊢ Toset = {𝑓 ∈ Poset ∣
[(Base‘𝑓) /
𝑏][(le‘𝑓) / 𝑟]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥)} |
cp0 16860 | class 0. |
cp1 16861 | class 1. |
df-p0 16862 | ⊢ 0. = (𝑝 ∈ V ↦ ((glb‘𝑝)‘(Base‘𝑝))) |
df-p1 16863 | ⊢ 1. = (𝑝 ∈ V ↦ ((lub‘𝑝)‘(Base‘𝑝))) |
clat 16868 | class Lat |
df-lat 16869 | ⊢ Lat = {𝑝 ∈ Poset ∣ (dom (join‘𝑝) = ((Base‘𝑝) × (Base‘𝑝)) ∧ dom (meet‘𝑝) = ((Base‘𝑝) × (Base‘𝑝)))} |
ccla 16930 | class CLat |
df-clat 16931 | ⊢ CLat = {𝑝 ∈ Poset ∣ (dom (lub‘𝑝) = 𝒫 (Base‘𝑝) ∧ dom (glb‘𝑝) = 𝒫 (Base‘𝑝))} |
codu 16951 | class ODual |
df-odu 16952 | ⊢ ODual = (𝑤 ∈ V ↦ (𝑤 sSet 〈(le‘ndx), ◡(le‘𝑤)〉)) |
cipo 16974 | class toInc |
df-ipo 16975 | ⊢ toInc = (𝑓 ∈ V ↦ ⦋{〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑓 ∧ 𝑥 ⊆ 𝑦)} / 𝑜⦌({〈(Base‘ndx),
𝑓〉,
〈(TopSet‘ndx), (ordTop‘𝑜)〉} ∪ {〈(le‘ndx), 𝑜〉, 〈(oc‘ndx),
(𝑥 ∈ 𝑓 ↦ ∪ {𝑦
∈ 𝑓 ∣ (𝑦 ∩ 𝑥) = ∅})〉})) |
cdlat 17014 | class DLat |
df-dlat 17015 | ⊢ DLat = {𝑘 ∈ Lat ∣ [(Base‘𝑘) / 𝑏][(join‘𝑘) / 𝑗][(meet‘𝑘) / 𝑚]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧))} |
cps 17021 | class PosetRel |
ctsr 17022 | class TosetRel |
df-ps 17023 | ⊢ PosetRel = {𝑟 ∣ (Rel 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟 ∧ (𝑟 ∩ ◡𝑟) = ( I ↾ ∪
∪ 𝑟))} |
df-tsr 17024 | ⊢ TosetRel = {𝑟 ∈ PosetRel ∣ (dom 𝑟 × dom 𝑟) ⊆ (𝑟 ∪ ◡𝑟)} |
cdir 17051 | class DirRel |
ctail 17052 | class tail |
df-dir 17053 | ⊢ DirRel = {𝑟 ∣ ((Rel 𝑟 ∧ ( I ↾ ∪ ∪ 𝑟) ⊆ 𝑟) ∧ ((𝑟 ∘ 𝑟) ⊆ 𝑟 ∧ (∪ ∪ 𝑟
× ∪ ∪ 𝑟) ⊆ (◡𝑟 ∘ 𝑟)))} |
df-tail 17054 | ⊢ tail = (𝑟 ∈ DirRel ↦ (𝑥 ∈ ∪ ∪ 𝑟
↦ (𝑟 “ {𝑥}))) |
cplusf 17062 | class
+𝑓 |
cmgm 17063 | class Mgm |
df-plusf 17064 | ⊢ +𝑓 = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)𝑦))) |
df-mgm 17065 | ⊢ Mgm = {𝑔 ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑜]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑜𝑦) ∈ 𝑏} |
csgrp 17106 | class SGrp |
df-sgrp 17107 | ⊢ SGrp = {𝑔 ∈ Mgm ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑜]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))} |
cmnd 17117 | class Mnd |
df-mnd 17118 | ⊢ Mnd = {𝑔 ∈ SGrp ∣
[(Base‘𝑔) /
𝑏][(+g‘𝑔) / 𝑝]∃𝑒 ∈ 𝑏 ∀𝑥 ∈ 𝑏 ((𝑒𝑝𝑥) = 𝑥 ∧ (𝑥𝑝𝑒) = 𝑥)} |
cmhm 17156 | class MndHom |
csubmnd 17157 | class SubMnd |
df-mhm 17158 | ⊢ MndHom = (𝑠 ∈ Mnd, 𝑡 ∈ Mnd ↦ {𝑓 ∈ ((Base‘𝑡) ↑𝑚
(Base‘𝑠)) ∣
(∀𝑥 ∈
(Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g‘𝑠)𝑦)) = ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝑠)) = (0g‘𝑡))}) |
df-submnd 17159 | ⊢ SubMnd = (𝑠 ∈ Mnd ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣
((0g‘𝑠)
∈ 𝑡 ∧
∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑠)𝑦) ∈ 𝑡)}) |
cfrmd 17207 | class freeMnd |
cvrmd 17208 | class
varFMnd |
df-frmd 17209 | ⊢ freeMnd = (𝑖 ∈ V ↦ {〈(Base‘ndx),
Word 𝑖〉,
〈(+g‘ndx), ( ++ ↾ (Word 𝑖 × Word 𝑖))〉}) |
df-vrmd 17210 | ⊢ varFMnd = (𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ 〈“𝑗”〉)) |
cgrp 17245 | class Grp |
cminusg 17246 | class invg |
csg 17247 | class -g |
df-grp 17248 | ⊢ Grp = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∃𝑚 ∈ (Base‘𝑔)(𝑚(+g‘𝑔)𝑎) = (0g‘𝑔)} |
df-minusg 17249 | ⊢ invg = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ (℩𝑤 ∈ (Base‘𝑔)(𝑤(+g‘𝑔)𝑥) = (0g‘𝑔)))) |
df-sbg 17250 | ⊢ -g = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)((invg‘𝑔)‘𝑦)))) |
cmg 17363 | class .g |
df-mulg 17364 | ⊢ .g = (𝑔 ∈ V ↦ (𝑛 ∈ ℤ, 𝑥 ∈ (Base‘𝑔) ↦ if(𝑛 = 0, (0g‘𝑔),
⦋seq1((+g‘𝑔), (ℕ × {𝑥})) / 𝑠⦌if(0 < 𝑛, (𝑠‘𝑛), ((invg‘𝑔)‘(𝑠‘-𝑛)))))) |
csubg 17411 | class SubGrp |
cnsg 17412 | class NrmSGrp |
cqg 17413 | class
~QG |
df-subg 17414 | ⊢ SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp}) |
df-nsg 17415 | ⊢ NrmSGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ (SubGrp‘𝑤) ∣ [(Base‘𝑤) / 𝑏][(+g‘𝑤) / 𝑝]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ((𝑥𝑝𝑦) ∈ 𝑠 ↔ (𝑦𝑝𝑥) ∈ 𝑠)}) |
df-eqg 17416 | ⊢ ~QG = (𝑟 ∈ V, 𝑖 ∈ V ↦ {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑟) ∧ (((invg‘𝑟)‘𝑥)(+g‘𝑟)𝑦) ∈ 𝑖)}) |
cghm 17480 | class GrpHom |
df-ghm 17481 | ⊢ GrpHom = (𝑠 ∈ Grp, 𝑡 ∈ Grp ↦ {𝑔 ∣ [(Base‘𝑠) / 𝑤](𝑔:𝑤⟶(Base‘𝑡) ∧ ∀𝑥 ∈ 𝑤 ∀𝑦 ∈ 𝑤 (𝑔‘(𝑥(+g‘𝑠)𝑦)) = ((𝑔‘𝑥)(+g‘𝑡)(𝑔‘𝑦)))}) |
cgim 17522 | class GrpIso |
cgic 17523 | class
≃𝑔 |
df-gim 17524 | ⊢ GrpIso = (𝑠 ∈ Grp, 𝑡 ∈ Grp ↦ {𝑔 ∈ (𝑠 GrpHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)}) |
df-gic 17525 | ⊢ ≃𝑔 = (◡ GrpIso “ (V ∖
1𝑜)) |
cga 17545 | class GrpAct |
df-ga 17546 | ⊢ GrpAct = (𝑔 ∈ Grp, 𝑠 ∈ V ↦
⦋(Base‘𝑔) / 𝑏⦌{𝑚 ∈ (𝑠 ↑𝑚 (𝑏 × 𝑠)) ∣ ∀𝑥 ∈ 𝑠 (((0g‘𝑔)𝑚𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑦(+g‘𝑔)𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))}) |
ccntz 17571 | class Cntz |
ccntr 17572 | class Cntr |
df-cntz 17573 | ⊢ Cntz = (𝑚 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑚) ↦ {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑚)𝑦) = (𝑦(+g‘𝑚)𝑥)})) |
df-cntr 17574 | ⊢ Cntr = (𝑚 ∈ V ↦ ((Cntz‘𝑚)‘(Base‘𝑚))) |
coppg 17598 | class
oppg |
df-oppg 17599 | ⊢ oppg = (𝑤 ∈ V ↦ (𝑤 sSet 〈(+g‘ndx), tpos
(+g‘𝑤)〉)) |
csymg 17620 | class SymGrp |
df-symg 17621 | ⊢ SymGrp = (𝑥 ∈ V ↦ ⦋{ℎ ∣ ℎ:𝑥–1-1-onto→𝑥} / 𝑏⦌{〈(Base‘ndx), 𝑏〉,
〈(+g‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓 ∘ 𝑔))〉, 〈(TopSet‘ndx),
(∏t‘(𝑥 × {𝒫 𝑥}))〉}) |
cpmtr 17684 | class pmTrsp |
df-pmtr 17685 | ⊢ pmTrsp = (𝑑 ∈ V ↦ (𝑝 ∈ {𝑦 ∈ 𝒫 𝑑 ∣ 𝑦 ≈ 2𝑜} ↦
(𝑧 ∈ 𝑑 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))) |
cpsgn 17732 | class pmSgn |
cevpm 17733 | class pmEven |
df-psgn 17734 | ⊢ pmSgn = (𝑑 ∈ V ↦ (𝑥 ∈ {𝑝 ∈ (Base‘(SymGrp‘𝑑)) ∣ dom (𝑝 ∖ I ) ∈ Fin} ↦
(℩𝑠∃𝑤 ∈ Word ran
(pmTrsp‘𝑑)(𝑥 = ((SymGrp‘𝑑) Σg
𝑤) ∧ 𝑠 = (-1↑(#‘𝑤)))))) |
df-evpm 17735 | ⊢ pmEven = (𝑑 ∈ V ↦ (◡(pmSgn‘𝑑) “ {1})) |
cod 17767 | class od |
cgex 17768 | class gEx |
cpgp 17769 | class pGrp |
cslw 17770 | class pSyl |
df-od 17771 | ⊢ od = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ ⦋{𝑛 ∈ ℕ ∣ (𝑛(.g‘𝑔)𝑥) = (0g‘𝑔)} / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))) |
df-gex 17772 | ⊢ gEx = (𝑔 ∈ V ↦ ⦋{𝑛 ∈ ℕ ∣
∀𝑥 ∈
(Base‘𝑔)(𝑛(.g‘𝑔)𝑥) = (0g‘𝑔)} / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))) |
df-pgp 17773 | ⊢ pGrp = {〈𝑝, 𝑔〉 ∣ ((𝑝 ∈ ℙ ∧ 𝑔 ∈ Grp) ∧ ∀𝑥 ∈ (Base‘𝑔)∃𝑛 ∈ ℕ0 ((od‘𝑔)‘𝑥) = (𝑝↑𝑛))} |
df-slw 17774 | ⊢ pSyl = (𝑝 ∈ ℙ, 𝑔 ∈ Grp ↦ {ℎ ∈ (SubGrp‘𝑔) ∣ ∀𝑘 ∈ (SubGrp‘𝑔)((ℎ ⊆ 𝑘 ∧ 𝑝 pGrp (𝑔 ↾s 𝑘)) ↔ ℎ = 𝑘)}) |
clsm 17872 | class LSSum |
cpj1 17873 | class
proj1 |
df-lsm 17874 | ⊢ LSSum = (𝑤 ∈ V ↦ (𝑡 ∈ 𝒫 (Base‘𝑤), 𝑢 ∈ 𝒫 (Base‘𝑤) ↦ ran (𝑥 ∈ 𝑡, 𝑦 ∈ 𝑢 ↦ (𝑥(+g‘𝑤)𝑦)))) |
df-pj1 17875 | ⊢ proj1 = (𝑤 ∈ V ↦ (𝑡 ∈ 𝒫 (Base‘𝑤), 𝑢 ∈ 𝒫 (Base‘𝑤) ↦ (𝑧 ∈ (𝑡(LSSum‘𝑤)𝑢) ↦ (℩𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝑢 𝑧 = (𝑥(+g‘𝑤)𝑦))))) |
cefg 17942 | class
~FG |
cfrgp 17943 | class freeGrp |
cvrgp 17944 | class
varFGrp |
df-efg 17945 | ⊢ ~FG = (𝑖 ∈ V ↦ ∩ {𝑟
∣ (𝑟 Er Word (𝑖 × 2𝑜)
∧ ∀𝑥 ∈ Word
(𝑖 ×
2𝑜)∀𝑛 ∈ (0...(#‘𝑥))∀𝑦 ∈ 𝑖 ∀𝑧 ∈ 2𝑜 𝑥𝑟(𝑥 splice 〈𝑛, 𝑛, 〈“〈𝑦, 𝑧〉〈𝑦, (1𝑜 ∖ 𝑧)〉”〉〉))}) |
df-frgp 17946 | ⊢ freeGrp = (𝑖 ∈ V ↦ ((freeMnd‘(𝑖 × 2𝑜))
/s ( ~FG ‘𝑖))) |
df-vrgp 17947 | ⊢ varFGrp = (𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ [〈“〈𝑗, ∅〉”〉](
~FG ‘𝑖))) |
ccmn 18016 | class CMnd |
cabl 18017 | class Abel |
df-cmn 18018 | ⊢ CMnd = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∀𝑏 ∈ (Base‘𝑔)(𝑎(+g‘𝑔)𝑏) = (𝑏(+g‘𝑔)𝑎)} |
df-abl 18019 | ⊢ Abel = (Grp ∩ CMnd) |
ccyg 18102 | class CycGrp |
df-cyg 18103 | ⊢ CycGrp = {𝑔 ∈ Grp ∣ ∃𝑥 ∈ (Base‘𝑔)ran (𝑛 ∈ ℤ ↦ (𝑛(.g‘𝑔)𝑥)) = (Base‘𝑔)} |
cdprd 18215 | class DProd |
cdpj 18216 | class dProj |
df-dprd 18217 | ⊢ DProd = (𝑔 ∈ Grp, 𝑠 ∈ {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝑔) ∧ ∀𝑥 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ ((Cntz‘𝑔)‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ
“ (dom ℎ ∖
{𝑥})))) =
{(0g‘𝑔)}))} ↦ ran (𝑓 ∈ {ℎ ∈ X𝑥 ∈ dom 𝑠(𝑠‘𝑥) ∣ ℎ finSupp (0g‘𝑔)} ↦ (𝑔 Σg 𝑓))) |
df-dpj 18218 | ⊢ dProj = (𝑔 ∈ Grp, 𝑠 ∈ (dom DProd “ {𝑔}) ↦ (𝑖 ∈ dom 𝑠 ↦ ((𝑠‘𝑖)(proj1‘𝑔)(𝑔 DProd (𝑠 ↾ (dom 𝑠 ∖ {𝑖})))))) |
cmgp 18312 | class mulGrp |
df-mgp 18313 | ⊢ mulGrp = (𝑤 ∈ V ↦ (𝑤 sSet 〈(+g‘ndx),
(.r‘𝑤)〉)) |
cur 18324 | class 1r |
df-ur 18325 | ⊢ 1r = (0g
∘ mulGrp) |
csrg 18328 | class SRing |
df-srg 18329 | ⊢ SRing = {𝑓 ∈ CMnd ∣ ((mulGrp‘𝑓) ∈ Mnd ∧
[(Base‘𝑓) /
𝑟][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡][(0g‘𝑓) / 𝑛]∀𝑥 ∈ 𝑟 (∀𝑦 ∈ 𝑟 ∀𝑧 ∈ 𝑟 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)))} |
crg 18370 | class Ring |
ccrg 18371 | class CRing |
df-ring 18372 | ⊢ Ring = {𝑓 ∈ Grp ∣ ((mulGrp‘𝑓) ∈ Mnd ∧
[(Base‘𝑓) /
𝑟][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡]∀𝑥 ∈ 𝑟 ∀𝑦 ∈ 𝑟 ∀𝑧 ∈ 𝑟 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))))} |
df-cring 18373 | ⊢ CRing = {𝑓 ∈ Ring ∣ (mulGrp‘𝑓) ∈ CMnd} |
coppr 18445 | class
oppr |
df-oppr 18446 | ⊢ oppr = (𝑓 ∈ V ↦ (𝑓 sSet 〈(.r‘ndx), tpos
(.r‘𝑓)〉)) |
cdsr 18461 | class
∥r |
cui 18462 | class Unit |
cir 18463 | class Irred |
df-dvdsr 18464 | ⊢ ∥r = (𝑤 ∈ V ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r‘𝑤)𝑥) = 𝑦)}) |
df-unit 18465 | ⊢ Unit = (𝑤 ∈ V ↦ (◡((∥r‘𝑤) ∩
(∥r‘(oppr‘𝑤))) “
{(1r‘𝑤)})) |
df-irred 18466 | ⊢ Irred = (𝑤 ∈ V ↦
⦋((Base‘𝑤) ∖ (Unit‘𝑤)) / 𝑏⦌{𝑧 ∈ 𝑏 ∣ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(.r‘𝑤)𝑦) ≠ 𝑧}) |
cinvr 18494 | class invr |
df-invr 18495 | ⊢ invr = (𝑟 ∈ V ↦
(invg‘((mulGrp‘𝑟) ↾s (Unit‘𝑟)))) |
cdvr 18505 | class /r |
df-dvr 18506 | ⊢ /r = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦)))) |
crh 18535 | class RingHom |
crs 18536 | class RingIso |
cric 18537 | class
≃𝑟 |
df-rnghom 18538 | ⊢ RingHom = (𝑟 ∈ Ring, 𝑠 ∈ Ring ↦
⦋(Base‘𝑟) / 𝑣⦌⦋(Base‘𝑠) / 𝑤⦌{𝑓 ∈ (𝑤 ↑𝑚 𝑣) ∣ ((𝑓‘(1r‘𝑟)) = (1r‘𝑠) ∧ ∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦))))}) |
df-rngiso 18539 | ⊢ RingIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RingHom 𝑠) ∣ ◡𝑓 ∈ (𝑠 RingHom 𝑟)}) |
df-ric 18541 | ⊢ ≃𝑟 = (◡ RingIso “ (V ∖
1𝑜)) |
cdr 18570 | class DivRing |
cfield 18571 | class Field |
df-drng 18572 | ⊢ DivRing = {𝑟 ∈ Ring ∣ (Unit‘𝑟) = ((Base‘𝑟) ∖
{(0g‘𝑟)})} |
df-field 18573 | ⊢ Field = (DivRing ∩
CRing) |
csubrg 18599 | class SubRing |
crgspn 18600 | class RingSpan |
df-subrg 18601 | ⊢ SubRing = (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾s 𝑠) ∈ Ring ∧
(1r‘𝑤)
∈ 𝑠)}) |
df-rgspn 18602 | ⊢ RingSpan = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡
∈ (SubRing‘𝑤)
∣ 𝑠 ⊆ 𝑡})) |
cabv 18639 | class AbsVal |
df-abv 18640 | ⊢ AbsVal = (𝑟 ∈ Ring ↦ {𝑓 ∈ ((0[,)+∞)
↑𝑚 (Base‘𝑟)) ∣ ∀𝑥 ∈ (Base‘𝑟)(((𝑓‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑟)) ∧ ∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝑟)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))))}) |
cstf 18666 | class
*rf |
csr 18667 | class *-Ring |
df-staf 18668 | ⊢ *rf = (𝑓 ∈ V ↦ (𝑥 ∈ (Base‘𝑓) ↦ ((*𝑟‘𝑓)‘𝑥))) |
df-srng 18669 | ⊢ *-Ring = {𝑓 ∣
[(*rf‘𝑓) / 𝑖](𝑖 ∈ (𝑓 RingHom (oppr‘𝑓)) ∧ 𝑖 = ◡𝑖)} |
clmod 18686 | class LMod |
cscaf 18687 | class
·sf |
df-lmod 18688 | ⊢ LMod = {𝑔 ∈ Grp ∣ [(Base‘𝑔) / 𝑣][(+g‘𝑔) / 𝑎][(Scalar‘𝑔) / 𝑓][(
·𝑠 ‘𝑔) / 𝑠][(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤)))} |
df-scaf 18689 | ⊢ ·sf = (𝑔 ∈ V ↦ (𝑥 ∈
(Base‘(Scalar‘𝑔)), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥( ·𝑠
‘𝑔)𝑦))) |
clss 18753 | class LSubSp |
df-lss 18754 | ⊢ LSubSp = (𝑤 ∈ V ↦ {𝑠 ∈ (𝒫 (Base‘𝑤) ∖ {∅}) ∣
∀𝑥 ∈
(Base‘(Scalar‘𝑤))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥( ·𝑠
‘𝑤)𝑎)(+g‘𝑤)𝑏) ∈ 𝑠}) |
clspn 18792 | class LSpan |
df-lsp 18793 | ⊢ LSpan = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡
∈ (LSubSp‘𝑤)
∣ 𝑠 ⊆ 𝑡})) |
clmhm 18840 | class LMHom |
clmim 18841 | class LMIso |
clmic 18842 | class
≃𝑚 |
df-lmhm 18843 | ⊢ LMHom = (𝑠 ∈ LMod, 𝑡 ∈ LMod ↦ {𝑓 ∈ (𝑠 GrpHom 𝑡) ∣ [(Scalar‘𝑠) / 𝑤]((Scalar‘𝑡) = 𝑤 ∧ ∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥( ·𝑠
‘𝑠)𝑦)) = (𝑥( ·𝑠
‘𝑡)(𝑓‘𝑦)))}) |
df-lmim 18844 | ⊢ LMIso = (𝑠 ∈ LMod, 𝑡 ∈ LMod ↦ {𝑔 ∈ (𝑠 LMHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)}) |
df-lmic 18845 | ⊢ ≃𝑚 = (◡ LMIso “ (V ∖
1𝑜)) |
clbs 18895 | class LBasis |
df-lbs 18896 | ⊢ LBasis = (𝑤 ∈ V ↦ {𝑏 ∈ 𝒫 (Base‘𝑤) ∣
[(LSpan‘𝑤) /
𝑛][(Scalar‘𝑤) / 𝑠]((𝑛‘𝑏) = (Base‘𝑤) ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ ((Base‘𝑠) ∖ {(0g‘𝑠)}) ¬ (𝑦( ·𝑠
‘𝑤)𝑥) ∈ (𝑛‘(𝑏 ∖ {𝑥})))}) |
clvec 18923 | class LVec |
df-lvec 18924 | ⊢ LVec = {𝑓 ∈ LMod ∣ (Scalar‘𝑓) ∈
DivRing} |
csra 18989 | class subringAlg |
crglmod 18990 | class ringLMod |
clidl 18991 | class LIdeal |
crsp 18992 | class RSpan |
df-sra 18993 | ⊢ subringAlg = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ (((𝑤 sSet 〈(Scalar‘ndx), (𝑤 ↾s 𝑠)〉) sSet 〈(
·𝑠 ‘ndx), (.r‘𝑤)〉) sSet
〈(·𝑖‘ndx),
(.r‘𝑤)〉))) |
df-rgmod 18994 | ⊢ ringLMod = (𝑤 ∈ V ↦ ((subringAlg ‘𝑤)‘(Base‘𝑤))) |
df-lidl 18995 | ⊢ LIdeal = (LSubSp ∘
ringLMod) |
df-rsp 18996 | ⊢ RSpan = (LSpan ∘
ringLMod) |
c2idl 19052 | class 2Ideal |
df-2idl 19053 | ⊢ 2Ideal = (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩
(LIdeal‘(oppr‘𝑟)))) |
clpidl 19062 | class LPIdeal |
clpir 19063 | class LPIR |
df-lpidl 19064 | ⊢ LPIdeal = (𝑤 ∈ Ring ↦ ∪ 𝑔 ∈ (Base‘𝑤){((RSpan‘𝑤)‘{𝑔})}) |
df-lpir 19065 | ⊢ LPIR = {𝑤 ∈ Ring ∣ (LIdeal‘𝑤) = (LPIdeal‘𝑤)} |
cnzr 19078 | class NzRing |
df-nzr 19079 | ⊢ NzRing = {𝑟 ∈ Ring ∣
(1r‘𝑟)
≠ (0g‘𝑟)} |
crlreg 19100 | class RLReg |
cdomn 19101 | class Domn |
cidom 19102 | class IDomn |
cpid 19103 | class PID |
df-rlreg 19104 | ⊢ RLReg = (𝑟 ∈ V ↦ {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) → 𝑦 = (0g‘𝑟))}) |
df-domn 19105 | ⊢ Domn = {𝑟 ∈ NzRing ∣
[(Base‘𝑟) /
𝑏][(0g‘𝑟) / 𝑧]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ((𝑥(.r‘𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧 ∨ 𝑦 = 𝑧))} |
df-idom 19106 | ⊢ IDomn = (CRing ∩ Domn) |
df-pid 19107 | ⊢ PID = (IDomn ∩ LPIR) |
casa 19130 | class AssAlg |
casp 19131 | class AlgSpan |
cascl 19132 | class algSc |
df-assa 19133 | ⊢ AssAlg = {𝑤 ∈ (LMod ∩ Ring) ∣
[(Scalar‘𝑤) /
𝑓](𝑓 ∈ CRing ∧
∀𝑟 ∈
(Base‘𝑓)∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)[(
·𝑠 ‘𝑤) / 𝑠][(.r‘𝑤) / 𝑡](((𝑟𝑠𝑥)𝑡𝑦) = (𝑟𝑠(𝑥𝑡𝑦)) ∧ (𝑥𝑡(𝑟𝑠𝑦)) = (𝑟𝑠(𝑥𝑡𝑦))))} |
df-asp 19134 | ⊢ AlgSpan = (𝑤 ∈ AssAlg ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡
∈ ((SubRing‘𝑤)
∩ (LSubSp‘𝑤))
∣ 𝑠 ⊆ 𝑡})) |
df-ascl 19135 | ⊢ algSc = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)) ↦ (𝑥( ·𝑠
‘𝑤)(1r‘𝑤)))) |
cmps 19172 | class mPwSer |
cmvr 19173 | class mVar |
cmpl 19174 | class mPoly |
cltb 19175 | class
<bag |
copws 19176 | class ordPwSer |
df-psr 19177 | ⊢ mPwSer = (𝑖 ∈ V, 𝑟 ∈ V ↦ ⦋{ℎ ∈ (ℕ0
↑𝑚 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} / 𝑑⦌⦋((Base‘𝑟) ↑𝑚 𝑑) / 𝑏⦌({〈(Base‘ndx), 𝑏〉,
〈(+g‘ndx), ( ∘𝑓
(+g‘𝑟)
↾ (𝑏 × 𝑏))〉,
〈(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑘 ∈ 𝑑 ↦ (𝑟 Σg (𝑥 ∈ {𝑦 ∈ 𝑑 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥)(.r‘𝑟)(𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))〉} ∪
{〈(Scalar‘ndx), 𝑟〉, 〈(
·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑟), 𝑓 ∈ 𝑏 ↦ ((𝑑 × {𝑥}) ∘𝑓
(.r‘𝑟)𝑓))〉, 〈(TopSet‘ndx),
(∏t‘(𝑑
× {(TopOpen‘𝑟)}))〉})) |
df-mvr 19178 | ⊢ mVar = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑥 ∈ 𝑖 ↦ (𝑓 ∈ {ℎ ∈ (ℕ0
↑𝑚 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦 ∈ 𝑖 ↦ if(𝑦 = 𝑥, 1, 0)), (1r‘𝑟), (0g‘𝑟))))) |
df-mpl 19179 | ⊢ mPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ ⦋(𝑖 mPwSer 𝑟) / 𝑤⦌(𝑤 ↾s {𝑓 ∈ (Base‘𝑤) ∣ 𝑓 finSupp (0g‘𝑟)})) |
df-ltbag 19180 | ⊢ <bag = (𝑟 ∈ V, 𝑖 ∈ V ↦ {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ {ℎ ∈ (ℕ0
↑𝑚 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ∧
∃𝑧 ∈ 𝑖 ((𝑥‘𝑧) < (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝑖 (𝑧𝑟𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))))}) |
df-opsr 19181 | ⊢ ordPwSer = (𝑖 ∈ V, 𝑠 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑖 × 𝑖) ↦ ⦋(𝑖 mPwSer 𝑠) / 𝑝⦌(𝑝 sSet 〈(le‘ndx), {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑝) ∧ ([{ℎ ∈ (ℕ0
↑𝑚 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} / 𝑑]∃𝑧 ∈ 𝑑 ((𝑥‘𝑧)(lt‘𝑠)(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝑑 (𝑤(𝑟 <bag 𝑖)𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤))) ∨ 𝑥 = 𝑦))}〉))) |
ces 19325 | class evalSub |
cevl 19326 | class eval |
df-evls 19327 | ⊢ evalSub = (𝑖 ∈ V, 𝑠 ∈ CRing ↦
⦋(Base‘𝑠) / 𝑏⦌(𝑟 ∈ (SubRing‘𝑠) ↦ ⦋(𝑖 mPoly (𝑠 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑s (𝑏 ↑𝑚
𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑𝑚 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑𝑚 𝑖) ↦ (𝑔‘𝑥))))))) |
df-evl 19328 | ⊢ eval = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((𝑖 evalSub 𝑟)‘(Base‘𝑟))) |
cmhp 19358 | class mHomP |
cpsd 19359 | class mPSDer |
cslv 19360 | class selectVars |
cai 19361 | class AlgInd |
df-mhp 19362 | ⊢ mHomP = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ∣ (𝑓 supp (0g‘𝑟)) ⊆ {𝑔 ∈ {ℎ ∈ (ℕ0
↑𝑚 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣
Σ𝑗 ∈
ℕ0 (𝑔‘𝑗) = 𝑛}})) |
df-psd 19363 | ⊢ mPSDer = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑥 ∈ 𝑖 ↦ (𝑓 ∈ (Base‘(𝑖 mPwSer 𝑟)) ↦ (𝑘 ∈ {ℎ ∈ (ℕ0
↑𝑚 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((𝑘‘𝑥) + 1)(.g‘𝑟)(𝑓‘(𝑘 ∘𝑓 + (𝑦 ∈ 𝑖 ↦ if(𝑦 = 𝑥, 1, 0))))))))) |
df-selv 19364 | ⊢ selectVars = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑓 ∈ (𝑖 mPoly 𝑟) ↦ ⦋((𝑖 ∖ 𝑗) mPoly 𝑟) / 𝑠⦌⦋(𝑥 ∈ (Scalar‘𝑠) ↦ (𝑥( ·𝑠
‘𝑠)(1r‘𝑠))) / 𝑐⦌((((𝑖 evalSub 𝑠)‘(𝑐 “s 𝑟))‘(𝑐 ∘ 𝑓))‘(𝑥 ∈ 𝑖 ↦ if(𝑥 ∈ 𝑗, ((𝑗 mVar ((𝑖 ∖ 𝑗) mPoly 𝑟))‘𝑥), (𝑐 ∘ (((𝑖 ∖ 𝑗) mVar 𝑟)‘𝑥)))))))) |
df-algind 19365 | ⊢ AlgInd = (𝑤 ∈ V, 𝑘 ∈ 𝒫 (Base‘𝑤) ↦ {𝑣 ∈ 𝒫 (Base‘𝑤) ∣ Fun ◡(𝑓 ∈ (Base‘(𝑣 mPoly (𝑤 ↾s 𝑘))) ↦ ((((𝑣 evalSub 𝑤)‘𝑘)‘𝑓)‘( I ↾ 𝑣)))}) |
cps1 19366 | class
PwSer1 |
cv1 19367 | class var1 |
cpl1 19368 | class
Poly1 |
cco1 19369 | class coe1 |
ctp1 19370 | class
toPoly1 |
df-psr1 19371 | ⊢ PwSer1 = (𝑟 ∈ V ↦ ((1𝑜
ordPwSer 𝑟)‘∅)) |
df-vr1 19372 | ⊢ var1 = (𝑟 ∈ V ↦ ((1𝑜
mVar 𝑟)‘∅)) |
df-ply1 19373 | ⊢ Poly1 = (𝑟 ∈ V ↦
((PwSer1‘𝑟) ↾s
(Base‘(1𝑜 mPoly 𝑟)))) |
df-coe1 19374 | ⊢ coe1 = (𝑓 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ (𝑓‘(1𝑜
× {𝑛})))) |
df-toply1 19375 | ⊢ toPoly1 = (𝑓 ∈ V ↦ (𝑛 ∈ (ℕ0
↑𝑚 1𝑜) ↦ (𝑓‘(𝑛‘∅)))) |
ces1 19499 | class
evalSub1 |
ce1 19500 | class
eval1 |
df-evls1 19501 | ⊢ evalSub1 = (𝑠 ∈ V, 𝑟 ∈ 𝒫 (Base‘𝑠) ↦
⦋(Base‘𝑠) / 𝑏⦌((𝑥 ∈ (𝑏 ↑𝑚 (𝑏 ↑𝑚
1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1𝑜 ×
{𝑦})))) ∘
((1𝑜 evalSub 𝑠)‘𝑟))) |
df-evl1 19502 | ⊢ eval1 = (𝑟 ∈ V ↦
⦋(Base‘𝑟) / 𝑏⦌((𝑥 ∈ (𝑏 ↑𝑚 (𝑏 ↑𝑚
1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1𝑜 ×
{𝑦})))) ∘
(1𝑜 eval 𝑟))) |
cpsmet 19551 | class PsMet |
cxmt 19552 | class ∞Met |
cme 19553 | class Met |
cbl 19554 | class ball |
cfbas 19555 | class fBas |
cfg 19556 | class filGen |
cmopn 19557 | class MetOpen |
cmetu 19558 | class metUnif |
df-psmet 19559 | ⊢ PsMet = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ*
↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ((𝑦𝑑𝑦) = 0 ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))}) |
df-xmet 19560 | ⊢ ∞Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ*
↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))}) |
df-met 19561 | ⊢ Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ ↑𝑚
(𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) + (𝑤𝑑𝑧)))}) |
df-bl 19562 | ⊢ ball = (𝑑 ∈ V ↦ (𝑥 ∈ dom dom 𝑑, 𝑧 ∈ ℝ* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑧})) |
df-mopn 19563 | ⊢ MetOpen = (𝑑 ∈ ∪ ran
∞Met ↦ (topGen‘ran (ball‘𝑑))) |
df-fbas 19564 | ⊢ fBas = (𝑤 ∈ V ↦ {𝑥 ∈ 𝒫 𝒫 𝑤 ∣ (𝑥 ≠ ∅ ∧ ∅ ∉ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑥 ∩ 𝒫 (𝑦 ∩ 𝑧)) ≠ ∅)}) |
df-fg 19565 | ⊢ filGen = (𝑤 ∈ V, 𝑥 ∈ (fBas‘𝑤) ↦ {𝑦 ∈ 𝒫 𝑤 ∣ (𝑥 ∩ 𝒫 𝑦) ≠ ∅}) |
df-metu 19566 | ⊢ metUnif = (𝑑 ∈ ∪ ran
PsMet ↦ ((dom dom 𝑑
× dom dom 𝑑)filGenran
(𝑎 ∈
ℝ+ ↦ (◡𝑑 “ (0[,)𝑎))))) |
ccnfld 19567 | class
ℂfld |
df-cnfld 19568 | ⊢ ℂfld =
(({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), +
〉, 〈(.r‘ndx), · 〉} ∪
{〈(*𝑟‘ndx), ∗〉}) ∪
({〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉,
〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ −
)〉} ∪ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ −
))〉})) |
zring 19637 | class
ℤring |
df-zring 19638 | ⊢ ℤring =
(ℂfld ↾s ℤ) |
czrh 19667 | class ℤRHom |
czlm 19668 | class ℤMod |
cchr 19669 | class chr |
czn 19670 | class
ℤ/nℤ |
df-zrh 19671 | ⊢ ℤRHom = (𝑟 ∈ V ↦ ∪ (ℤring RingHom 𝑟)) |
df-zlm 19672 | ⊢ ℤMod = (𝑔 ∈ V ↦ ((𝑔 sSet 〈(Scalar‘ndx),
ℤring〉) sSet 〈( ·𝑠
‘ndx), (.g‘𝑔)〉)) |
df-chr 19673 | ⊢ chr = (𝑔 ∈ V ↦ ((od‘𝑔)‘(1r‘𝑔))) |
df-zn 19674 | ⊢ ℤ/nℤ = (𝑛 ∈ ℕ0
↦ ⦋ℤring / 𝑧⦌⦋(𝑧 /s (𝑧 ~QG
((RSpan‘𝑧)‘{𝑛}))) / 𝑠⦌(𝑠 sSet 〈(le‘ndx),
⦋((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) / 𝑓⦌((𝑓 ∘ ≤ ) ∘ ◡𝑓)〉)) |
crefld 19769 | class
ℝfld |
df-refld 19770 | ⊢ ℝfld =
(ℂfld ↾s ℝ) |
cphl 19788 | class PreHil |
cipf 19789 | class
·if |
df-phl 19790 | ⊢ PreHil = {𝑔 ∈ LVec ∣
[(Base‘𝑔) /
𝑣][(·𝑖‘𝑔) / ℎ][(Scalar‘𝑔) / 𝑓](𝑓
∈ *-Ring ∧ ∀𝑥 ∈ 𝑣 ((𝑦 ∈ 𝑣
↦ (𝑦ℎ𝑥)) ∈
(𝑔 LMHom (ringLMod‘𝑓)) ∧ ((𝑥ℎ𝑥) = (0g‘𝑓) → 𝑥 =
(0g‘𝑔)) ∧ ∀𝑦 ∈ 𝑣 ((*𝑟‘𝑓)‘(𝑥ℎ𝑦)) = (𝑦ℎ𝑥)))} |
df-ipf 19791 | ⊢ ·if = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦))) |
cocv 19823 | class ocv |
ccss 19824 | class CSubSp |
cthl 19825 | class toHL |
df-ocv 19826 | ⊢ ocv = (ℎ ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘ℎ) ↦ {𝑥 ∈ (Base‘ℎ) ∣ ∀𝑦 ∈ 𝑠 (𝑥(·𝑖‘ℎ)𝑦) = (0g‘(Scalar‘ℎ))})) |
df-css 19827 | ⊢ CSubSp = (ℎ ∈ V ↦ {𝑠 ∣ 𝑠 = ((ocv‘ℎ)‘((ocv‘ℎ)‘𝑠))}) |
df-thl 19828 | ⊢ toHL = (ℎ ∈ V ↦
((toInc‘(CSubSp‘ℎ)) sSet 〈(oc‘ndx), (ocv‘ℎ)〉)) |
cpj 19863 | class proj |
chs 19864 | class Hil |
cobs 19865 | class OBasis |
df-pj 19866 | ⊢ proj = (ℎ ∈ V ↦ ((𝑥 ∈ (LSubSp‘ℎ) ↦ (𝑥(proj1‘ℎ)((ocv‘ℎ)‘𝑥))) ∩ (V × ((Base‘ℎ) ↑𝑚
(Base‘ℎ))))) |
df-hil 19867 | ⊢ Hil = {ℎ ∈ PreHil ∣ dom (proj‘ℎ) = (CSubSp‘ℎ)} |
df-obs 19868 | ⊢ OBasis = (ℎ ∈ PreHil ↦ {𝑏 ∈ 𝒫 (Base‘ℎ) ∣ (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(·𝑖‘ℎ)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘ℎ)),
(0g‘(Scalar‘ℎ))) ∧ ((ocv‘ℎ)‘𝑏) = {(0g‘ℎ)})}) |
cdsmm 19894 | class
⊕m |
df-dsmm 19895 | ⊢ ⊕m = (𝑠 ∈ V, 𝑟 ∈ V ↦ ((𝑠Xs𝑟) ↾s {𝑓 ∈ X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ∣ {𝑥 ∈ dom 𝑟 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑟‘𝑥))} ∈ Fin})) |
cfrlm 19909 | class freeLMod |
df-frlm 19910 | ⊢ freeLMod = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟 ⊕m (𝑖 × {(ringLMod‘𝑟)}))) |
cuvc 19940 | class unitVec |
df-uvc 19941 | ⊢ unitVec = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ (𝑘 ∈ 𝑖 ↦ if(𝑘 = 𝑗, (1r‘𝑟), (0g‘𝑟))))) |
clindf 19962 | class LIndF |
clinds 19963 | class LIndS |
df-lindf 19964 | ⊢ LIndF = {〈𝑓, 𝑤〉 ∣ (𝑓:dom 𝑓⟶(Base‘𝑤) ∧ [(Scalar‘𝑤) / 𝑠]∀𝑥 ∈ dom 𝑓∀𝑘 ∈ ((Base‘𝑠) ∖ {(0g‘𝑠)}) ¬ (𝑘( ·𝑠
‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))))} |
df-linds 19965 | ⊢ LIndS = (𝑤 ∈ V ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ( I ↾ 𝑠) LIndF 𝑤}) |
cmmul 20008 | class maMul |
df-mamu 20009 | ⊢ maMul = (𝑟 ∈ V, 𝑜 ∈ V ↦
⦋(1st ‘(1st ‘𝑜)) / 𝑚⦌⦋(2nd
‘(1st ‘𝑜)) / 𝑛⦌⦋(2nd
‘𝑜) / 𝑝⦌(𝑥 ∈ ((Base‘𝑟) ↑𝑚
(𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑𝑚 (𝑛 × 𝑝)) ↦ (𝑖 ∈ 𝑚, 𝑘 ∈ 𝑝 ↦ (𝑟 Σg (𝑗 ∈ 𝑛 ↦ ((𝑖𝑥𝑗)(.r‘𝑟)(𝑗𝑦𝑘))))))) |
cmat 20032 | class Mat |
df-mat 20033 | ⊢ Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet 〈(.r‘ndx),
(𝑟 maMul 〈𝑛, 𝑛, 𝑛〉)〉)) |
cdmat 20113 | class DMat |
cscmat 20114 | class ScMat |
df-dmat 20115 | ⊢ DMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = (0g‘𝑟))}) |
df-scmat 20116 | ⊢ ScMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ⦋(𝑛 Mat 𝑟) / 𝑎⦌{𝑚 ∈ (Base‘𝑎) ∣ ∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠
‘𝑎)(1r‘𝑎))}) |
cmvmul 20165 | class maVecMul |
df-mvmul 20166 | ⊢ maVecMul = (𝑟 ∈ V, 𝑜 ∈ V ↦
⦋(1st ‘𝑜) / 𝑚⦌⦋(2nd
‘𝑜) / 𝑛⦌(𝑥 ∈ ((Base‘𝑟) ↑𝑚
(𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑𝑚 𝑛) ↦ (𝑖 ∈ 𝑚 ↦ (𝑟 Σg (𝑗 ∈ 𝑛 ↦ ((𝑖𝑥𝑗)(.r‘𝑟)(𝑦‘𝑗))))))) |
cmarrep 20181 | class matRRep |
cmatrepV 20182 | class matRepV |
df-marrep 20183 | ⊢ matRRep = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)), 𝑠 ∈ (Base‘𝑟) ↦ (𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, (0g‘𝑟)), (𝑖𝑚𝑗)))))) |
df-marepv 20184 | ⊢ matRepV = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)), 𝑣 ∈ ((Base‘𝑟) ↑𝑚 𝑛) ↦ (𝑘 ∈ 𝑛 ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑗 = 𝑘, (𝑣‘𝑖), (𝑖𝑚𝑗)))))) |
csubma 20201 | class subMat |
df-subma 20202 | ⊢ subMat = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗))))) |
cmdat 20209 | class maDet |
df-mdet 20210 | ⊢ maDet = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑟 Σg (𝑝 ∈
(Base‘(SymGrp‘𝑛)) ↦ ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r‘𝑟)((mulGrp‘𝑟) Σg (𝑥 ∈ 𝑛 ↦ ((𝑝‘𝑥)𝑚𝑥)))))))) |
cmadu 20257 | class maAdju |
cminmar1 20258 | class minMatR1 |
df-madu 20259 | ⊢ maAdju = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ ((𝑛 maDet 𝑟)‘(𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑟), (0g‘𝑟)), (𝑘𝑚𝑙))))))) |
df-minmar1 20260 | ⊢ minMatR1 = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, (1r‘𝑟), (0g‘𝑟)), (𝑖𝑚𝑗)))))) |
ccpmat 20327 | class ConstPolyMat |
cmat2pmat 20328 | class matToPolyMat |
ccpmat2mat 20329 | class cPolyMatToMat |
df-cpmat 20330 | ⊢ ConstPolyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ {𝑚 ∈ (Base‘(𝑛 Mat (Poly1‘𝑟))) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 ∀𝑘 ∈ ℕ
((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g‘𝑟)}) |
df-mat2pmat 20331 | ⊢ matToPolyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦
((algSc‘(Poly1‘𝑟))‘(𝑥𝑚𝑦))))) |
df-cpmat2mat 20332 | ⊢ cPolyMatToMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))) |
cdecpmat 20386 | class decompPMat |
df-decpmat 20387 | ⊢ decompPMat = (𝑚 ∈ V, 𝑘 ∈ ℕ0 ↦ (𝑖 ∈ dom dom 𝑚, 𝑗 ∈ dom dom 𝑚 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘))) |
cpm2mp 20416 | class pMatToMatPoly |
df-pm2mp 20417 | ⊢ pMatToMatPoly = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat (Poly1‘𝑟))) ↦
⦋(𝑛 Mat 𝑟) / 𝑎⦌⦋(Poly1‘𝑎) / 𝑞⦌(𝑞 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠 ‘𝑞)(𝑘(.g‘(mulGrp‘𝑞))(var1‘𝑎))))))) |
cchpmat 20450 | class CharPlyMat |
df-chpmat 20451 | ⊢ CharPlyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ ((𝑛 maDet (Poly1‘𝑟))‘(((var1‘𝑟)(
·𝑠 ‘(𝑛 Mat (Poly1‘𝑟)))(1r‘(𝑛 Mat
(Poly1‘𝑟))))(-g‘(𝑛 Mat (Poly1‘𝑟)))((𝑛 matToPolyMat 𝑟)‘𝑚))))) |
ctop 20517 | class Top |
ctopon 20518 | class TopOn |
ctps 20519 | class TopSp |
ctb 20520 | class TopBases |
df-top 20521 | ⊢ Top = {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥)} |
df-bases 20522 | ⊢ TopBases = {𝑥 ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ⊆ ∪ (𝑥 ∩ 𝒫 (𝑦 ∩ 𝑧))} |
df-topon 20523 | ⊢ TopOn = (𝑏 ∈ V ↦ {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗}) |
df-topsp 20524 | ⊢ TopSp = {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))} |
ccld 20630 | class Clsd |
cnt 20631 | class int |
ccl 20632 | class cls |
df-cld 20633 | ⊢ Clsd = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 ∪ 𝑗
∣ (∪ 𝑗 ∖ 𝑥) ∈ 𝑗}) |
df-ntr 20634 | ⊢ int = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗
↦ ∪ (𝑗 ∩ 𝒫 𝑥))) |
df-cls 20635 | ⊢ cls = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗
↦ ∩ {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥 ⊆ 𝑦})) |
cnei 20711 | class nei |
df-nei 20712 | ⊢ nei = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗
↦ {𝑦 ∈ 𝒫
∪ 𝑗 ∣ ∃𝑔 ∈ 𝑗 (𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦)})) |
clp 20748 | class limPt |
cperf 20749 | class Perf |
df-lp 20750 | ⊢ limPt = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗
↦ {𝑦 ∣ 𝑦 ∈ ((cls‘𝑗)‘(𝑥 ∖ {𝑦}))})) |
df-perf 20751 | ⊢ Perf = {𝑗 ∈ Top ∣ ((limPt‘𝑗)‘∪ 𝑗) =
∪ 𝑗} |
ccn 20838 | class Cn |
ccnp 20839 | class CnP |
clm 20840 | class
⇝𝑡 |
df-cn 20841 | ⊢ Cn = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (∪ 𝑘 ↑𝑚
∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 (◡𝑓 “ 𝑦) ∈ 𝑗}) |
df-cnp 20842 | ⊢ CnP = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ (𝑥 ∈ ∪ 𝑗 ↦ {𝑓 ∈ (∪ 𝑘 ↑𝑚
∪ 𝑗) ∣ ∀𝑦 ∈ 𝑘 ((𝑓‘𝑥) ∈ 𝑦 → ∃𝑔 ∈ 𝑗 (𝑥 ∈ 𝑔 ∧ (𝑓 “ 𝑔) ⊆ 𝑦))})) |
df-lm 20843 | ⊢ ⇝𝑡 =
(𝑗 ∈ Top ↦
{〈𝑓, 𝑥〉 ∣ (𝑓 ∈ (∪ 𝑗
↑pm ℂ) ∧ 𝑥 ∈ ∪ 𝑗 ∧ ∀𝑢 ∈ 𝑗 (𝑥 ∈ 𝑢 → ∃𝑦 ∈ ran ℤ≥(𝑓 ↾ 𝑦):𝑦⟶𝑢))}) |
ct0 20920 | class Kol2 |
ct1 20921 | class Fre |
cha 20922 | class Haus |
creg 20923 | class Reg |
cnrm 20924 | class Nrm |
ccnrm 20925 | class CNrm |
cpnrm 20926 | class PNrm |
df-t0 20927 | ⊢ Kol2 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗(∀𝑜 ∈ 𝑗 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)} |
df-t1 20928 | ⊢ Fre = {𝑥 ∈ Top ∣ ∀𝑎 ∈ ∪ 𝑥{𝑎} ∈ (Clsd‘𝑥)} |
df-haus 20929 | ⊢ Haus = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗(𝑥 ≠ 𝑦 → ∃𝑛 ∈ 𝑗 ∃𝑚 ∈ 𝑗 (𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))} |
df-reg 20930 | ⊢ Reg = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝑗 (𝑦 ∈ 𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)} |
df-nrm 20931 | ⊢ Nrm = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ ((Clsd‘𝑗) ∩ 𝒫 𝑥)∃𝑧 ∈ 𝑗 (𝑦 ⊆ 𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)} |
df-cnrm 20932 | ⊢ CNrm = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝒫 ∪ 𝑗(𝑗 ↾t 𝑥) ∈ Nrm} |
df-pnrm 20933 | ⊢ PNrm = {𝑗 ∈ Nrm ∣ (Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗 ↑𝑚 ℕ) ↦
∩ ran 𝑓)} |
ccmp 20999 | class Comp |
df-cmp 21000 | ⊢ Comp = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥(∪
𝑥 = ∪ 𝑦
→ ∃𝑧 ∈
(𝒫 𝑦 ∩
Fin)∪ 𝑥 = ∪ 𝑧)} |
ccon 21024 | class Con |
df-con 21025 | ⊢ Con = {𝑗 ∈ Top ∣ (𝑗 ∩ (Clsd‘𝑗)) = {∅, ∪
𝑗}} |
c1stc 21050 | class
1st𝜔 |
c2ndc 21051 | class
2nd𝜔 |
df-1stc 21052 | ⊢ 1st𝜔 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∃𝑦 ∈ 𝒫 𝑗(𝑦 ≼ ω ∧ ∀𝑧 ∈ 𝑗 (𝑥 ∈ 𝑧 → 𝑥 ∈ ∪ (𝑦 ∩ 𝒫 𝑧)))} |
df-2ndc 21053 | ⊢ 2nd𝜔 = {𝑗 ∣ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝑗)} |
clly 21077 | class Locally 𝐴 |
cnlly 21078 | class 𝑛-Locally 𝐴 |
df-lly 21079 | ⊢ Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝑗 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝑗 ↾t 𝑢) ∈ 𝐴)} |
df-nlly 21080 | ⊢ 𝑛-Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝑗 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗 ↾t 𝑢) ∈ 𝐴} |
cref 21115 | class Ref |
cptfin 21116 | class PtFin |
clocfin 21117 | class LocFin |
df-ref 21118 | ⊢ Ref = {〈𝑥, 𝑦〉 ∣ (∪
𝑦 = ∪ 𝑥
∧ ∀𝑧 ∈
𝑥 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)} |
df-ptfin 21119 | ⊢ PtFin = {𝑥 ∣ ∀𝑦 ∈ ∪ 𝑥{𝑧 ∈ 𝑥 ∣ 𝑦 ∈ 𝑧} ∈ Fin} |
df-locfin 21120 | ⊢ LocFin = (𝑥 ∈ Top ↦ {𝑦 ∣ (∪ 𝑥 = ∪
𝑦 ∧ ∀𝑝 ∈ ∪ 𝑥∃𝑛 ∈ 𝑥 (𝑝 ∈ 𝑛 ∧ {𝑠 ∈ 𝑦 ∣ (𝑠 ∩ 𝑛) ≠ ∅} ∈
Fin))}) |
ckgen 21146 | class 𝑘Gen |
df-kgen 21147 | ⊢ 𝑘Gen = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 ∪ 𝑗
∣ ∀𝑘 ∈
𝒫 ∪ 𝑗((𝑗 ↾t 𝑘) ∈ Comp → (𝑥 ∩ 𝑘) ∈ (𝑗 ↾t 𝑘))}) |
ctx 21173 | class
×t |
cxko 21174 | class
^ko |
df-tx 21175 | ⊢ ×t = (𝑟 ∈ V, 𝑠 ∈ V ↦ (topGen‘ran (𝑥 ∈ 𝑟, 𝑦 ∈ 𝑠 ↦ (𝑥 × 𝑦)))) |
df-xko 21176 | ⊢ ^ko = (𝑠 ∈ Top, 𝑟 ∈ Top ↦
(topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑟
∣ (𝑟
↾t 𝑥)
∈ Comp}, 𝑣 ∈
𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})))) |
ckq 21306 | class KQ |
df-kq 21307 | ⊢ KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦}))) |
chmeo 21366 | class Homeo |
chmph 21367 | class ≃ |
df-hmeo 21368 | ⊢ Homeo = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (𝑗 Cn 𝑘) ∣ ◡𝑓 ∈ (𝑘 Cn 𝑗)}) |
df-hmph 21369 | ⊢ ≃ = (◡Homeo “ (V ∖
1𝑜)) |
cfil 21459 | class Fil |
df-fil 21460 | ⊢ Fil = (𝑧 ∈ V ↦ {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓)}) |
cufil 21513 | class UFil |
cufl 21514 | class UFL |
df-ufil 21515 | ⊢ UFil = (𝑔 ∈ V ↦ {𝑓 ∈ (Fil‘𝑔) ∣ ∀𝑥 ∈ 𝒫 𝑔(𝑥 ∈ 𝑓 ∨ (𝑔 ∖ 𝑥) ∈ 𝑓)}) |
df-ufl 21516 | ⊢ UFL = {𝑥 ∣ ∀𝑓 ∈ (Fil‘𝑥)∃𝑔 ∈ (UFil‘𝑥)𝑓 ⊆ 𝑔} |
cfm 21547 | class FilMap |
cflim 21548 | class fLim |
cflf 21549 | class fLimf |
cfcls 21550 | class fClus |
cfcf 21551 | class fClusf |
df-fm 21552 | ⊢ FilMap = (𝑥 ∈ V, 𝑓 ∈ V ↦ (𝑦 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡))))) |
df-flim 21553 | ⊢ fLim = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil
↦ {𝑥 ∈ ∪ 𝑗
∣ (((nei‘𝑗)‘{𝑥}) ⊆ 𝑓 ∧ 𝑓 ⊆ 𝒫 ∪ 𝑗)}) |
df-flf 21554 | ⊢ fLimf = (𝑥 ∈ Top, 𝑦 ∈ ∪ ran Fil
↦ (𝑓 ∈ (∪ 𝑥
↑𝑚 ∪ 𝑦) ↦ (𝑥 fLim ((∪ 𝑥 FilMap 𝑓)‘𝑦)))) |
df-fcls 21555 | ⊢ fClus = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil
↦ if(∪ 𝑗 = ∪ 𝑓, ∩ 𝑥 ∈ 𝑓 ((cls‘𝑗)‘𝑥), ∅)) |
df-fcf 21556 | ⊢ fClusf = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil
↦ (𝑔 ∈ (∪ 𝑗
↑𝑚 ∪ 𝑓) ↦ (𝑗 fClus ((∪ 𝑗 FilMap 𝑔)‘𝑓)))) |
ccnext 21673 | class CnExt |
df-cnext 21674 | ⊢ CnExt = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ (𝑓 ∈ (∪ 𝑘 ↑pm
∪ 𝑗) ↦ ∪
𝑥 ∈ ((cls‘𝑗)‘dom 𝑓)({𝑥} × ((𝑘 fLimf (((nei‘𝑗)‘{𝑥}) ↾t dom 𝑓))‘𝑓)))) |
ctmd 21684 | class TopMnd |
ctgp 21685 | class TopGrp |
df-tmd 21686 | ⊢ TopMnd = {𝑓 ∈ (Mnd ∩ TopSp) ∣
[(TopOpen‘𝑓) /
𝑗](+𝑓‘𝑓) ∈ ((𝑗 ×t 𝑗) Cn 𝑗)} |
df-tgp 21687 | ⊢ TopGrp = {𝑓 ∈ (Grp ∩ TopMnd) ∣
[(TopOpen‘𝑓) /
𝑗](invg‘𝑓) ∈ (𝑗 Cn 𝑗)} |
ctsu 21739 | class tsums |
df-tsms 21740 | ⊢ tsums = (𝑤 ∈ V, 𝑓 ∈ V ↦ ⦋(𝒫
dom 𝑓 ∩ Fin) / 𝑠⦌(((TopOpen‘𝑤) fLimf (𝑠filGenran (𝑧 ∈ 𝑠 ↦ {𝑦 ∈ 𝑠 ∣ 𝑧 ⊆ 𝑦})))‘(𝑦 ∈ 𝑠 ↦ (𝑤 Σg (𝑓 ↾ 𝑦))))) |
ctrg 21769 | class TopRing |
ctdrg 21770 | class TopDRing |
ctlm 21771 | class TopMod |
ctvc 21772 | class TopVec |
df-trg 21773 | ⊢ TopRing = {𝑟 ∈ (TopGrp ∩ Ring) ∣
(mulGrp‘𝑟) ∈
TopMnd} |
df-tdrg 21774 | ⊢ TopDRing = {𝑟 ∈ (TopRing ∩ DivRing) ∣
((mulGrp‘𝑟)
↾s (Unit‘𝑟)) ∈ TopGrp} |
df-tlm 21775 | ⊢ TopMod = {𝑤 ∈ (TopMnd ∩ LMod) ∣
((Scalar‘𝑤) ∈
TopRing ∧ ( ·sf ‘𝑤) ∈ (((TopOpen‘(Scalar‘𝑤)) ×t
(TopOpen‘𝑤)) Cn
(TopOpen‘𝑤)))} |
df-tvc 21776 | ⊢ TopVec = {𝑤 ∈ TopMod ∣ (Scalar‘𝑤) ∈
TopDRing} |
cust 21813 | class UnifOn |
df-ust 21814 | ⊢ UnifOn = (𝑥 ∈ V ↦ {𝑢 ∣ (𝑢 ⊆ 𝒫 (𝑥 × 𝑥) ∧ (𝑥 × 𝑥) ∈ 𝑢 ∧ ∀𝑣 ∈ 𝑢 (∀𝑤 ∈ 𝒫 (𝑥 × 𝑥)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑢) ∧ ∀𝑤 ∈ 𝑢 (𝑣 ∩ 𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑥) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑢 ∧ ∃𝑤 ∈ 𝑢 (𝑤 ∘ 𝑤) ⊆ 𝑣)))}) |
cutop 21844 | class unifTop |
df-utop 21845 | ⊢ unifTop = (𝑢 ∈ ∪ ran
UnifOn ↦ {𝑎 ∈
𝒫 dom ∪ 𝑢 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑢 (𝑣 “ {𝑥}) ⊆ 𝑎}) |
cuss 21867 | class UnifSt |
cusp 21868 | class UnifSp |
ctus 21869 | class toUnifSp |
df-uss 21870 | ⊢ UnifSt = (𝑓 ∈ V ↦ ((UnifSet‘𝑓) ↾t
((Base‘𝑓) ×
(Base‘𝑓)))) |
df-usp 21871 | ⊢ UnifSp = {𝑓 ∣ ((UnifSt‘𝑓) ∈ (UnifOn‘(Base‘𝑓)) ∧ (TopOpen‘𝑓) =
(unifTop‘(UnifSt‘𝑓)))} |
df-tus 21872 | ⊢ toUnifSp = (𝑢 ∈ ∪ ran
UnifOn ↦ ({〈(Base‘ndx), dom ∪ 𝑢〉,
〈(UnifSet‘ndx), 𝑢〉} sSet 〈(TopSet‘ndx),
(unifTop‘𝑢)〉)) |
cucn 21889 | class Cnu |
df-ucn 21890 | ⊢ Cnu = (𝑢 ∈ ∪ ran
UnifOn, 𝑣 ∈ ∪ ran UnifOn ↦ {𝑓 ∈ (dom ∪
𝑣
↑𝑚 dom ∪ 𝑢) ∣ ∀𝑠 ∈ 𝑣 ∃𝑟 ∈ 𝑢 ∀𝑥 ∈ dom ∪
𝑢∀𝑦 ∈ dom ∪
𝑢(𝑥𝑟𝑦 → (𝑓‘𝑥)𝑠(𝑓‘𝑦))}) |
ccfilu 21900 | class
CauFilu |
df-cfilu 21901 | ⊢ CauFilu = (𝑢 ∈ ∪ ran
UnifOn ↦ {𝑓 ∈
(fBas‘dom ∪ 𝑢) ∣ ∀𝑣 ∈ 𝑢 ∃𝑎 ∈ 𝑓 (𝑎 × 𝑎) ⊆ 𝑣}) |
ccusp 21911 | class CUnifSp |
df-cusp 21912 | ⊢ CUnifSp = {𝑤 ∈ UnifSp ∣ ∀𝑐 ∈
(Fil‘(Base‘𝑤))(𝑐 ∈
(CauFilu‘(UnifSt‘𝑤)) → ((TopOpen‘𝑤) fLim 𝑐) ≠ ∅)} |
cxme 21932 | class ∞MetSp |
cmt 21933 | class MetSp |
ctmt 21934 | class toMetSp |
df-xms 21935 | ⊢ ∞MetSp = {𝑓 ∈ TopSp ∣ (TopOpen‘𝑓) =
(MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))} |
df-ms 21936 | ⊢ MetSp = {𝑓 ∈ ∞MetSp ∣
((dist‘𝑓) ↾
((Base‘𝑓) ×
(Base‘𝑓))) ∈
(Met‘(Base‘𝑓))} |
df-tms 21937 | ⊢ toMetSp = (𝑑 ∈ ∪ ran
∞Met ↦ ({〈(Base‘ndx), dom dom 𝑑〉, 〈(dist‘ndx), 𝑑〉} sSet
〈(TopSet‘ndx), (MetOpen‘𝑑)〉)) |
cnm 22191 | class norm |
cngp 22192 | class NrmGrp |
ctng 22193 | class toNrmGrp |
cnrg 22194 | class NrmRing |
cnlm 22195 | class NrmMod |
cnvc 22196 | class NrmVec |
df-nm 22197 | ⊢ norm = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤) ↦ (𝑥(dist‘𝑤)(0g‘𝑤)))) |
df-ngp 22198 | ⊢ NrmGrp = {𝑔 ∈ (Grp ∩ MetSp) ∣
((norm‘𝑔) ∘
(-g‘𝑔))
⊆ (dist‘𝑔)} |
df-tng 22199 | ⊢ toNrmGrp = (𝑔 ∈ V, 𝑓 ∈ V ↦ ((𝑔 sSet 〈(dist‘ndx), (𝑓 ∘
(-g‘𝑔))〉) sSet 〈(TopSet‘ndx),
(MetOpen‘(𝑓 ∘
(-g‘𝑔)))〉)) |
df-nrg 22200 | ⊢ NrmRing = {𝑤 ∈ NrmGrp ∣ (norm‘𝑤) ∈ (AbsVal‘𝑤)} |
df-nlm 22201 | ⊢ NrmMod = {𝑤 ∈ (NrmGrp ∩ LMod) ∣
[(Scalar‘𝑤) /
𝑓](𝑓 ∈ NrmRing ∧
∀𝑥 ∈
(Base‘𝑓)∀𝑦 ∈ (Base‘𝑤)((norm‘𝑤)‘(𝑥( ·𝑠
‘𝑤)𝑦)) = (((norm‘𝑓)‘𝑥) · ((norm‘𝑤)‘𝑦)))} |
df-nvc 22202 | ⊢ NrmVec = (NrmMod ∩ LVec) |
cnmo 22319 | class normOp |
cnghm 22320 | class NGHom |
cnmhm 22321 | class NMHom |
df-nmo 22322 | ⊢ normOp = (𝑠 ∈ NrmGrp, 𝑡 ∈ NrmGrp ↦ (𝑓 ∈ (𝑠 GrpHom 𝑡) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ (Base‘𝑠)((norm‘𝑡)‘(𝑓‘𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥))}, ℝ*, <
))) |
df-nghm 22323 | ⊢ NGHom = (𝑠 ∈ NrmGrp, 𝑡 ∈ NrmGrp ↦ (◡(𝑠 normOp 𝑡) “ ℝ)) |
df-nmhm 22324 | ⊢ NMHom = (𝑠 ∈ NrmMod, 𝑡 ∈ NrmMod ↦ ((𝑠 LMHom 𝑡) ∩ (𝑠 NGHom 𝑡))) |
cii 22486 | class II |
ccncf 22487 | class –cn→ |
df-ii 22488 | ⊢ II = (MetOpen‘((abs
∘ − ) ↾ ((0[,]1) × (0[,]1)))) |
df-cncf 22489 | ⊢ –cn→ = (𝑎 ∈ 𝒫 ℂ, 𝑏 ∈ 𝒫 ℂ ↦ {𝑓 ∈ (𝑏 ↑𝑚 𝑎) ∣ ∀𝑥 ∈ 𝑎 ∀𝑒 ∈ ℝ+ ∃𝑑 ∈ ℝ+
∀𝑦 ∈ 𝑎 ((abs‘(𝑥 − 𝑦)) < 𝑑 → (abs‘((𝑓‘𝑥) − (𝑓‘𝑦))) < 𝑒)}) |
chtpy 22574 | class Htpy |
cphtpy 22575 | class PHtpy |
cphtpc 22576 | class
≃ph |
df-htpy 22577 | ⊢ Htpy = (𝑥 ∈ Top, 𝑦 ∈ Top ↦ (𝑓 ∈ (𝑥 Cn 𝑦), 𝑔 ∈ (𝑥 Cn 𝑦) ↦ {ℎ ∈ ((𝑥 ×t II) Cn 𝑦) ∣ ∀𝑠 ∈ ∪ 𝑥((𝑠ℎ0) = (𝑓‘𝑠) ∧ (𝑠ℎ1) = (𝑔‘𝑠))})) |
df-phtpy 22578 | ⊢ PHtpy = (𝑥 ∈ Top ↦ (𝑓 ∈ (II Cn 𝑥), 𝑔 ∈ (II Cn 𝑥) ↦ {ℎ ∈ (𝑓(II Htpy 𝑥)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0ℎ𝑠) = (𝑓‘0) ∧ (1ℎ𝑠) = (𝑓‘1))})) |
df-phtpc 22599 | ⊢ ≃ph = (𝑥 ∈ Top ↦ {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ (II Cn 𝑥) ∧ (𝑓(PHtpy‘𝑥)𝑔) ≠ ∅)}) |
cpco 22608 | class
*𝑝 |
comi 22609 | class
Ω1 |
comn 22610 | class
Ω𝑛 |
cpi1 22611 | class
π1 |
cpin 22612 | class
πn |
df-pco 22613 | ⊢ *𝑝 = (𝑗 ∈ Top ↦ (𝑓 ∈ (II Cn 𝑗), 𝑔 ∈ (II Cn 𝑗) ↦ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝑓‘(2 · 𝑥)), (𝑔‘((2 · 𝑥) − 1)))))) |
df-om1 22614 | ⊢ Ω1 = (𝑗 ∈ Top, 𝑦 ∈ ∪ 𝑗 ↦
{〈(Base‘ndx), {𝑓
∈ (II Cn 𝑗) ∣
((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)}〉, 〈(+g‘ndx),
(*𝑝‘𝑗)〉, 〈(TopSet‘ndx), (𝑗 ^ko
II)〉}) |
df-omn 22615 | ⊢ Ω𝑛 = (𝑗 ∈ Top, 𝑦 ∈ ∪ 𝑗 ↦ seq0(((𝑥 ∈ V, 𝑝 ∈ V ↦
〈((TopOpen‘(1st ‘𝑥)) Ω1 (2nd
‘𝑥)), ((0[,]1)
× {(2nd ‘𝑥)})〉) ∘ 1st ),
〈{〈(Base‘ndx), ∪ 𝑗〉, 〈(TopSet‘ndx), 𝑗〉}, 𝑦〉)) |
df-pi1 22616 | ⊢ π1 = (𝑗 ∈ Top, 𝑦 ∈ ∪ 𝑗 ↦ ((𝑗 Ω1 𝑦) /s (
≃ph‘𝑗))) |
df-pin 22617 | ⊢ πn = (𝑗 ∈ Top, 𝑝 ∈ ∪ 𝑗 ↦ (𝑛 ∈ ℕ0 ↦
((1st ‘((𝑗
Ω𝑛 𝑝)‘𝑛)) /s if(𝑛 = 0, {〈𝑥, 𝑦〉 ∣ ∃𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)}, (
≃ph‘(TopOpen‘(1st ‘((𝑗 Ω𝑛
𝑝)‘(𝑛 −
1))))))))) |
cclm 22670 | class ℂMod |
df-clm 22671 | ⊢ ℂMod = {𝑤 ∈ LMod ∣
[(Scalar‘𝑤) /
𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s
𝑘) ∧ 𝑘 ∈
(SubRing‘ℂfld))} |
ccvs 22731 | class ℂVec |
df-cvs 22732 | ⊢ ℂVec = (ℂMod ∩
LVec) |
ccph 22774 | class ℂPreHil |
ctch 22775 | class toℂHil |
df-cph 22776 | ⊢ ℂPreHil = {𝑤 ∈ (PreHil ∩ NrmMod) ∣
[(Scalar‘𝑤) /
𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s
𝑘) ∧ (√ “
(𝑘 ∩ (0[,)+∞)))
⊆ 𝑘 ∧
(norm‘𝑤) = (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥))))} |
df-tch 22777 | ⊢ toℂHil = (𝑤 ∈ V ↦ (𝑤 toNrmGrp (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥))))) |
ccfil 22858 | class CauFil |
cca 22859 | class Cau |
cms 22860 | class CMet |
df-cfil 22861 | ⊢ CauFil = (𝑑 ∈ ∪ ran
∞Met ↦ {𝑓
∈ (Fil‘dom dom 𝑑) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝑓 (𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)}) |
df-cau 22862 | ⊢ Cau = (𝑑 ∈ ∪ ran
∞Met ↦ {𝑓
∈ (dom dom 𝑑
↑pm ℂ) ∣ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ (𝑓 ↾
(ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶((𝑓‘𝑗)(ball‘𝑑)𝑥)}) |
df-cmet 22863 | ⊢ CMet = (𝑥 ∈ V ↦ {𝑑 ∈ (Met‘𝑥) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅}) |
ccms 22937 | class CMetSp |
cbn 22938 | class Ban |
chl 22939 | class ℂHil |
df-cms 22940 | ⊢ CMetSp = {𝑤 ∈ MetSp ∣
[(Base‘𝑤) /
𝑏]((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏)} |
df-bn 22941 | ⊢ Ban = {𝑤 ∈ (NrmVec ∩ CMetSp) ∣
(Scalar‘𝑤) ∈
CMetSp} |
df-hl 22942 | ⊢ ℂHil = (Ban ∩
ℂPreHil) |
crrx 22979 | class ℝ^ |
cehl 22980 | class
𝔼hil |
df-rrx 22981 | ⊢ ℝ^ = (𝑖 ∈ V ↦
(toℂHil‘(ℝfld freeLMod 𝑖))) |
df-ehl 22982 | ⊢ 𝔼hil = (𝑛 ∈ ℕ0 ↦
(ℝ^‘(1...𝑛))) |
covol 23038 | class vol* |
cvol 23039 | class vol |
df-ovol 23040 | ⊢ vol* = (𝑥 ∈ 𝒫 ℝ ↦ inf({𝑦 ∈ ℝ*
∣ ∃𝑓 ∈ ((
≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝑥 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − )
∘ 𝑓)),
ℝ*, < ))}, ℝ*, < )) |
df-vol 23041 | ⊢ vol = (vol* ↾ {𝑥 ∣ ∀𝑦 ∈ (◡vol* “ ℝ)(vol*‘𝑦) = ((vol*‘(𝑦 ∩ 𝑥)) + (vol*‘(𝑦 ∖ 𝑥)))}) |
cmbf 23189 | class MblFn |
citg1 23190 | class
∫1 |
citg2 23191 | class
∫2 |
cibl 23192 | class
𝐿1 |
citg 23193 | class ∫𝐴𝐵 d𝑥 |
df-mbf 23194 | ⊢ MblFn = {𝑓 ∈ (ℂ ↑pm
ℝ) ∣ ∀𝑥
∈ ran (,)((◡(ℜ ∘ 𝑓) “ 𝑥) ∈ dom vol ∧ (◡(ℑ ∘ 𝑓) “ 𝑥) ∈ dom vol)} |
df-itg1 23195 | ⊢ ∫1 = (𝑓 ∈ {𝑔 ∈ MblFn ∣ (𝑔:ℝ⟶ℝ ∧ ran 𝑔 ∈ Fin ∧
(vol‘(◡𝑔 “ (ℝ ∖ {0}))) ∈
ℝ)} ↦ Σ𝑥
∈ (ran 𝑓 ∖
{0})(𝑥 ·
(vol‘(◡𝑓 “ {𝑥})))) |
df-itg2 23196 | ⊢ ∫2 = (𝑓 ∈ ((0[,]+∞)
↑𝑚 ℝ) ↦ sup({𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘𝑟
≤ 𝑓 ∧ 𝑥 =
(∫1‘𝑔))}, ℝ*, <
)) |
df-ibl 23197 | ⊢ 𝐿1 = {𝑓 ∈ MblFn ∣ ∀𝑘 ∈
(0...3)(∫2‘(𝑥 ∈ ℝ ↦
⦋(ℜ‘((𝑓‘𝑥) / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ dom 𝑓 ∧ 0 ≤ 𝑦), 𝑦, 0))) ∈ ℝ} |
df-itg 23198 | ⊢ ∫𝐴𝐵 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦
⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))) |
c0p 23242 | class
0𝑝 |
df-0p 23243 | ⊢ 0𝑝 = (ℂ
× {0}) |
cdit 23416 | class ⨜[𝐴 → 𝐵]𝐶 d𝑥 |
df-ditg 23417 | ⊢ ⨜[𝐴 → 𝐵]𝐶 d𝑥 = if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥) |
climc 23432 | class
limℂ |
cdv 23433 | class D |
cdvn 23434 | class
D𝑛 |
ccpn 23435 | class
Cn |
df-limc 23436 | ⊢ limℂ = (𝑓 ∈ (ℂ ↑pm
ℂ), 𝑥 ∈ ℂ
↦ {𝑦 ∣
[(TopOpen‘ℂfld) / 𝑗](𝑧 ∈ (dom 𝑓 ∪ {𝑥}) ↦ if(𝑧 = 𝑥, 𝑦, (𝑓‘𝑧))) ∈ (((𝑗 ↾t (dom 𝑓 ∪ {𝑥})) CnP 𝑗)‘𝑥)}) |
df-dv 23437 | ⊢ D = (𝑠 ∈ 𝒫 ℂ, 𝑓 ∈ (ℂ ↑pm
𝑠) ↦ ∪ 𝑥 ∈
((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom 𝑓)({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓‘𝑧) − (𝑓‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥))) |
df-dvn 23438 | ⊢ D𝑛 = (𝑠 ∈ 𝒫 ℂ, 𝑓 ∈ (ℂ ↑pm
𝑠) ↦ seq0(((𝑥 ∈ V ↦ (𝑠 D 𝑥)) ∘ 1st ),
(ℕ0 × {𝑓}))) |
df-cpn 23439 | ⊢ Cn = (𝑠 ∈ 𝒫 ℂ ↦ (𝑥 ∈ ℕ0
↦ {𝑓 ∈ (ℂ
↑pm 𝑠) ∣ ((𝑠 D𝑛 𝑓)‘𝑥) ∈ (dom 𝑓–cn→ℂ)})) |
cmdg 23617 | class mDeg |
cdg1 23618 | class
deg1 |
df-mdeg 23619 | ⊢ mDeg = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ↦ sup(ran (ℎ ∈ (𝑓 supp (0g‘𝑟)) ↦ (ℂfld
Σg ℎ)), ℝ*, <
))) |
df-deg1 23620 | ⊢ deg1 = (𝑟 ∈ V ↦ (1𝑜 mDeg
𝑟)) |
cmn1 23689 | class
Monic1p |
cuc1p 23690 | class
Unic1p |
cq1p 23691 | class
quot1p |
cr1p 23692 | class
rem1p |
cig1p 23693 | class
idlGen1p |
df-mon1 23694 | ⊢ Monic1p = (𝑟 ∈ V ↦ {𝑓 ∈
(Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠
(0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘(( deg1
‘𝑟)‘𝑓)) = (1r‘𝑟))}) |
df-uc1p 23695 | ⊢ Unic1p = (𝑟 ∈ V ↦ {𝑓 ∈
(Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠
(0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘(( deg1
‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))}) |
df-q1p 23696 | ⊢ quot1p = (𝑟 ∈ V ↦
⦋(Poly1‘𝑟) / 𝑝⦌⦋(Base‘𝑝) / 𝑏⦌(𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (℩𝑞 ∈ 𝑏 (( deg1 ‘𝑟)‘(𝑓(-g‘𝑝)(𝑞(.r‘𝑝)𝑔))) < (( deg1 ‘𝑟)‘𝑔)))) |
df-r1p 23697 | ⊢ rem1p = (𝑟 ∈ V ↦
⦋(Base‘(Poly1‘𝑟)) / 𝑏⦌(𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ (𝑓(-g‘(Poly1‘𝑟))((𝑓(quot1p‘𝑟)𝑔)(.r‘(Poly1‘𝑟))𝑔)))) |
df-ig1p 23698 | ⊢ idlGen1p = (𝑟 ∈ V ↦ (𝑖 ∈
(LIdeal‘(Poly1‘𝑟)) ↦ if(𝑖 =
{(0g‘(Poly1‘𝑟))},
(0g‘(Poly1‘𝑟)), (℩𝑔 ∈ (𝑖 ∩ (Monic1p‘𝑟))(( deg1
‘𝑟)‘𝑔) = inf((( deg1
‘𝑟) “ (𝑖 ∖
{(0g‘(Poly1‘𝑟))})), ℝ, < ))))) |
cply 23744 | class Poly |
cidp 23745 | class
Xp |
ccoe 23746 | class coeff |
cdgr 23747 | class deg |
df-ply 23748 | ⊢ Poly = (𝑥 ∈ 𝒫 ℂ ↦ {𝑓 ∣ ∃𝑛 ∈ ℕ0
∃𝑎 ∈ ((𝑥 ∪ {0})
↑𝑚 ℕ0)𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))}) |
df-idp 23749 | ⊢ Xp = ( I ↾
ℂ) |
df-coe 23750 | ⊢ coeff = (𝑓 ∈ (Poly‘ℂ) ↦
(℩𝑎 ∈
(ℂ ↑𝑚 ℕ0)∃𝑛 ∈ ℕ0
((𝑎 “
(ℤ≥‘(𝑛 + 1))) = {0} ∧ 𝑓 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑛)((𝑎‘𝑘) · (𝑧↑𝑘)))))) |
df-dgr 23751 | ⊢ deg = (𝑓 ∈ (Poly‘ℂ) ↦
sup((◡(coeff‘𝑓) “ (ℂ ∖ {0})),
ℕ0, < )) |
cquot 23849 | class quot |
df-quot 23850 | ⊢ quot = (𝑓 ∈ (Poly‘ℂ), 𝑔 ∈ ((Poly‘ℂ)
∖ {0𝑝}) ↦ (℩𝑞 ∈ (Poly‘ℂ)[(𝑓 ∘𝑓
− (𝑔
∘𝑓 · 𝑞)) / 𝑟](𝑟 = 0𝑝 ∨
(deg‘𝑟) <
(deg‘𝑔)))) |
caa 23873 | class 𝔸 |
df-aa 23874 | ⊢ 𝔸 = ∪ 𝑓 ∈ ((Poly‘ℤ) ∖
{0𝑝})(◡𝑓 “ {0}) |
ctayl 23911 | class Tayl |
cana 23912 | class Ana |
df-tayl 23913 | ⊢ Tayl = (𝑠 ∈ {ℝ, ℂ}, 𝑓 ∈ (ℂ
↑pm 𝑠) ↦ (𝑛 ∈ (ℕ0 ∪
{+∞}), 𝑎 ∈
∩ 𝑘 ∈ ((0[,]𝑛) ∩ ℤ)dom ((𝑠 D𝑛 𝑓)‘𝑘) ↦ ∪
𝑥 ∈ ℂ ({𝑥} × (ℂfld
tsums (𝑘 ∈ ((0[,]𝑛) ∩ ℤ) ↦
(((((𝑠
D𝑛 𝑓)‘𝑘)‘𝑎) / (!‘𝑘)) · ((𝑥 − 𝑎)↑𝑘))))))) |
df-ana 23914 | ⊢ Ana = (𝑠 ∈ {ℝ, ℂ} ↦ {𝑓 ∈ (ℂ
↑pm 𝑠) ∣ ∀𝑥 ∈ dom 𝑓 𝑥 ∈
((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom (𝑓 ∩ (+∞(𝑠 Tayl 𝑓)𝑥)))}) |
culm 23934 | class
⇝𝑢 |
df-ulm 23935 | ⊢ ⇝𝑢 = (𝑠 ∈ V ↦ {〈𝑓, 𝑦〉 ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑𝑚 𝑠) ∧ 𝑦:𝑠⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑠 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)}) |
clog 24105 | class log |
ccxp 24106 | class
↑𝑐 |
df-log 24107 | ⊢ log = ◡(exp ↾ (◡ℑ “
(-π(,]π))) |
df-cxp 24108 | ⊢ ↑𝑐 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ if(𝑥 = 0, if(𝑦 = 0, 1, 0), (exp‘(𝑦 · (log‘𝑥))))) |
clogb 24302 | class
logb |
df-logb 24303 | ⊢ logb = (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0})
↦ ((log‘𝑦) /
(log‘𝑥))) |
casin 24389 | class arcsin |
cacos 24390 | class arccos |
catan 24391 | class arctan |
df-asin 24392 | ⊢ arcsin = (𝑥 ∈ ℂ ↦ (-i ·
(log‘((i · 𝑥)
+ (√‘(1 − (𝑥↑2))))))) |
df-acos 24393 | ⊢ arccos = (𝑥 ∈ ℂ ↦ ((π / 2) −
(arcsin‘𝑥))) |
df-atan 24394 | ⊢ arctan = (𝑥 ∈ (ℂ ∖ {-i, i}) ↦ ((i
/ 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥)))))) |
carea 24482 | class area |
df-area 24483 | ⊢ area = (𝑠 ∈ {𝑡 ∈ 𝒫 (ℝ × ℝ)
∣ (∀𝑥 ∈
ℝ (𝑡 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦
(vol‘(𝑡 “
{𝑥}))) ∈
𝐿1)} ↦ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥) |
cem 24518 | class γ |
df-em 24519 | ⊢ γ = Σ𝑘 ∈ ℕ ((1 / 𝑘) − (log‘(1 + (1 /
𝑘)))) |
czeta 24539 | class ζ |
df-zeta 24540 | ⊢ ζ = (℩𝑓 ∈ ((ℂ ∖ {1})–cn→ℂ)∀𝑠 ∈ (ℂ ∖ {1})((1 −
(2↑𝑐(1 − 𝑠))) · (𝑓‘𝑠)) = Σ𝑛 ∈ ℕ0 (Σ𝑘 ∈ (0...𝑛)(((-1↑𝑘) · (𝑛C𝑘)) · ((𝑘 + 1)↑𝑐𝑠)) / (2↑(𝑛 + 1)))) |
clgam 24542 | class log Γ |
cgam 24543 | class Γ |
cigam 24544 | class 1/Γ |
df-lgam 24545 | ⊢ log Γ = (𝑧 ∈ (ℂ ∖ (ℤ ∖
ℕ)) ↦ (Σ𝑚
∈ ℕ ((𝑧 ·
(log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧))) |
df-gam 24546 | ⊢ Γ = (exp ∘ log
Γ) |
df-igam 24547 | ⊢ 1/Γ = (𝑥 ∈ ℂ ↦ if(𝑥 ∈ (ℤ ∖ ℕ), 0, (1 /
(Γ‘𝑥)))) |
ccht 24617 | class θ |
cvma 24618 | class Λ |
cchp 24619 | class ψ |
cppi 24620 | class π |
cmu 24621 | class μ |
csgm 24622 | class σ |
df-cht 24623 | ⊢ θ = (𝑥 ∈ ℝ ↦ Σ𝑝 ∈ ((0[,]𝑥) ∩ ℙ)(log‘𝑝)) |
df-vma 24624 | ⊢ Λ = (𝑥 ∈ ℕ ↦ ⦋{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥} / 𝑠⦌if((#‘𝑠) = 1, (log‘∪ 𝑠),
0)) |
df-chp 24625 | ⊢ ψ = (𝑥 ∈ ℝ ↦ Σ𝑛 ∈
(1...(⌊‘𝑥))(Λ‘𝑛)) |
df-ppi 24626 | ⊢ π = (𝑥 ∈ ℝ ↦ (#‘((0[,]𝑥) ∩
ℙ))) |
df-mu 24627 | ⊢ μ = (𝑥 ∈ ℕ ↦ if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝑥, 0, (-1↑(#‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑥})))) |
df-sgm 24628 | ⊢ σ = (𝑥 ∈ ℂ, 𝑛 ∈ ℕ ↦ Σ𝑘 ∈ {𝑝 ∈ ℕ ∣ 𝑝 ∥ 𝑛} (𝑘↑𝑐𝑥)) |
cdchr 24757 | class DChr |
df-dchr 24758 | ⊢ DChr = (𝑛 ∈ ℕ ↦
⦋(ℤ/nℤ‘𝑛) / 𝑧⦌⦋{𝑥 ∈ ((mulGrp‘𝑧) MndHom
(mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥} / 𝑏⦌{〈(Base‘ndx), 𝑏〉,
〈(+g‘ndx), ( ∘𝑓 · ↾
(𝑏 × 𝑏))〉}) |
clgs 24819 | class
/L |
df-lgs 24820 | ⊢ /L = (𝑎 ∈ ℤ, 𝑛 ∈ ℤ ↦ if(𝑛 = 0, if((𝑎↑2) = 1, 1, 0), (if((𝑛 < 0 ∧ 𝑎 < 0), -1, 1) · (seq1( · ,
(𝑚 ∈ ℕ ↦
if(𝑚 ∈ ℙ,
(if(𝑚 = 2, if(2 ∥
𝑎, 0, if((𝑎 mod 8) ∈ {1, 7}, 1, -1)),
((((𝑎↑((𝑚 − 1) / 2)) + 1) mod 𝑚) − 1))↑(𝑚 pCnt 𝑛)), 1)))‘(abs‘𝑛))))) |
cstrkg 25129 | class TarskiG |
cstrkgc 25130 | class
TarskiGC |
cstrkgb 25131 | class
TarskiGB |
cstrkgcb 25132 | class
TarskiGCB |
cstrkgld 25133 | class
DimTarskiG≥ |
cstrkge 25134 | class
TarskiGE |
citv 25135 | class Itv |
clng 25136 | class LineG |
df-itv 25137 | ⊢ Itv = Slot ;16 |
df-lng 25138 | ⊢ LineG = Slot ;17 |
df-trkgc 25147 | ⊢ TarskiGC = {𝑓 ∣ [(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑](∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 (𝑥𝑑𝑦) = (𝑦𝑑𝑥) ∧ ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ((𝑥𝑑𝑦) = (𝑧𝑑𝑧) → 𝑥 = 𝑦))} |
df-trkgb 25148 | ⊢ TarskiGB = {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 (𝑦 ∈ (𝑥𝑖𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ∀𝑢 ∈ 𝑝 ∀𝑣 ∈ 𝑝 ((𝑢 ∈ (𝑥𝑖𝑧) ∧ 𝑣 ∈ (𝑦𝑖𝑧)) → ∃𝑎 ∈ 𝑝 (𝑎 ∈ (𝑢𝑖𝑦) ∧ 𝑎 ∈ (𝑣𝑖𝑥))) ∧ ∀𝑠 ∈ 𝒫 𝑝∀𝑡 ∈ 𝒫 𝑝(∃𝑎 ∈ 𝑝 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑥 ∈ (𝑎𝑖𝑦) → ∃𝑏 ∈ 𝑝 ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑡 𝑏 ∈ (𝑥𝑖𝑦)))} |
df-trkgcb 25149 | ⊢ TarskiGCB = {𝑓 ∣ [(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑][(Itv‘𝑓) / 𝑖](∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ∀𝑢 ∈ 𝑝 ∀𝑎 ∈ 𝑝 ∀𝑏 ∈ 𝑝 ∀𝑐 ∈ 𝑝 ∀𝑣 ∈ 𝑝 (((𝑥 ≠ 𝑦 ∧ 𝑦 ∈ (𝑥𝑖𝑧) ∧ 𝑏 ∈ (𝑎𝑖𝑐)) ∧ (((𝑥𝑑𝑦) = (𝑎𝑑𝑏) ∧ (𝑦𝑑𝑧) = (𝑏𝑑𝑐)) ∧ ((𝑥𝑑𝑢) = (𝑎𝑑𝑣) ∧ (𝑦𝑑𝑢) = (𝑏𝑑𝑣)))) → (𝑧𝑑𝑢) = (𝑐𝑑𝑣)) ∧ ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑎 ∈ 𝑝 ∀𝑏 ∈ 𝑝 ∃𝑧 ∈ 𝑝 (𝑦 ∈ (𝑥𝑖𝑧) ∧ (𝑦𝑑𝑧) = (𝑎𝑑𝑏)))} |
df-trkge 25150 | ⊢ TarskiGE = {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖]∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ∀𝑢 ∈ 𝑝 ∀𝑣 ∈ 𝑝 ((𝑢 ∈ (𝑥𝑖𝑣) ∧ 𝑢 ∈ (𝑦𝑖𝑧) ∧ 𝑥 ≠ 𝑢) → ∃𝑎 ∈ 𝑝 ∃𝑏 ∈ 𝑝 (𝑦 ∈ (𝑥𝑖𝑎) ∧ 𝑧 ∈ (𝑥𝑖𝑏) ∧ 𝑣 ∈ (𝑎𝑖𝑏)))} |
df-trkgld 25151 | ⊢ DimTarskiG≥ = {〈𝑔, 𝑛〉 ∣ [(Base‘𝑔) / 𝑝][(dist‘𝑔) / 𝑑][(Itv‘𝑔) / 𝑖]∃𝑓(𝑓:(1..^𝑛)–1-1→𝑝 ∧ ∃𝑥 ∈ 𝑝 ∃𝑦 ∈ 𝑝 ∃𝑧 ∈ 𝑝 (∀𝑗 ∈ (2..^𝑛)(((𝑓‘1)𝑑𝑥) = ((𝑓‘𝑗)𝑑𝑥) ∧ ((𝑓‘1)𝑑𝑦) = ((𝑓‘𝑗)𝑑𝑦) ∧ ((𝑓‘1)𝑑𝑧) = ((𝑓‘𝑗)𝑑𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))))} |
df-trkg 25152 | ⊢ TarskiG = ((TarskiGC ∩
TarskiGB) ∩ (TarskiGCB ∩ {𝑓 ∣ [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥 ∈ 𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧 ∈ 𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})) |
ccgrg 25205 | class cgrG |
df-cgrg 25206 | ⊢ cgrG = (𝑔 ∈ V ↦ {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ ((Base‘𝑔) ↑pm ℝ) ∧
𝑏 ∈ ((Base‘𝑔) ↑pm
ℝ)) ∧ (dom 𝑎 =
dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗)) = ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗))))}) |
cismt 25227 | class Ismt |
df-ismt 25228 | ⊢ Ismt = (𝑔 ∈ V, ℎ ∈ V ↦ {𝑓 ∣ (𝑓:(Base‘𝑔)–1-1-onto→(Base‘ℎ) ∧ ∀𝑎 ∈ (Base‘𝑔)∀𝑏 ∈ (Base‘𝑔)((𝑓‘𝑎)(dist‘ℎ)(𝑓‘𝑏)) = (𝑎(dist‘𝑔)𝑏))}) |
cleg 25277 | class ≤G |
df-leg 25278 | ⊢ ≤G = (𝑔 ∈ V ↦ {〈𝑒, 𝑓〉 ∣ [(Base‘𝑔) / 𝑝][(dist‘𝑔) / 𝑑][(Itv‘𝑔) / 𝑖]∃𝑥 ∈ 𝑝 ∃𝑦 ∈ 𝑝 (𝑓 = (𝑥𝑑𝑦) ∧ ∃𝑧 ∈ 𝑝 (𝑧 ∈ (𝑥𝑖𝑦) ∧ 𝑒 = (𝑥𝑑𝑧)))}) |
chlg 25295 | class hlG |
df-hlg 25296 | ⊢ hlG = (𝑔 ∈ V ↦ (𝑐 ∈ (Base‘𝑔) ↦ {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (Base‘𝑔) ∧ 𝑏 ∈ (Base‘𝑔)) ∧ (𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐 ∧ (𝑎 ∈ (𝑐(Itv‘𝑔)𝑏) ∨ 𝑏 ∈ (𝑐(Itv‘𝑔)𝑎))))})) |
cmir 25347 | class pInvG |
df-mir 25348 | ⊢ pInvG = (𝑔 ∈ V ↦ (𝑚 ∈ (Base‘𝑔) ↦ (𝑎 ∈ (Base‘𝑔) ↦ (℩𝑏 ∈ (Base‘𝑔)((𝑚(dist‘𝑔)𝑏) = (𝑚(dist‘𝑔)𝑎) ∧ 𝑚 ∈ (𝑏(Itv‘𝑔)𝑎)))))) |
crag 25388 | class ∟G |
df-rag 25389 | ⊢ ∟G = (𝑔 ∈ V ↦ {𝑤 ∈ Word (Base‘𝑔) ∣ ((#‘𝑤) = 3 ∧ ((𝑤‘0)(dist‘𝑔)(𝑤‘2)) = ((𝑤‘0)(dist‘𝑔)(((pInvG‘𝑔)‘(𝑤‘1))‘(𝑤‘2))))}) |
cperpg 25390 | class ⟂G |
df-perpg 25391 | ⊢ ⟂G = (𝑔 ∈ V ↦ {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ ran (LineG‘𝑔) ∧ 𝑏 ∈ ran (LineG‘𝑔)) ∧ ∃𝑥 ∈ (𝑎 ∩ 𝑏)∀𝑢 ∈ 𝑎 ∀𝑣 ∈ 𝑏 〈“𝑢𝑥𝑣”〉 ∈ (∟G‘𝑔))}) |
chpg 25449 | class hpG |
df-hpg 25450 | ⊢ hpG = (𝑔 ∈ V ↦ (𝑑 ∈ ran (LineG‘𝑔) ↦ {〈𝑎, 𝑏〉 ∣ [(Base‘𝑔) / 𝑝][(Itv‘𝑔) / 𝑖]∃𝑐 ∈ 𝑝 (((𝑎 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑎𝑖𝑐)) ∧ ((𝑏 ∈ (𝑝 ∖ 𝑑) ∧ 𝑐 ∈ (𝑝 ∖ 𝑑)) ∧ ∃𝑡 ∈ 𝑑 𝑡 ∈ (𝑏𝑖𝑐)))})) |
cmid 25464 | class midG |
clmi 25465 | class lInvG |
df-mid 25466 | ⊢ midG = (𝑔 ∈ V ↦ (𝑎 ∈ (Base‘𝑔), 𝑏 ∈ (Base‘𝑔) ↦ (℩𝑚 ∈ (Base‘𝑔)𝑏 = (((pInvG‘𝑔)‘𝑚)‘𝑎)))) |
df-lmi 25467 | ⊢ lInvG = (𝑔 ∈ V ↦ (𝑚 ∈ ran (LineG‘𝑔) ↦ (𝑎 ∈ (Base‘𝑔) ↦ (℩𝑏 ∈ (Base‘𝑔)((𝑎(midG‘𝑔)𝑏) ∈ 𝑚 ∧ (𝑚(⟂G‘𝑔)(𝑎(LineG‘𝑔)𝑏) ∨ 𝑎 = 𝑏)))))) |
ccgra 25499 | class cgrA |
df-cgra 25500 | ⊢ cgrA = (𝑔 ∈ V ↦ {〈𝑎, 𝑏〉 ∣ [(Base‘𝑔) / 𝑝][(hlG‘𝑔) / 𝑘]((𝑎 ∈ (𝑝 ↑𝑚 (0..^3)) ∧
𝑏 ∈ (𝑝 ↑𝑚
(0..^3))) ∧ ∃𝑥
∈ 𝑝 ∃𝑦 ∈ 𝑝 (𝑎(cgrG‘𝑔)〈“𝑥(𝑏‘1)𝑦”〉 ∧ 𝑥(𝑘‘(𝑏‘1))(𝑏‘0) ∧ 𝑦(𝑘‘(𝑏‘1))(𝑏‘2)))}) |
cinag 25526 | class inA |
cleag 25527 | class
≤∠ |
df-inag 25528 | ⊢ inA = (𝑔 ∈ V ↦ {〈𝑝, 𝑡〉 ∣ ((𝑝 ∈ (Base‘𝑔) ∧ 𝑡 ∈ ((Base‘𝑔) ↑𝑚 (0..^3))) ∧
(((𝑡‘0) ≠ (𝑡‘1) ∧ (𝑡‘2) ≠ (𝑡‘1) ∧ 𝑝 ≠ (𝑡‘1)) ∧ ∃𝑥 ∈ (Base‘𝑔)(𝑥 ∈ ((𝑡‘0)(Itv‘𝑔)(𝑡‘2)) ∧ (𝑥 = (𝑡‘1) ∨ 𝑥((hlG‘𝑔)‘(𝑡‘1))𝑝))))}) |
df-leag 25532 | ⊢ ≤∠ = (𝑔 ∈ V ↦ {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ ((Base‘𝑔) ↑𝑚 (0..^3)) ∧
𝑏 ∈ ((Base‘𝑔) ↑𝑚
(0..^3))) ∧ ∃𝑥
∈ (Base‘𝑔)(𝑥(inA‘𝑔)〈“(𝑏‘0)(𝑏‘1)(𝑏‘2)”〉 ∧
〈“(𝑎‘0)(𝑎‘1)(𝑎‘2)”〉(cgrA‘𝑔)〈“(𝑏‘0)(𝑏‘1)𝑥”〉))}) |
ceqlg 25545 | class eqltrG |
df-eqlg 25546 | ⊢ eqltrG = (𝑔 ∈ V ↦ {𝑥 ∈ ((Base‘𝑔) ↑𝑚 (0..^3)) ∣
𝑥(cgrG‘𝑔)〈“(𝑥‘1)(𝑥‘2)(𝑥‘0)”〉}) |
cttg 25553 | class toTG |
df-ttg 25554 | ⊢ toTG = (𝑤 ∈ V ↦ ⦋(𝑥 ∈ (Base‘𝑤), 𝑦 ∈ (Base‘𝑤) ↦ {𝑧 ∈ (Base‘𝑤) ∣ ∃𝑘 ∈ (0[,]1)(𝑧(-g‘𝑤)𝑥) = (𝑘( ·𝑠
‘𝑤)(𝑦(-g‘𝑤)𝑥))}) / 𝑖⦌((𝑤 sSet 〈(Itv‘ndx), 𝑖〉) sSet
〈(LineG‘ndx), (𝑥
∈ (Base‘𝑤),
𝑦 ∈ (Base‘𝑤) ↦ {𝑧 ∈ (Base‘𝑤) ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})〉)) |
cee 25568 | class 𝔼 |
cbtwn 25569 | class Btwn |
ccgr 25570 | class Cgr |
df-ee 25571 | ⊢ 𝔼 = (𝑛 ∈ ℕ ↦ (ℝ
↑𝑚 (1...𝑛))) |
df-btwn 25572 | ⊢ Btwn = ◡{〈〈𝑥, 𝑧〉, 𝑦〉 ∣ ∃𝑛 ∈ ℕ ((𝑥 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑦 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝑦‘𝑖) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑧‘𝑖))))} |
df-cgr 25573 | ⊢ Cgr = {〈𝑥, 𝑦〉 ∣ ∃𝑛 ∈ ℕ ((𝑥 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑦 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))) ∧ Σ𝑖 ∈ (1...𝑛)((((1st ‘𝑥)‘𝑖) − ((2nd ‘𝑥)‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑛)((((1st ‘𝑦)‘𝑖) − ((2nd ‘𝑦)‘𝑖))↑2))} |
ceeng 25657 | class EEG |
df-eeng 25658 | ⊢ EEG = (𝑛 ∈ ℕ ↦
({〈(Base‘ndx), (𝔼‘𝑛)〉, 〈(dist‘ndx), (𝑥 ∈ (𝔼‘𝑛), 𝑦 ∈ (𝔼‘𝑛) ↦ Σ𝑖 ∈ (1...𝑛)(((𝑥‘𝑖) − (𝑦‘𝑖))↑2))〉} ∪
{〈(Itv‘ndx), (𝑥
∈ (𝔼‘𝑛),
𝑦 ∈
(𝔼‘𝑛) ↦
{𝑧 ∈
(𝔼‘𝑛) ∣
𝑧 Btwn 〈𝑥, 𝑦〉})〉, 〈(LineG‘ndx),
(𝑥 ∈
(𝔼‘𝑛), 𝑦 ∈ ((𝔼‘𝑛) ∖ {𝑥}) ↦ {𝑧 ∈ (𝔼‘𝑛) ∣ (𝑧 Btwn 〈𝑥, 𝑦〉 ∨ 𝑥 Btwn 〈𝑧, 𝑦〉 ∨ 𝑦 Btwn 〈𝑥, 𝑧〉)})〉})) |
cedgf 25667 | class .ef |
df-edgf 25668 | ⊢ .ef = Slot ;18 |
cvtx 25673 | class Vtx |
ciedg 25674 | class iEdg |
df-vtx 25675 | ⊢ Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st
‘𝑔),
(Base‘𝑔))) |
df-iedg 25676 | ⊢ iEdg = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2nd
‘𝑔), (.ef‘𝑔))) |
cuhgr 25722 | class UHGraph |
cushgr 25723 | class USHGraph |
df-uhgr 25724 | ⊢ UHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})} |
df-ushgr 25725 | ⊢ USHGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})} |
cupgr 25747 | class UPGraph |
cumgr 25748 | class UMGraph |
df-upgr 25749 | ⊢ UPGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}} |
df-umgr 25750 | ⊢ UMGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}} |
cedga 25792 | class Edg |
df-edga 25793 | ⊢ Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔)) |
cuhg 25819 | class UHGrph |
cushg 25820 | class USHGrph |
df-uhgra 25821 | ⊢ UHGrph = {〈𝑣, 𝑒〉 ∣ 𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})} |
df-ushgra 25822 | ⊢ USHGrph = {〈𝑣, 𝑒〉 ∣ 𝑒:dom 𝑒–1-1→(𝒫 𝑣 ∖ {∅})} |
cumg 25841 | class UMGrph |
df-umgra 25842 | ⊢ UMGrph = {〈𝑣, 𝑒〉 ∣ 𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}} |
cuslg 25858 | class USLGrph |
cusg 25859 | class USGrph |
cedg 25860 | class Edges |
df-uslgra 25861 | ⊢ USLGrph = {〈𝑣, 𝑒〉 ∣ 𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}} |
df-usgra 25862 | ⊢ USGrph = {〈𝑣, 𝑒〉 ∣ 𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}} |
df-edg 25865 | ⊢ Edges = (𝑔 ∈ V ↦ ran (2nd
‘𝑔)) |
cnbgra 25946 | class Neighbors |
ccusgra 25947 | class ComplUSGrph |
cuvtx 25948 | class UnivVertex |
df-nbgra 25949 | ⊢ Neighbors = (𝑔 ∈ V, 𝑘 ∈ (1st ‘𝑔) ↦ {𝑛 ∈ (1st ‘𝑔) ∣ {𝑘, 𝑛} ∈ ran (2nd ‘𝑔)}) |
df-cusgra 25950 | ⊢ ComplUSGrph = {〈𝑣, 𝑒〉 ∣ (𝑣 USGrph 𝑒 ∧ ∀𝑘 ∈ 𝑣 ∀𝑛 ∈ (𝑣 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝑒)} |
df-uvtx 25951 | ⊢ UnivVertex = (𝑣 ∈ V, 𝑒 ∈ V ↦ {𝑛 ∈ 𝑣 ∣ ∀𝑘 ∈ (𝑣 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝑒}) |
cwalk 26026 | class Walks |
ctrail 26027 | class Trails |
cpath 26028 | class Paths |
cspath 26029 | class SPaths |
cwlkon 26030 | class WalkOn |
ctrlon 26031 | class TrailOn |
cpthon 26032 | class PathOn |
cspthon 26033 | class SPathOn |
ccrct 26034 | class Circuits |
ccycl 26035 | class Cycles |
df-wlk 26036 | ⊢ Walks = (𝑣 ∈ V, 𝑒 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓 ∈ Word dom 𝑒 ∧ 𝑝:(0...(#‘𝑓))⟶𝑣 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝑒‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})}) |
df-trail 26037 | ⊢ Trails = (𝑣 ∈ V, 𝑒 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑣 Walks 𝑒)𝑝 ∧ Fun ◡𝑓)}) |
df-pth 26038 | ⊢ Paths = (𝑣 ∈ V, 𝑒 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑣 Trails 𝑒)𝑝 ∧ Fun ◡(𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)}) |
df-spth 26039 | ⊢ SPaths = (𝑣 ∈ V, 𝑒 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑣 Trails 𝑒)𝑝 ∧ Fun ◡𝑝)}) |
df-crct 26040 | ⊢ Circuits = (𝑣 ∈ V, 𝑒 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑣 Trails 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))}) |
df-cycl 26041 | ⊢ Cycles = (𝑣 ∈ V, 𝑒 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑣 Paths 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))}) |
df-wlkon 26042 | ⊢ WalkOn = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑎 ∈ 𝑣, 𝑏 ∈ 𝑣 ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑣 Walks 𝑒)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏)})) |
df-trlon 26043 | ⊢ TrailOn = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑎 ∈ 𝑣, 𝑏 ∈ 𝑣 ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑎(𝑣 WalkOn 𝑒)𝑏)𝑝 ∧ 𝑓(𝑣 Trails 𝑒)𝑝)})) |
df-pthon 26044 | ⊢ PathOn = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑎 ∈ 𝑣, 𝑏 ∈ 𝑣 ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑎(𝑣 WalkOn 𝑒)𝑏)𝑝 ∧ 𝑓(𝑣 Paths 𝑒)𝑝)})) |
df-spthon 26045 | ⊢ SPathOn = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑎 ∈ 𝑣, 𝑏 ∈ 𝑣 ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑎(𝑣 WalkOn 𝑒)𝑏)𝑝 ∧ 𝑓(𝑣 SPaths 𝑒)𝑝)})) |
cconngra 26197 | class ConnGrph |
df-conngra 26198 | ⊢ ConnGrph = {〈𝑣, 𝑒〉 ∣ ∀𝑘 ∈ 𝑣 ∀𝑛 ∈ 𝑣 ∃𝑓∃𝑝 𝑓(𝑘(𝑣 PathOn 𝑒)𝑛)𝑝} |
cwwlk 26205 | class WWalks |
cwwlkn 26206 | class WWalksN |
df-wwlk 26207 | ⊢ WWalks = (𝑣 ∈ V, 𝑒 ∈ V ↦ {𝑤 ∈ Word 𝑣 ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝑒)}) |
df-wwlkn 26208 | ⊢ WWalksN = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑣 WWalks 𝑒) ∣ (#‘𝑤) = (𝑛 + 1)})) |
cclwlk 26275 | class ClWalks |
cclwwlk 26276 | class ClWWalks |
cclwwlkn 26277 | class ClWWalksN |
df-clwlk 26278 | ⊢ ClWalks = (𝑣 ∈ V, 𝑒 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑣 Walks 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))}) |
df-clwwlk 26279 | ⊢ ClWWalks = (𝑣 ∈ V, 𝑒 ∈ V ↦ {𝑤 ∈ Word 𝑣 ∣ (∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝑒 ∧ {( lastS ‘𝑤), (𝑤‘0)} ∈ ran 𝑒)}) |
df-clwwlkn 26280 | ⊢ ClWWalksN = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑣 ClWWalks 𝑒) ∣ (#‘𝑤) = 𝑛})) |
c2wlkot 26381 | class 2WalksOt |
c2wlkonot 26382 | class 2WalksOnOt |
c2spthot 26383 | class 2SPathsOt |
c2pthonot 26384 | class 2SPathOnOt |
df-2wlkonot 26385 | ⊢ 2WalksOnOt = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑎 ∈ 𝑣, 𝑏 ∈ 𝑣 ↦ {𝑡 ∈ ((𝑣 × 𝑣) × 𝑣) ∣ ∃𝑓∃𝑝(𝑓(𝑎(𝑣 WalkOn 𝑒)𝑏)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st
‘(1st ‘𝑡)) = 𝑎 ∧ (2nd ‘(1st
‘𝑡)) = (𝑝‘1) ∧ (2nd
‘𝑡) = 𝑏))})) |
df-2wlksot 26386 | ⊢ 2WalksOt = (𝑣 ∈ V, 𝑒 ∈ V ↦ {𝑡 ∈ ((𝑣 × 𝑣) × 𝑣) ∣ ∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 𝑡 ∈ (𝑎(𝑣 2WalksOnOt 𝑒)𝑏)}) |
df-2spthonot 26387 | ⊢ 2SPathOnOt = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑎 ∈ 𝑣, 𝑏 ∈ 𝑣 ↦ {𝑡 ∈ ((𝑣 × 𝑣) × 𝑣) ∣ ∃𝑓∃𝑝(𝑓(𝑎(𝑣 SPathOn 𝑒)𝑏)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st
‘(1st ‘𝑡)) = 𝑎 ∧ (2nd ‘(1st
‘𝑡)) = (𝑝‘1) ∧ (2nd
‘𝑡) = 𝑏))})) |
df-2spthsot 26388 | ⊢ 2SPathsOt = (𝑣 ∈ V, 𝑒 ∈ V ↦ {𝑡 ∈ ((𝑣 × 𝑣) × 𝑣) ∣ ∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 𝑡 ∈ (𝑎(𝑣 2SPathOnOt 𝑒)𝑏)}) |
cvdg 26420 | class VDeg |
df-vdgr 26421 | ⊢ VDeg = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑢 ∈ 𝑣 ↦ ((#‘{𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}})))) |
crgra 26449 | class RegGrph |
crusgra 26450 | class RegUSGrph |
df-rgra 26451 | ⊢ RegGrph = {〈〈𝑣, 𝑒〉, 𝑘〉 ∣ (𝑘 ∈ ℕ0 ∧
∀𝑝 ∈ 𝑣 ((𝑣 VDeg 𝑒)‘𝑝) = 𝑘)} |
df-rusgra 26452 | ⊢ RegUSGrph = {〈〈𝑣, 𝑒〉, 𝑘〉 ∣ (𝑣 USGrph 𝑒 ∧ 〈𝑣, 𝑒〉 RegGrph 𝑘)} |
ceup 26489 | class EulPaths |
df-eupa 26490 | ⊢ EulPaths = (𝑣 ∈ V, 𝑒 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑣 UMGrph 𝑒 ∧ ∃𝑛 ∈ ℕ0 (𝑓:(1...𝑛)–1-1-onto→dom
𝑒 ∧ 𝑝:(0...𝑛)⟶𝑣 ∧ ∀𝑘 ∈ (1...𝑛)(𝑒‘(𝑓‘𝑘)) = {(𝑝‘(𝑘 − 1)), (𝑝‘𝑘)}))}) |
cfrgra 26515 | class FriendGrph |
df-frgra 26516 | ⊢ FriendGrph = {〈𝑣, 𝑒〉 ∣ (𝑣 USGrph 𝑒 ∧ ∀𝑘 ∈ 𝑣 ∀𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥 ∈ 𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝑒)} |
cplig 26718 | class Plig |
df-plig 26719 | ⊢ Plig = {𝑥 ∣ (∀𝑎 ∈ ∪ 𝑥∀𝑏 ∈ ∪ 𝑥(𝑎 ≠ 𝑏 → ∃!𝑙 ∈ 𝑥 (𝑎 ∈ 𝑙 ∧ 𝑏 ∈ 𝑙)) ∧ ∀𝑙 ∈ 𝑥 ∃𝑎 ∈ ∪ 𝑥∃𝑏 ∈ ∪ 𝑥(𝑎 ≠ 𝑏 ∧ 𝑎 ∈ 𝑙 ∧ 𝑏 ∈ 𝑙) ∧ ∃𝑎 ∈ ∪ 𝑥∃𝑏 ∈ ∪ 𝑥∃𝑐 ∈ ∪ 𝑥∀𝑙 ∈ 𝑥 ¬ (𝑎 ∈ 𝑙 ∧ 𝑏 ∈ 𝑙 ∧ 𝑐 ∈ 𝑙))} |
crpm 26723 | class RPrime |
df-rprm 26724 | ⊢ RPrime = (𝑤 ∈ V ↦
⦋(Base‘𝑤) / 𝑏⦌{𝑝 ∈ (𝑏 ∖ ((Unit‘𝑤) ∪ {(0g‘𝑤)})) ∣ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏
[(∥r‘𝑤) / 𝑑](𝑝𝑑(𝑥(.r‘𝑤)𝑦) → (𝑝𝑑𝑥 ∨ 𝑝𝑑𝑦))}) |
cgr 26727 | class GrpOp |
cgi 26728 | class GId |
cgn 26729 | class inv |
cgs 26730 | class
/𝑔 |
df-grpo 26731 | ⊢ GrpOp = {𝑔 ∣ ∃𝑡(𝑔:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 ∀𝑧 ∈ 𝑡 ((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ∧ ∃𝑢 ∈ 𝑡 ∀𝑥 ∈ 𝑡 ((𝑢𝑔𝑥) = 𝑥 ∧ ∃𝑦 ∈ 𝑡 (𝑦𝑔𝑥) = 𝑢))} |
df-gid 26732 | ⊢ GId = (𝑔 ∈ V ↦ (℩𝑢 ∈ ran 𝑔∀𝑥 ∈ ran 𝑔((𝑢𝑔𝑥) = 𝑥 ∧ (𝑥𝑔𝑢) = 𝑥))) |
df-ginv 26733 | ⊢ inv = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔 ↦ (℩𝑧 ∈ ran 𝑔(𝑧𝑔𝑥) = (GId‘𝑔)))) |
df-gdiv 26734 | ⊢ /𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦)))) |
cablo 26782 | class AbelOp |
df-ablo 26783 | ⊢ AbelOp = {𝑔 ∈ GrpOp ∣ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥)} |
cvc 26797 | class
CVecOLD |
df-vc 26798 | ⊢ CVecOLD =
{〈𝑔, 𝑠〉 ∣ (𝑔 ∈ AbelOp ∧ 𝑠:(ℂ × ran 𝑔)⟶ran 𝑔 ∧ ∀𝑥 ∈ ran 𝑔((1𝑠𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ ran 𝑔(𝑦𝑠(𝑥𝑔𝑧)) = ((𝑦𝑠𝑥)𝑔(𝑦𝑠𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝑔(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥))))))} |
cnv 26823 | class NrmCVec |
cpv 26824 | class
+𝑣 |
cba 26825 | class BaseSet |
cns 26826 | class
·𝑠OLD |
cn0v 26827 | class 0vec |
cnsb 26828 | class
−𝑣 |
cnmcv 26829 | class
normCV |
cims 26830 | class IndMet |
df-nv 26831 | ⊢ NrmCVec = {〈〈𝑔, 𝑠〉, 𝑛〉 ∣ (〈𝑔, 𝑠〉 ∈ CVecOLD ∧ 𝑛:ran 𝑔⟶ℝ ∧ ∀𝑥 ∈ ran 𝑔(((𝑛‘𝑥) = 0 → 𝑥 = (GId‘𝑔)) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))))} |
df-va 26834 | ⊢ +𝑣 =
(1st ∘ 1st ) |
df-ba 26835 | ⊢ BaseSet = (𝑥 ∈ V ↦ ran ( +𝑣
‘𝑥)) |
df-sm 26836 | ⊢
·𝑠OLD = (2nd ∘ 1st
) |
df-0v 26837 | ⊢ 0vec = (GId ∘
+𝑣 ) |
df-vs 26838 | ⊢ −𝑣 = (
/𝑔 ∘ +𝑣 ) |
df-nmcv 26839 | ⊢ normCV =
2nd |
df-ims 26840 | ⊢ IndMet = (𝑢 ∈ NrmCVec ↦
((normCV‘𝑢) ∘ ( −𝑣
‘𝑢))) |
cdip 26939 | class
·𝑖OLD |
df-dip 26940 | ⊢ ·𝑖OLD =
(𝑢 ∈ NrmCVec ↦
(𝑥 ∈
(BaseSet‘𝑢), 𝑦 ∈ (BaseSet‘𝑢) ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) ·
(((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD
‘𝑢)𝑦)))↑2)) / 4))) |
css 26960 | class SubSp |
df-ssp 26961 | ⊢ SubSp = (𝑢 ∈ NrmCVec ↦ {𝑤 ∈ NrmCVec ∣ ((
+𝑣 ‘𝑤) ⊆ ( +𝑣
‘𝑢) ∧ (
·𝑠OLD ‘𝑤) ⊆ (
·𝑠OLD ‘𝑢) ∧ (normCV‘𝑤) ⊆
(normCV‘𝑢))}) |
clno 26979 | class LnOp |
cnmoo 26980 | class
normOpOLD |
cblo 26981 | class BLnOp |
c0o 26982 | class 0op |
df-lno 26983 | ⊢ LnOp = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {𝑡 ∈ ((BaseSet‘𝑤) ↑𝑚
(BaseSet‘𝑢)) ∣
∀𝑥 ∈ ℂ
∀𝑦 ∈
(BaseSet‘𝑢)∀𝑧 ∈ (BaseSet‘𝑢)(𝑡‘((𝑥( ·𝑠OLD
‘𝑢)𝑦)( +𝑣 ‘𝑢)𝑧)) = ((𝑥( ·𝑠OLD
‘𝑤)(𝑡‘𝑦))( +𝑣 ‘𝑤)(𝑡‘𝑧))}) |
df-nmoo 26984 | ⊢ normOpOLD = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ (𝑡 ∈ ((BaseSet‘𝑤) ↑𝑚
(BaseSet‘𝑢)) ↦
sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV‘𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV‘𝑤)‘(𝑡‘𝑧)))}, ℝ*, <
))) |
df-blo 26985 | ⊢ BLnOp = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {𝑡 ∈ (𝑢 LnOp 𝑤) ∣ ((𝑢 normOpOLD 𝑤)‘𝑡) < +∞}) |
df-0o 26986 | ⊢ 0op = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦
((BaseSet‘𝑢) ×
{(0vec‘𝑤)})) |
caj 26987 | class adj |
chmo 26988 | class HmOp |
df-aj 26989 | ⊢ adj = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {〈𝑡, 𝑠〉 ∣ (𝑡:(BaseSet‘𝑢)⟶(BaseSet‘𝑤) ∧ 𝑠:(BaseSet‘𝑤)⟶(BaseSet‘𝑢) ∧ ∀𝑥 ∈ (BaseSet‘𝑢)∀𝑦 ∈ (BaseSet‘𝑤)((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦) = (𝑥(·𝑖OLD‘𝑢)(𝑠‘𝑦)))}) |
df-hmo 26990 | ⊢ HmOp = (𝑢 ∈ NrmCVec ↦ {𝑡 ∈ dom (𝑢adj𝑢) ∣ ((𝑢adj𝑢)‘𝑡) = 𝑡}) |
ccphlo 27051 | class
CPreHilOLD |
df-ph 27052 | ⊢ CPreHilOLD = (NrmCVec
∩ {〈〈𝑔, 𝑠〉, 𝑛〉 ∣ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛‘𝑥)↑2) + ((𝑛‘𝑦)↑2)))}) |
ccbn 27102 | class CBan |
df-cbn 27103 | ⊢ CBan = {𝑢 ∈ NrmCVec ∣ (IndMet‘𝑢) ∈
(CMet‘(BaseSet‘𝑢))} |
chlo 27125 | class
CHilOLD |
df-hlo 27126 | ⊢ CHilOLD = (CBan ∩
CPreHilOLD) |
The
list of syntax, axioms (ax-) and definitions (df-) for the Hilbert Space Explorer starts here |
chil 27160 | class ℋ |
cva 27161 | class
+ℎ |
csm 27162 | class
·ℎ |
csp 27163 | class
·ih |
cno 27164 | class
normℎ |
c0v 27165 | class
0ℎ |
cmv 27166 | class
−ℎ |
ccau 27167 | class Cauchy |
chli 27168 | class
⇝𝑣 |
csh 27169 | class
Sℋ |
cch 27170 | class
Cℋ |
cort 27171 | class ⊥ |
cph 27172 | class
+ℋ |
cspn 27173 | class span |
chj 27174 | class
∨ℋ |
chsup 27175 | class ∨ℋ |
c0h 27176 | class
0ℋ |
ccm 27177 | class
𝐶ℋ |
cpjh 27178 | class
projℎ |
chos 27179 | class +op |
chot 27180 | class
·op |
chod 27181 | class
−op |
chfs 27182 | class +fn |
chft 27183 | class
·fn |
ch0o 27184 | class
0hop |
chio 27185 | class Iop |
cnop 27186 | class
normop |
ccop 27187 | class ConOp |
clo 27188 | class LinOp |
cbo 27189 | class BndLinOp |
cuo 27190 | class UniOp |
cho 27191 | class HrmOp |
cnmf 27192 | class
normfn |
cnl 27193 | class null |
ccnfn 27194 | class ConFn |
clf 27195 | class LinFn |
cado 27196 | class
adjℎ |
cbr 27197 | class bra |
ck 27198 | class ketbra |
cleo 27199 | class
≤op |
cei 27200 | class eigvec |
cel 27201 | class eigval |
cspc 27202 | class Lambda |
cst 27203 | class States |
chst 27204 | class CHStates |
ccv 27205 | class
⋖ℋ |
cat 27206 | class HAtoms |
cmd 27207 | class
𝑀ℋ |
cdmd 27208 | class
𝑀ℋ* |
df-hnorm 27209 | ⊢ normℎ = (𝑥 ∈ dom dom
·ih ↦ (√‘(𝑥 ·ih 𝑥))) |
df-hba 27210 | ⊢ ℋ = (BaseSet‘〈〈
+ℎ , ·ℎ 〉,
normℎ〉) |
df-h0v 27211 | ⊢ 0ℎ =
(0vec‘〈〈 +ℎ ,
·ℎ 〉,
normℎ〉) |
df-hvsub 27212 | ⊢ −ℎ = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑥 +ℎ (-1
·ℎ 𝑦))) |
df-hlim 27213 | ⊢ ⇝𝑣 = {〈𝑓, 𝑤〉 ∣ ((𝑓:ℕ⟶ ℋ ∧ 𝑤 ∈ ℋ) ∧
∀𝑥 ∈
ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝑓‘𝑧) −ℎ 𝑤)) < 𝑥)} |
df-hcau 27214 | ⊢ Cauchy = {𝑓 ∈ ( ℋ ↑𝑚
ℕ) ∣ ∀𝑥
∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝑓‘𝑦) −ℎ (𝑓‘𝑧))) < 𝑥} |
ax-hilex 27240 | ⊢ ℋ ∈ V |
ax-hfvadd 27241 | ⊢ +ℎ :( ℋ ×
ℋ)⟶ ℋ |
ax-hvcom 27242 | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴)) |
ax-hvass 27243 | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = (𝐴 +ℎ (𝐵 +ℎ 𝐶))) |
ax-hv0cl 27244 | ⊢ 0ℎ ∈
ℋ |
ax-hvaddid 27245 | ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ)
= 𝐴) |
ax-hfvmul 27246 | ⊢ ·ℎ :(ℂ
× ℋ)⟶ ℋ |
ax-hvmulid 27247 | ⊢ (𝐴 ∈ ℋ → (1
·ℎ 𝐴) = 𝐴) |
ax-hvmulass 27248 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) ·ℎ 𝐶) = (𝐴 ·ℎ (𝐵
·ℎ 𝐶))) |
ax-hvdistr1 27249 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ℎ (𝐵 +ℎ 𝐶)) = ((𝐴 ·ℎ 𝐵) +ℎ (𝐴
·ℎ 𝐶))) |
ax-hvdistr2 27250 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) ·ℎ 𝐶) = ((𝐴 ·ℎ 𝐶) +ℎ (𝐵
·ℎ 𝐶))) |
ax-hvmul0 27251 | ⊢ (𝐴 ∈ ℋ → (0
·ℎ 𝐴) = 0ℎ) |
ax-hfi 27320 | ⊢ ·ih :(
ℋ × ℋ)⟶ℂ |
ax-his1 27323 | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) = (∗‘(𝐵
·ih 𝐴))) |
ax-his2 27324 | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) ·ih 𝐶) = ((𝐴 ·ih 𝐶) + (𝐵 ·ih 𝐶))) |
ax-his3 27325 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·ℎ 𝐵)
·ih 𝐶) = (𝐴 · (𝐵 ·ih 𝐶))) |
ax-his4 27326 | ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 <
(𝐴
·ih 𝐴)) |
ax-hcompl 27443 | ⊢ (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝𝑣
𝑥) |
df-sh 27448 | ⊢
Sℋ = {ℎ ∈ 𝒫 ℋ ∣
(0ℎ ∈ ℎ ∧ ( +ℎ “ (ℎ × ℎ)) ⊆ ℎ ∧ ( ·ℎ
“ (ℂ × ℎ))
⊆ ℎ)} |
df-ch 27462 | ⊢
Cℋ = {ℎ ∈ Sℋ
∣ ( ⇝𝑣 “ (ℎ ↑𝑚 ℕ)) ⊆
ℎ} |
df-oc 27493 | ⊢ ⊥ = (𝑥 ∈ 𝒫 ℋ ↦ {𝑦 ∈ ℋ ∣
∀𝑧 ∈ 𝑥 (𝑦 ·ih 𝑧) = 0}) |
df-ch0 27494 | ⊢ 0ℋ =
{0ℎ} |
df-shs 27551 | ⊢ +ℋ = (𝑥 ∈ Sℋ ,
𝑦 ∈
Sℋ ↦ ( +ℎ “ (𝑥 × 𝑦))) |
df-span 27552 | ⊢ span = (𝑥 ∈ 𝒫 ℋ ↦ ∩ {𝑦
∈ Sℋ ∣ 𝑥 ⊆ 𝑦}) |
df-chj 27553 | ⊢ ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦
(⊥‘(⊥‘(𝑥 ∪ 𝑦)))) |
df-chsup 27554 | ⊢ ∨ℋ = (𝑥 ∈ 𝒫 𝒫 ℋ ↦
(⊥‘(⊥‘∪ 𝑥))) |
df-pjh 27638 | ⊢ projℎ = (ℎ ∈ Cℋ
↦ (𝑥 ∈ ℋ
↦ (℩𝑧
∈ ℎ ∃𝑦 ∈ (⊥‘ℎ)𝑥 = (𝑧 +ℎ 𝑦)))) |
df-cm 27826 | ⊢ 𝐶ℋ =
{〈𝑥, 𝑦〉 ∣ ((𝑥 ∈
Cℋ ∧ 𝑦 ∈ Cℋ )
∧ 𝑥 = ((𝑥 ∩ 𝑦) ∨ℋ (𝑥 ∩ (⊥‘𝑦))))} |
df-hosum 27973 | ⊢ +op = (𝑓 ∈ ( ℋ ↑𝑚
ℋ), 𝑔 ∈ (
ℋ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) +ℎ (𝑔‘𝑥)))) |
df-homul 27974 | ⊢ ·op = (𝑓 ∈ ℂ, 𝑔 ∈ ( ℋ
↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 ·ℎ (𝑔‘𝑥)))) |
df-hodif 27975 | ⊢ −op = (𝑓 ∈ ( ℋ ↑𝑚
ℋ), 𝑔 ∈ (
ℋ ↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) −ℎ (𝑔‘𝑥)))) |
df-hfsum 27976 | ⊢ +fn = (𝑓 ∈ (ℂ ↑𝑚
ℋ), 𝑔 ∈ (ℂ
↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ ((𝑓‘𝑥) + (𝑔‘𝑥)))) |
df-hfmul 27977 | ⊢ ·fn = (𝑓 ∈ ℂ, 𝑔 ∈ (ℂ
↑𝑚 ℋ) ↦ (𝑥 ∈ ℋ ↦ (𝑓 · (𝑔‘𝑥)))) |
df-h0op 27991 | ⊢ 0hop =
(projℎ‘0ℋ) |
df-iop 27992 | ⊢ Iop =
(projℎ‘ ℋ) |
df-nmop 28082 | ⊢ normop = (𝑡 ∈ ( ℋ ↑𝑚
ℋ) ↦ sup({𝑥
∣ ∃𝑧 ∈
ℋ ((normℎ‘𝑧) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑡‘𝑧)))}, ℝ*, <
)) |
df-cnop 28083 | ⊢ ConOp = {𝑡 ∈ ( ℋ ↑𝑚
ℋ) ∣ ∀𝑥
∈ ℋ ∀𝑦
∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ ℋ
((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 →
(normℎ‘((𝑡‘𝑤) −ℎ (𝑡‘𝑥))) < 𝑦)} |
df-lnop 28084 | ⊢ LinOp = {𝑡 ∈ ( ℋ ↑𝑚
ℋ) ∣ ∀𝑥
∈ ℂ ∀𝑦
∈ ℋ ∀𝑧
∈ ℋ (𝑡‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ (𝑡‘𝑦)) +ℎ (𝑡‘𝑧))} |
df-bdop 28085 | ⊢ BndLinOp = {𝑡 ∈ LinOp ∣
(normop‘𝑡)
< +∞} |
df-unop 28086 | ⊢ UniOp = {𝑡 ∣ (𝑡: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑡‘𝑥) ·ih (𝑡‘𝑦)) = (𝑥 ·ih 𝑦))} |
df-hmop 28087 | ⊢ HrmOp = {𝑡 ∈ ( ℋ ↑𝑚
ℋ) ∣ ∀𝑥
∈ ℋ ∀𝑦
∈ ℋ (𝑥
·ih (𝑡‘𝑦)) = ((𝑡‘𝑥) ·ih 𝑦)} |
df-nmfn 28088 | ⊢ normfn = (𝑡 ∈ (ℂ ↑𝑚
ℋ) ↦ sup({𝑥
∣ ∃𝑧 ∈
ℋ ((normℎ‘𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡‘𝑧)))}, ℝ*, <
)) |
df-nlfn 28089 | ⊢ null = (𝑡 ∈ (ℂ ↑𝑚
ℋ) ↦ (◡𝑡 “ {0})) |
df-cnfn 28090 | ⊢ ConFn = {𝑡 ∈ (ℂ ↑𝑚
ℋ) ∣ ∀𝑥
∈ ℋ ∀𝑦
∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ ℋ
((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 → (abs‘((𝑡‘𝑤) − (𝑡‘𝑥))) < 𝑦)} |
df-lnfn 28091 | ⊢ LinFn = {𝑡 ∈ (ℂ ↑𝑚
ℋ) ∣ ∀𝑥
∈ ℂ ∀𝑦
∈ ℋ ∀𝑧
∈ ℋ (𝑡‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (𝑡‘𝑦)) + (𝑡‘𝑧))} |
df-adjh 28092 | ⊢ adjℎ = {〈𝑡, 𝑢〉 ∣ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧
∀𝑥 ∈ ℋ
∀𝑦 ∈ ℋ
((𝑡‘𝑥)
·ih 𝑦) = (𝑥 ·ih (𝑢‘𝑦)))} |
df-bra 28093 | ⊢ bra = (𝑥 ∈ ℋ ↦ (𝑦 ∈ ℋ ↦ (𝑦 ·ih 𝑥))) |
df-kb 28094 | ⊢ ketbra = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑧 ∈ ℋ ↦ ((𝑧 ·ih 𝑦)
·ℎ 𝑥))) |
df-leop 28095 | ⊢ ≤op = {〈𝑡, 𝑢〉 ∣ ((𝑢 −op 𝑡) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢 −op 𝑡)‘𝑥) ·ih 𝑥))} |
df-eigvec 28096 | ⊢ eigvec = (𝑡 ∈ ( ℋ ↑𝑚
ℋ) ↦ {𝑥 ∈
( ℋ ∖ 0ℋ) ∣ ∃𝑧 ∈ ℂ (𝑡‘𝑥) = (𝑧 ·ℎ 𝑥)}) |
df-eigval 28097 | ⊢ eigval = (𝑡 ∈ ( ℋ ↑𝑚
ℋ) ↦ (𝑥 ∈
(eigvec‘𝑡) ↦
(((𝑡‘𝑥)
·ih 𝑥) / ((normℎ‘𝑥)↑2)))) |
df-spec 28098 | ⊢ Lambda = (𝑡 ∈ ( ℋ ↑𝑚
ℋ) ↦ {𝑥 ∈
ℂ ∣ ¬ (𝑡
−op (𝑥
·op ( I ↾ ℋ))): ℋ–1-1→ ℋ}) |
df-st 28454 | ⊢ States = {𝑓 ∈ ((0[,]1) ↑𝑚
Cℋ ) ∣ ((𝑓‘ ℋ) = 1 ∧ ∀𝑥 ∈
Cℋ ∀𝑦 ∈ Cℋ
(𝑥 ⊆
(⊥‘𝑦) →
(𝑓‘(𝑥 ∨ℋ 𝑦)) = ((𝑓‘𝑥) + (𝑓‘𝑦))))} |
df-hst 28455 | ⊢ CHStates = {𝑓 ∈ ( ℋ ↑𝑚
Cℋ ) ∣
((normℎ‘(𝑓‘ ℋ)) = 1 ∧ ∀𝑥 ∈
Cℋ ∀𝑦 ∈ Cℋ
(𝑥 ⊆
(⊥‘𝑦) →
(((𝑓‘𝑥)
·ih (𝑓‘𝑦)) = 0 ∧ (𝑓‘(𝑥 ∨ℋ 𝑦)) = ((𝑓‘𝑥) +ℎ (𝑓‘𝑦)))))} |
df-cv 28522 | ⊢ ⋖ℋ =
{〈𝑥, 𝑦〉 ∣ ((𝑥 ∈
Cℋ ∧ 𝑦 ∈ Cℋ )
∧ (𝑥 ⊊ 𝑦 ∧ ¬ ∃𝑧 ∈
Cℋ (𝑥 ⊊ 𝑧 ∧ 𝑧 ⊊ 𝑦)))} |
df-md 28523 | ⊢ 𝑀ℋ =
{〈𝑥, 𝑦〉 ∣ ((𝑥 ∈
Cℋ ∧ 𝑦 ∈ Cℋ )
∧ ∀𝑧 ∈
Cℋ (𝑧 ⊆ 𝑦 → ((𝑧 ∨ℋ 𝑥) ∩ 𝑦) = (𝑧 ∨ℋ (𝑥 ∩ 𝑦))))} |
df-dmd 28524 | ⊢ 𝑀ℋ* =
{〈𝑥, 𝑦〉 ∣ ((𝑥 ∈
Cℋ ∧ 𝑦 ∈ Cℋ )
∧ ∀𝑧 ∈
Cℋ (𝑦 ⊆ 𝑧 → ((𝑧 ∩ 𝑥) ∨ℋ 𝑦) = (𝑧 ∩ (𝑥 ∨ℋ 𝑦))))} |
df-at 28581 | ⊢ HAtoms = {𝑥 ∈ Cℋ
∣ 0ℋ ⋖ℋ 𝑥} |
The
list of syntax, axioms (ax-) and definitions (df-) for the User Mathboxes starts here |
cxdiv 28956 | class
/𝑒 |
df-xdiv 28957 | ⊢ /𝑒 = (𝑥 ∈ ℝ*, 𝑦 ∈ (ℝ ∖ {0})
↦ (℩𝑧
∈ ℝ* (𝑦 ·e 𝑧) = 𝑥)) |
ax-xrssca 29004 | ⊢ ℝfld =
(Scalar‘ℝ*𝑠) |
ax-xrsvsca 29005 | ⊢ ·e = (
·𝑠
‘ℝ*𝑠) |
comnd 29028 | class oMnd |
cogrp 29029 | class oGrp |
df-omnd 29030 | ⊢ oMnd = {𝑔 ∈ Mnd ∣ [(Base‘𝑔) / 𝑣][(+g‘𝑔) / 𝑝][(le‘𝑔) / 𝑙](𝑔 ∈ Toset ∧ ∀𝑎 ∈ 𝑣 ∀𝑏 ∈ 𝑣 ∀𝑐 ∈ 𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)))} |
df-ogrp 29031 | ⊢ oGrp = (Grp ∩ oMnd) |
csgns 29056 | class sgns |
df-sgns 29057 | ⊢ sgns = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟) ↦ if(𝑥 = (0g‘𝑟), 0, if((0g‘𝑟)(lt‘𝑟)𝑥, 1, -1)))) |
cinftm 29061 | class ⋘ |
carchi 29062 | class Archi |
df-inftm 29063 | ⊢ ⋘ = (𝑤 ∈ V ↦ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ∧ ((0g‘𝑤)(lt‘𝑤)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g‘𝑤)𝑥)(lt‘𝑤)𝑦))}) |
df-archi 29064 | ⊢ Archi = {𝑤 ∣ (⋘‘𝑤) = ∅} |
cslmd 29084 | class SLMod |
df-slmd 29085 | ⊢ SLMod = {𝑔 ∈ CMnd ∣
[(Base‘𝑔) /
𝑣][(+g‘𝑔) / 𝑎][(
·𝑠 ‘𝑔) / 𝑠][(Scalar‘𝑔) / 𝑓][(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ SRing ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤 ∧ ((0g‘𝑓)𝑠𝑤) = (0g‘𝑔))))} |
corng 29126 | class oRing |
cofld 29127 | class oField |
df-orng 29128 | ⊢ oRing = {𝑟 ∈ (Ring ∩ oGrp) ∣
[(Base‘𝑟) /
𝑣][(0g‘𝑟) / 𝑧][(.r‘𝑟) / 𝑡][(le‘𝑟) / 𝑙]∀𝑎 ∈ 𝑣 ∀𝑏 ∈ 𝑣 ((𝑧𝑙𝑎 ∧ 𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏))} |
df-ofld 29129 | ⊢ oField = (Field ∩ oRing) |
cresv 29155 | class
↾v |
df-resv 29156 | ⊢ ↾v = (𝑤 ∈ V, 𝑥 ∈ V ↦
if((Base‘(Scalar‘𝑤)) ⊆ 𝑥, 𝑤, (𝑤 sSet 〈(Scalar‘ndx),
((Scalar‘𝑤)
↾s 𝑥)〉))) |
csmat 29187 | class subMat1 |
df-smat 29188 | ⊢ subMat1 = (𝑚 ∈ V ↦ (𝑘 ∈ ℕ, 𝑙 ∈ ℕ ↦ (𝑚 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ 〈if(𝑖 < 𝑘, 𝑖, (𝑖 + 1)), if(𝑗 < 𝑙, 𝑗, (𝑗 + 1))〉)))) |
clmat 29205 | class litMat |
df-lmat 29206 | ⊢ litMat = (𝑚 ∈ V ↦ (𝑖 ∈ (1...(#‘𝑚)), 𝑗 ∈ (1...(#‘(𝑚‘0))) ↦ ((𝑚‘(𝑖 − 1))‘(𝑗 − 1)))) |
ccref 29237 | class CovHasRef𝐴 |
df-cref 29238 | ⊢ CovHasRef𝐴 = {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗(∪
𝑗 = ∪ 𝑦
→ ∃𝑧 ∈
(𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦)} |
cldlf 29247 | class Ldlf |
df-ldlf 29248 | ⊢ Ldlf = CovHasRef{𝑥 ∣ 𝑥 ≼ ω} |
cpcmp 29250 | class Paracomp |
df-pcmp 29251 | ⊢ Paracomp = {𝑗 ∣ 𝑗 ∈ CovHasRef(LocFin‘𝑗)} |
cmetid 29257 | class ~Met |
cpstm 29258 | class pstoMet |
df-metid 29259 | ⊢ ~Met = (𝑑 ∈ ∪ ran
PsMet ↦ {〈𝑥,
𝑦〉 ∣ ((𝑥 ∈ dom dom 𝑑 ∧ 𝑦 ∈ dom dom 𝑑) ∧ (𝑥𝑑𝑦) = 0)}) |
df-pstm 29260 | ⊢ pstoMet = (𝑑 ∈ ∪ ran
PsMet ↦ (𝑎 ∈
(dom dom 𝑑 /
(~Met‘𝑑)),
𝑏 ∈ (dom dom 𝑑 /
(~Met‘𝑑))
↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝑑𝑦)})) |
chcmp 29330 | class HCmp |
df-hcmp 29331 | ⊢ HCmp = {〈𝑢, 𝑤〉 ∣ ((𝑢 ∈ ∪ ran
UnifOn ∧ 𝑤 ∈
CUnifSp) ∧ ((UnifSt‘𝑤) ↾t dom ∪ 𝑢) =
𝑢 ∧
((cls‘(TopOpen‘𝑤))‘dom ∪
𝑢) = (Base‘𝑤))} |
cqqh 29344 | class ℚHom |
df-qqh 29345 | ⊢ ℚHom = (𝑟 ∈ V ↦ ran (𝑥 ∈ ℤ, 𝑦 ∈ (◡(ℤRHom‘𝑟) “ (Unit‘𝑟)) ↦ 〈(𝑥 / 𝑦), (((ℤRHom‘𝑟)‘𝑥)(/r‘𝑟)((ℤRHom‘𝑟)‘𝑦))〉)) |
crrh 29365 | class ℝHom |
crrext 29366 | class ℝExt |
df-rrh 29367 | ⊢ ℝHom = (𝑟 ∈ V ↦ (((topGen‘ran
(,))CnExt(TopOpen‘𝑟))‘(ℚHom‘𝑟))) |
df-rrext 29371 | ⊢ ℝExt = {𝑟 ∈ (NrmRing ∩ DivRing) ∣
(((ℤMod‘𝑟)
∈ NrmMod ∧ (chr‘𝑟) = 0) ∧ (𝑟 ∈ CUnifSp ∧ (UnifSt‘𝑟) =
(metUnif‘((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟))))))} |
cxrh 29388 | class
ℝ*Hom |
df-xrh 29389 | ⊢ ℝ*Hom = (𝑟 ∈ V ↦ (𝑥 ∈ ℝ* ↦ if(𝑥 ∈ ℝ,
((ℝHom‘𝑟)‘𝑥), if(𝑥 = +∞, ((lub‘𝑟)‘((ℝHom‘𝑟) “ ℝ)), ((glb‘𝑟)‘((ℝHom‘𝑟) “
ℝ)))))) |
cmntop 29394 | class ManTop |
df-mntop 29395 | ⊢ ManTop = {〈𝑛, 𝑗〉 ∣ (𝑛 ∈ ℕ0 ∧ (𝑗 ∈ 2nd𝜔
∧ 𝑗 ∈ Haus ∧
𝑗 ∈ Locally
[(TopOpen‘(𝔼hil‘𝑛))] ≃ ))} |
cind 29400 | class 𝟭 |
df-ind 29401 | ⊢ 𝟭 = (𝑜 ∈ V ↦ (𝑎 ∈ 𝒫 𝑜 ↦ (𝑥 ∈ 𝑜 ↦ if(𝑥 ∈ 𝑎, 1, 0)))) |
cesum 29416 | class Σ*𝑘 ∈ 𝐴𝐵 |
df-esum 29417 | ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪
((ℝ*𝑠 ↾s (0[,]+∞))
tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) |
cofc 29484 | class
∘𝑓/𝑐𝑅 |
df-ofc 29485 | ⊢ ∘𝑓/𝑐𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐))) |
csiga 29497 | class sigAlgebra |
df-siga 29498 | ⊢ sigAlgebra = (𝑜 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑜 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → ∪ 𝑥
∈ 𝑠)))}) |
csigagen 29528 | class sigaGen |
df-sigagen 29529 | ⊢ sigaGen = (𝑥 ∈ V ↦ ∩ {𝑠
∈ (sigAlgebra‘∪ 𝑥) ∣ 𝑥 ⊆ 𝑠}) |
cbrsiga 29571 | class
𝔅ℝ |
df-brsiga 29572 | ⊢ 𝔅ℝ =
(sigaGen‘(topGen‘ran (,))) |
csx 29578 | class
×s |
df-sx 29579 | ⊢ ×s = (𝑠 ∈ V, 𝑡 ∈ V ↦ (sigaGen‘ran (𝑥 ∈ 𝑠, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦)))) |
cmeas 29585 | class measures |
df-meas 29586 | ⊢ measures = (𝑠 ∈ ∪ ran
sigAlgebra ↦ {𝑚
∣ (𝑚:𝑠⟶(0[,]+∞) ∧
(𝑚‘∅) = 0 ∧
∀𝑥 ∈ 𝒫
𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑚‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑚‘𝑦)))}) |
cdde 29622 | class δ |
df-dde 29623 | ⊢ δ = (𝑎 ∈ 𝒫 ℝ ↦ if(0 ∈
𝑎, 1, 0)) |
cae 29627 | class a.e. |
cfae 29628 | class ~ a.e. |
df-ae 29629 | ⊢ a.e. = {〈𝑎, 𝑚〉 ∣ (𝑚‘(∪ dom
𝑚 ∖ 𝑎)) = 0} |
df-fae 29635 | ⊢ ~ a.e. = (𝑟 ∈ V, 𝑚 ∈ ∪ ran
measures ↦ {〈𝑓,
𝑔〉 ∣ ((𝑓 ∈ (dom 𝑟 ↑𝑚 ∪ dom 𝑚) ∧ 𝑔 ∈ (dom 𝑟 ↑𝑚 ∪ dom 𝑚)) ∧ {𝑥 ∈ ∪ dom
𝑚 ∣ (𝑓‘𝑥)𝑟(𝑔‘𝑥)}a.e.𝑚)}) |
cmbfm 29639 | class MblFnM |
df-mbfm 29640 | ⊢ MblFnM = (𝑠 ∈ ∪ ran
sigAlgebra, 𝑡 ∈ ∪ ran sigAlgebra ↦ {𝑓 ∈ (∪ 𝑡 ↑𝑚
∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠}) |
coms 29680 | class toOMeas |
df-oms 29681 | ⊢ toOMeas = (𝑟 ∈ V ↦ (𝑎 ∈ 𝒫 ∪ dom 𝑟 ↦ inf(ran (𝑥 ∈ {𝑧 ∈ 𝒫 dom 𝑟 ∣ (𝑎 ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω)} ↦
Σ*𝑦 ∈
𝑥(𝑟‘𝑦)), (0[,]+∞), < ))) |
ccarsg 29690 | class toCaraSiga |
df-carsg 29691 | ⊢ toCaraSiga = (𝑚 ∈ V ↦ {𝑎 ∈ 𝒫 ∪ dom 𝑚 ∣ ∀𝑒 ∈ 𝒫 ∪ dom 𝑚((𝑚‘(𝑒 ∩ 𝑎)) +𝑒 (𝑚‘(𝑒 ∖ 𝑎))) = (𝑚‘𝑒)}) |
citgm 29716 | class itgm |
csitm 29717 | class sitm |
csitg 29718 | class sitg |
df-sitg 29719 | ⊢ sitg = (𝑤 ∈ V, 𝑚 ∈ ∪ ran
measures ↦ (𝑓 ∈
{𝑔 ∈ (dom 𝑚MblFnM(sigaGen‘(TopOpen‘𝑤))) ∣ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ {(0g‘𝑤)})(𝑚‘(◡𝑔 “ {𝑥})) ∈ (0[,)+∞))} ↦ (𝑤 Σg
(𝑥 ∈ (ran 𝑓 ∖
{(0g‘𝑤)})
↦ (((ℝHom‘(Scalar‘𝑤))‘(𝑚‘(◡𝑓 “ {𝑥})))( ·𝑠
‘𝑤)𝑥))))) |
df-sitm 29720 | ⊢ sitm = (𝑤 ∈ V, 𝑚 ∈ ∪ ran
measures ↦ (𝑓 ∈
dom (𝑤sitg𝑚), 𝑔 ∈ dom (𝑤sitg𝑚) ↦
(((ℝ*𝑠 ↾s
(0[,]+∞))sitg𝑚)‘(𝑓 ∘𝑓
(dist‘𝑤)𝑔)))) |
df-itgm 29742 | ⊢ itgm = (𝑤 ∈ V, 𝑚 ∈ ∪ ran
measures ↦ (((metUnif‘(𝑤sitm𝑚))CnExt(UnifSt‘𝑤))‘(𝑤sitg𝑚))) |
csseq 29772 | class
seqstr |
df-sseq 29773 | ⊢ seqstr = (𝑚 ∈ V, 𝑓 ∈ V ↦ (𝑚 ∪ ( lastS ∘ seq(#‘𝑚)((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ++ 〈“(𝑓‘𝑥)”〉)), (ℕ0
× {(𝑚 ++
〈“(𝑓‘𝑚)”〉)}))))) |
cfib 29785 | class Fibci |
df-fib 29786 | ⊢ Fibci =
(〈“01”〉seqstr(𝑤 ∈ (Word ℕ0 ∩
(◡# “
(ℤ≥‘2))) ↦ ((𝑤‘((#‘𝑤) − 2)) + (𝑤‘((#‘𝑤) − 1))))) |
cprb 29796 | class Prob |
df-prob 29797 | ⊢ Prob = {𝑝 ∈ ∪ ran
measures ∣ (𝑝‘∪ dom 𝑝) = 1} |
ccprob 29820 | class cprob |
df-cndprob 29821 | ⊢ cprob = (𝑝 ∈ Prob ↦ (𝑎 ∈ dom 𝑝, 𝑏 ∈ dom 𝑝 ↦ ((𝑝‘(𝑎 ∩ 𝑏)) / (𝑝‘𝑏)))) |
crrv 29829 | class rRndVar |
df-rrv 29830 | ⊢ rRndVar = (𝑝 ∈ Prob ↦ (dom 𝑝MblFnM𝔅ℝ)) |
corvc 29844 | class
∘RV/𝑐𝑅 |
df-orvc 29845 | ⊢ ∘RV/𝑐𝑅 = (𝑥 ∈ {𝑥 ∣ Fun 𝑥}, 𝑎 ∈ V ↦ (◡𝑥 “ {𝑦 ∣ 𝑦𝑅𝑎})) |
cstrkg2d 29995 | class
TarskiG2D |
df-trkg2d 29996 | ⊢ TarskiG2D = {𝑓 ∣ [(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑][(Itv‘𝑓) / 𝑖](∃𝑥 ∈ 𝑝 ∃𝑦 ∈ 𝑝 ∃𝑧 ∈ 𝑝 ¬ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧)) ∧ ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ∀𝑢 ∈ 𝑝 ∀𝑣 ∈ 𝑝 ((((𝑥𝑑𝑢) = (𝑥𝑑𝑣) ∧ (𝑦𝑑𝑢) = (𝑦𝑑𝑣) ∧ (𝑧𝑑𝑢) = (𝑧𝑑𝑣)) ∧ 𝑢 ≠ 𝑣) → (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))))} |
cafs 30000 | class AFS |
df-afs 30001 | ⊢ AFS = (𝑔 ∈ TarskiG ↦ {〈𝑒, 𝑓〉 ∣ [(Base‘𝑔) / 𝑝][(dist‘𝑔) / ℎ][(Itv‘𝑔) / 𝑖]∃𝑎 ∈ 𝑝 ∃𝑏 ∈ 𝑝 ∃𝑐 ∈ 𝑝 ∃𝑑 ∈ 𝑝 ∃𝑥 ∈ 𝑝 ∃𝑦 ∈ 𝑝 ∃𝑧 ∈ 𝑝 ∃𝑤 ∈ 𝑝 (𝑒 = 〈〈𝑎, 𝑏〉, 〈𝑐, 𝑑〉〉 ∧ 𝑓 = 〈〈𝑥, 𝑦〉, 〈𝑧, 𝑤〉〉 ∧ ((𝑏 ∈ (𝑎𝑖𝑐) ∧ 𝑦 ∈ (𝑥𝑖𝑧)) ∧ ((𝑎ℎ𝑏) = (𝑥ℎ𝑦) ∧ (𝑏ℎ𝑐) = (𝑦ℎ𝑧)) ∧ ((𝑎ℎ𝑑) = (𝑥ℎ𝑤) ∧ (𝑏ℎ𝑑) = (𝑦ℎ𝑤))))}) |
w-bnj17 30005 | wff (𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) |
df-bnj17 30006 | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃)) |
c-bnj14 30007 | class pred(𝑋, 𝐴, 𝑅) |
df-bnj14 30008 | ⊢ pred(𝑋, 𝐴, 𝑅) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋} |
w-bnj13 30009 | wff 𝑅 Se 𝐴 |
df-bnj13 30010 | ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V) |
w-bnj15 30011 | wff 𝑅 FrSe 𝐴 |
df-bnj15 30012 | ⊢ (𝑅 FrSe 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴)) |
c-bnj18 30013 | class trCl(𝑋, 𝐴, 𝑅) |
df-bnj18 30014 | ⊢ trCl(𝑋, 𝐴, 𝑅) = ∪
𝑓 ∈ {𝑓 ∣ ∃𝑛 ∈ (ω ∖
{∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))}∪
𝑖 ∈ dom 𝑓(𝑓‘𝑖) |
w-bnj19 30015 | wff TrFo(𝐵, 𝐴, 𝑅) |
df-bnj19 30016 | ⊢ ( TrFo(𝐵, 𝐴, 𝑅) ↔ ∀𝑥 ∈ 𝐵 pred(𝑥, 𝐴, 𝑅) ⊆ 𝐵) |
ax-7d 30395 | ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) |
ax-8d 30396 | ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧)) |
ax-9d1 30397 | ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑥 |
ax-9d2 30398 | ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦 |
ax-10d 30399 | ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) |
ax-11d 30400 | ⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
cretr 30453 | class Retr |
df-retr 30454 | ⊢ Retr = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑟 ∈ (𝑗 Cn 𝑘) ∣ ∃𝑠 ∈ (𝑘 Cn 𝑗)((𝑟 ∘ 𝑠)(𝑗 Htpy 𝑗)( I ↾ ∪
𝑗)) ≠
∅}) |
cpcon 30455 | class PCon |
cscon 30456 | class SCon |
df-pcon 30457 | ⊢ PCon = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗∃𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)} |
df-scon 30458 | ⊢ SCon = {𝑗 ∈ PCon ∣ ∀𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝑗)((0[,]1) × {(𝑓‘0)}))} |
ccvm 30491 | class CovMap |
df-cvm 30492 | ⊢ CovMap = (𝑐 ∈ Top, 𝑗 ∈ Top ↦ {𝑓 ∈ (𝑐 Cn 𝑗) ∣ ∀𝑥 ∈ ∪ 𝑗∃𝑘 ∈ 𝑗 (𝑥 ∈ 𝑘 ∧ ∃𝑠 ∈ (𝒫 𝑐 ∖ {∅})(∪ 𝑠 =
(◡𝑓 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝑓 ↾ 𝑢) ∈ ((𝑐 ↾t 𝑢)Homeo(𝑗 ↾t 𝑘)))))}) |
cgoe 30569 | class
∈𝑔 |
cgna 30570 | class
⊼𝑔 |
cgol 30571 | class
∀𝑔𝑁𝑈 |
csat 30572 | class Sat |
cfmla 30573 | class Fmla |
csate 30574 | class
Sat∈ |
cprv 30575 | class ⊧ |
df-goel 30576 | ⊢ ∈𝑔 = (𝑥 ∈ (ω ×
ω) ↦ 〈∅, 𝑥〉) |
df-gona 30577 | ⊢ ⊼𝑔 = (𝑥 ∈ (V × V) ↦
〈1𝑜, 𝑥〉) |
df-goal 30578 | ⊢ ∀𝑔𝑁𝑈 = 〈2𝑜, 〈𝑁, 𝑈〉〉 |
df-sat 30579 | ⊢ Sat = (𝑚 ∈ V, 𝑒 ∈ V ↦ (rec((𝑓 ∈ V ↦ (𝑓 ∪ {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑚 ↑𝑚 ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑𝑚 ω) ∣
∀𝑧 ∈ 𝑚 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {〈𝑥, 𝑦〉 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑𝑚 ω) ∣
(𝑎‘𝑖)𝑒(𝑎‘𝑗)})}) ↾ suc ω)) |
df-sate 30580 | ⊢ Sat∈ = (𝑚 ∈ V, 𝑢 ∈ V ↦ (((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω)‘𝑢)) |
df-fmla 30581 | ⊢ Fmla = (𝑛 ∈ suc ω ↦ dom ((∅ Sat
∅)‘𝑛)) |
cgon 30582 | class ¬𝑔𝑈 |
cgoa 30583 | class
∧𝑔 |
cgoi 30584 | class
→𝑔 |
cgoo 30585 | class
∨𝑔 |
cgob 30586 | class
↔𝑔 |
cgoq 30587 | class
=𝑔 |
cgox 30588 | class ∃𝑔𝑁𝑈 |
df-gonot 30589 | ⊢ ¬𝑔𝑈 = (𝑈⊼𝑔𝑈) |
df-goan 30590 | ⊢ ∧𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦
¬𝑔(𝑢⊼𝑔𝑣)) |
df-goim 30591 | ⊢ →𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ (𝑢⊼𝑔¬𝑔𝑣)) |
df-goor 30592 | ⊢ ∨𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦
(¬𝑔𝑢
→𝑔 𝑣)) |
df-gobi 30593 | ⊢ ↔𝑔 = (𝑢 ∈ V, 𝑣 ∈ V ↦ ((𝑢 →𝑔 𝑣)∧𝑔(𝑣 →𝑔
𝑢))) |
df-goeq 30594 | ⊢ =𝑔 = (𝑢 ∈ ω, 𝑣 ∈ ω ↦ ⦋suc
(𝑢 ∪ 𝑣) / 𝑤⦌∀𝑔𝑤((𝑤∈𝑔𝑢) ↔𝑔 (𝑤∈𝑔𝑣))) |
df-goex 30595 | ⊢ ∃𝑔𝑁𝑈 =
¬𝑔∀𝑔𝑁¬𝑔𝑈 |
df-prv 30596 | ⊢ ⊧ = {〈𝑚, 𝑢〉 ∣ (𝑚 Sat∈ 𝑢) = (𝑚 ↑𝑚
ω)} |
cgze 30597 | class AxExt |
cgzr 30598 | class AxRep |
cgzp 30599 | class AxPow |
cgzu 30600 | class AxUn |
cgzg 30601 | class AxReg |
cgzi 30602 | class AxInf |
cgzf 30603 | class ZF |
df-gzext 30604 | ⊢ AxExt =
(∀𝑔2𝑜((2𝑜∈𝑔∅)
↔𝑔 (2𝑜∈𝑔1𝑜))
→𝑔 (∅=𝑔1𝑜)) |
df-gzrep 30605 | ⊢ AxRep = (𝑢 ∈ (Fmla‘ω) ↦
(∀𝑔3𝑜∃𝑔1𝑜∀𝑔2𝑜(∀𝑔1𝑜𝑢 →𝑔 (2𝑜=𝑔1𝑜)) →𝑔
∀𝑔1𝑜∀𝑔2𝑜((2𝑜∈𝑔1𝑜) ↔𝑔
∃𝑔3𝑜((3𝑜∈𝑔∅)∧𝑔∀𝑔1𝑜𝑢)))) |
df-gzpow 30606 | ⊢ AxPow =
∃𝑔1𝑜∀𝑔2𝑜(∀𝑔1𝑜((1𝑜∈𝑔2𝑜)
↔𝑔 (1𝑜∈𝑔∅)) →𝑔 (2𝑜∈𝑔1𝑜)) |
df-gzun 30607 | ⊢ AxUn =
∃𝑔1𝑜∀𝑔2𝑜(∃𝑔1𝑜((2𝑜∈𝑔1𝑜)∧𝑔(1𝑜∈𝑔∅))
→𝑔 (2𝑜∈𝑔1𝑜)) |
df-gzreg 30608 | ⊢ AxReg =
(∃𝑔1𝑜(1𝑜∈𝑔∅)
→𝑔
∃𝑔1𝑜((1𝑜∈𝑔∅)∧𝑔∀𝑔2𝑜((2𝑜∈𝑔1𝑜)
→𝑔 ¬𝑔(2𝑜∈𝑔∅)))) |
df-gzinf 30609 | ⊢ AxInf =
∃𝑔1𝑜((∅∈𝑔1𝑜)∧𝑔∀𝑔2𝑜((2𝑜∈𝑔1𝑜)
→𝑔 ∃𝑔∅((2𝑜∈𝑔∅)∧𝑔(∅∈𝑔1𝑜)))) |
df-gzf 30610 | ⊢ ZF =
{𝑚 ∣ ((Tr 𝑚 ∧ 𝑚⊧AxExt ∧ 𝑚⊧AxPow) ∧ (𝑚⊧AxUn ∧ 𝑚⊧AxReg ∧ 𝑚⊧AxInf) ∧ ∀𝑢 ∈
(Fmla‘ω)𝑚⊧(AxRep‘𝑢))} |
cmcn 30611 | class mCN |
cmvar 30612 | class mVR |
cmty 30613 | class mType |
cmvt 30614 | class mVT |
cmtc 30615 | class mTC |
cmax 30616 | class mAx |
cmrex 30617 | class mREx |
cmex 30618 | class mEx |
cmdv 30619 | class mDV |
cmvrs 30620 | class mVars |
cmrsub 30621 | class mRSubst |
cmsub 30622 | class mSubst |
cmvh 30623 | class mVH |
cmpst 30624 | class mPreSt |
cmsr 30625 | class mStRed |
cmsta 30626 | class mStat |
cmfs 30627 | class mFS |
cmcls 30628 | class mCls |
cmpps 30629 | class mPPSt |
cmthm 30630 | class mThm |
df-mcn 30631 | ⊢ mCN = Slot 1 |
df-mvar 30632 | ⊢ mVR = Slot 2 |
df-mty 30633 | ⊢ mType = Slot 3 |
df-mtc 30634 | ⊢ mTC = Slot 4 |
df-mmax 30635 | ⊢ mAx = Slot 5 |
df-mvt 30636 | ⊢ mVT = (𝑡 ∈ V ↦ ran (mType‘𝑡)) |
df-mrex 30637 | ⊢ mREx = (𝑡 ∈ V ↦ Word ((mCN‘𝑡) ∪ (mVR‘𝑡))) |
df-mex 30638 | ⊢ mEx = (𝑡 ∈ V ↦ ((mTC‘𝑡) × (mREx‘𝑡))) |
df-mdv 30639 | ⊢ mDV = (𝑡 ∈ V ↦ (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I )) |
df-mvrs 30640 | ⊢ mVars = (𝑡 ∈ V ↦ (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2nd ‘𝑒) ∩ (mVR‘𝑡)))) |
df-mrsub 30641 | ⊢ mRSubst = (𝑡 ∈ V ↦ (𝑓 ∈ ((mREx‘𝑡) ↑pm (mVR‘𝑡)) ↦ (𝑒 ∈ (mREx‘𝑡) ↦ ((freeMnd‘((mCN‘𝑡) ∪ (mVR‘𝑡))) Σg
((𝑣 ∈
((mCN‘𝑡) ∪
(mVR‘𝑡)) ↦
if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), 〈“𝑣”〉)) ∘ 𝑒))))) |
df-msub 30642 | ⊢ mSubst = (𝑡 ∈ V ↦ (𝑓 ∈ ((mREx‘𝑡) ↑pm (mVR‘𝑡)) ↦ (𝑒 ∈ (mEx‘𝑡) ↦ 〈(1st ‘𝑒), (((mRSubst‘𝑡)‘𝑓)‘(2nd ‘𝑒))〉))) |
df-mvh 30643 | ⊢ mVH = (𝑡 ∈ V ↦ (𝑣 ∈ (mVR‘𝑡) ↦ 〈((mType‘𝑡)‘𝑣), 〈“𝑣”〉〉)) |
df-mpst 30644 | ⊢ mPreSt = (𝑡 ∈ V ↦ (({𝑑 ∈ 𝒫 (mDV‘𝑡) ∣ ◡𝑑 = 𝑑} × (𝒫 (mEx‘𝑡) ∩ Fin)) ×
(mEx‘𝑡))) |
df-msr 30645 | ⊢ mStRed = (𝑡 ∈ V ↦ (𝑠 ∈ (mPreSt‘𝑡) ↦ ⦋(2nd
‘(1st ‘𝑠)) / ℎ⦌⦋(2nd
‘𝑠) / 𝑎⦌〈((1st
‘(1st ‘𝑠)) ∩ ⦋∪ ((mVars‘𝑡) “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎〉)) |
df-msta 30646 | ⊢ mStat = (𝑡 ∈ V ↦ ran (mStRed‘𝑡)) |
df-mfs 30647 | ⊢ mFS = {𝑡 ∣ ((((mCN‘𝑡) ∩ (mVR‘𝑡)) = ∅ ∧ (mType‘𝑡):(mVR‘𝑡)⟶(mTC‘𝑡)) ∧ ((mAx‘𝑡) ⊆ (mStat‘𝑡) ∧ ∀𝑣 ∈ (mVT‘𝑡) ¬ (◡(mType‘𝑡) “ {𝑣}) ∈ Fin))} |
df-mcls 30648 | ⊢ mCls = (𝑡 ∈ V ↦ (𝑑 ∈ 𝒫 (mDV‘𝑡), ℎ ∈ 𝒫 (mEx‘𝑡) ↦ ∩ {𝑐
∣ ((ℎ ∪ ran
(mVH‘𝑡)) ⊆
𝑐 ∧ ∀𝑚∀𝑜∀𝑝(〈𝑚, 𝑜, 𝑝〉 ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐)))})) |
df-mpps 30649 | ⊢ mPPSt = (𝑡 ∈ V ↦ {〈〈𝑑, ℎ〉, 𝑎〉 ∣ (〈𝑑, ℎ, 𝑎〉 ∈ (mPreSt‘𝑡) ∧ 𝑎 ∈ (𝑑(mCls‘𝑡)ℎ))}) |
df-mthm 30650 | ⊢ mThm = (𝑡 ∈ V ↦ (◡(mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡)))) |
cm0s 30736 | class m0St |
cmsa 30737 | class mSA |
cmwgfs 30738 | class mWGFS |
cmsy 30739 | class mSyn |
cmesy 30740 | class mESyn |
cmgfs 30741 | class mGFS |
cmtree 30742 | class mTree |
cmst 30743 | class mST |
cmsax 30744 | class mSAX |
cmufs 30745 | class mUFS |
df-m0s 30746 | ⊢ m0St = (𝑎 ∈ V ↦ 〈∅, ∅,
𝑎〉) |
df-msa 30747 | ⊢ mSA = (𝑡 ∈ V ↦ {𝑎 ∈ (mEx‘𝑡) ∣ ((m0St‘𝑎) ∈ (mAx‘𝑡) ∧ (1st ‘𝑎) ∈ (mVT‘𝑡) ∧ Fun (◡(2nd ‘𝑎) ↾ (mVR‘𝑡)))}) |
df-mwgfs 30748 | ⊢ mWGFS = {𝑡 ∈ mFS ∣ ∀𝑑∀ℎ∀𝑎((〈𝑑, ℎ, 𝑎〉 ∈ (mAx‘𝑡) ∧ (1st ‘𝑎) ∈ (mVT‘𝑡)) → ∃𝑠 ∈ ran (mSubst‘𝑡)𝑎 ∈ (𝑠 “ (mSA‘𝑡)))} |
df-msyn 30749 | ⊢ mSyn = Slot 6 |
df-mtree 30750 | ⊢ mTree = (𝑡 ∈ V ↦ (𝑑 ∈ 𝒫 (mDV‘𝑡), ℎ ∈ 𝒫 (mEx‘𝑡) ↦ ∩ {𝑟
∣ (∀𝑒 ∈
ran (mVH‘𝑡)𝑒𝑟〈(m0St‘𝑒), ∅〉 ∧ ∀𝑒 ∈ ℎ 𝑒𝑟〈((mStRed‘𝑡)‘〈𝑑, ℎ, 𝑒〉), ∅〉 ∧ ∀𝑚∀𝑜∀𝑝(〈𝑚, 𝑜, 𝑝〉 ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑) → ({(𝑠‘𝑝)} × X𝑒 ∈ (𝑜 ∪ ((mVH‘𝑡) “ ∪
((mVars‘𝑡) “
(𝑜 ∪ {𝑝}))))(𝑟 “ {(𝑠‘𝑒)})) ⊆ 𝑟)))})) |
df-mst 30751 | ⊢ mST = (𝑡 ∈ V ↦ ((∅(mTree‘𝑡)∅) ↾
((mEx‘𝑡) ↾
(mVT‘𝑡)))) |
df-msax 30752 | ⊢ mSAX = (𝑡 ∈ V ↦ (𝑝 ∈ (mSA‘𝑡) ↦ ((mVH‘𝑡) “ ((mVars‘𝑡)‘𝑝)))) |
df-mufs 30753 | ⊢ mUFS = {𝑡 ∈ mGFS ∣ Fun (mST‘𝑡)} |
cmuv 30754 | class mUV |
cmvl 30755 | class mVL |
cmvsb 30756 | class mVSubst |
cmfsh 30757 | class mFresh |
cmfr 30758 | class mFRel |
cmevl 30759 | class mEval |
cmdl 30760 | class mMdl |
cusyn 30761 | class mUSyn |
cgmdl 30762 | class mGMdl |
cmitp 30763 | class mItp |
cmfitp 30764 | class mFromItp |
df-muv 30765 | ⊢ mUV = Slot 7 |
df-mfsh 30766 | ⊢ mFresh = Slot 8 |
df-mevl 30767 | ⊢ mEval = Slot 9 |
df-mvl 30768 | ⊢ mVL = (𝑡 ∈ V ↦ X𝑣 ∈
(mVR‘𝑡)((mUV‘𝑡) “ {((mType‘𝑡)‘𝑣)})) |
df-mvsb 30769 | ⊢ mVSubst = (𝑡 ∈ V ↦ {〈〈𝑠, 𝑚〉, 𝑥〉 ∣ ((𝑠 ∈ ran (mSubst‘𝑡) ∧ 𝑚 ∈ (mVL‘𝑡)) ∧ ∀𝑣 ∈ (mVR‘𝑡)𝑚dom (mEval‘𝑡)(𝑠‘((mVH‘𝑡)‘𝑣)) ∧ 𝑥 = (𝑣 ∈ (mVR‘𝑡) ↦ (𝑚(mEval‘𝑡)(𝑠‘((mVH‘𝑡)‘𝑣)))))}) |
df-mfrel 30770 | ⊢ mFRel = (𝑡 ∈ V ↦ {𝑟 ∈ 𝒫 ((mUV‘𝑡) × (mUV‘𝑡)) ∣ (◡𝑟 = 𝑟 ∧ ∀𝑐 ∈ (mVT‘𝑡)∀𝑤 ∈ (𝒫 (mUV‘𝑡) ∩ Fin)∃𝑣 ∈ ((mUV‘𝑡) “ {𝑐})𝑤 ⊆ (𝑟 “ {𝑣}))}) |
df-mdl 30771 | ⊢ mMdl = {𝑡 ∈ mFS ∣ [(mUV‘𝑡) / 𝑢][(mEx‘𝑡) / 𝑥][(mVL‘𝑡) / 𝑣][(mEval‘𝑡) / 𝑛][(mFresh‘𝑡) / 𝑓]((𝑢 ⊆ ((mTC‘𝑡) × V) ∧ 𝑓 ∈ (mFRel‘𝑡) ∧ 𝑛 ∈ (𝑢 ↑pm (𝑣 × (mEx‘𝑡)))) ∧ ∀𝑚 ∈ 𝑣 ((∀𝑒 ∈ 𝑥 (𝑛 “ {〈𝑚, 𝑒〉}) ⊆ (𝑢 “ {(1st ‘𝑒)}) ∧ ∀𝑦 ∈ (mVR‘𝑡)〈𝑚, ((mVH‘𝑡)‘𝑦)〉𝑛(𝑚‘𝑦) ∧ ∀𝑑∀ℎ∀𝑎(〈𝑑, ℎ, 𝑎〉 ∈ (mAx‘𝑡) → ((∀𝑦∀𝑧(𝑦𝑑𝑧 → (𝑚‘𝑦)𝑓(𝑚‘𝑧)) ∧ ℎ ⊆ (dom 𝑛 “ {𝑚})) → 𝑚dom 𝑛 𝑎))) ∧ (∀𝑠 ∈ ran (mSubst‘𝑡)∀𝑒 ∈ (mEx‘𝑡)∀𝑦(〈𝑠, 𝑚〉(mVSubst‘𝑡)𝑦 → (𝑛 “ {〈𝑚, (𝑠‘𝑒)〉}) = (𝑛 “ {〈𝑦, 𝑒〉})) ∧ ∀𝑝 ∈ 𝑣 ∀𝑒 ∈ 𝑥 ((𝑚 ↾ ((mVars‘𝑡)‘𝑒)) = (𝑝 ↾ ((mVars‘𝑡)‘𝑒)) → (𝑛 “ {〈𝑚, 𝑒〉}) = (𝑛 “ {〈𝑝, 𝑒〉})) ∧ ∀𝑦 ∈ 𝑢 ∀𝑒 ∈ 𝑥 ((𝑚 “ ((mVars‘𝑡)‘𝑒)) ⊆ (𝑓 “ {𝑦}) → (𝑛 “ {〈𝑚, 𝑒〉}) ⊆ (𝑓 “ {𝑦})))))} |
df-musyn 30772 | ⊢ mUSyn = (𝑡 ∈ V ↦ (𝑣 ∈ (mUV‘𝑡) ↦ 〈((mSyn‘𝑡)‘(1st
‘𝑣)), (2nd
‘𝑣)〉)) |
df-gmdl 30773 | ⊢ mGMdl = {𝑡 ∈ (mGFS ∩ mMdl) ∣
(∀𝑐 ∈
(mTC‘𝑡)((mUV‘𝑡) “ {𝑐}) ⊆ ((mUV‘𝑡) “ {((mSyn‘𝑡)‘𝑐)}) ∧ ∀𝑣 ∈ (mUV‘𝑐)∀𝑤 ∈ (mUV‘𝑐)(𝑣(mFresh‘𝑡)𝑤 ↔ 𝑣(mFresh‘𝑡)((mUSyn‘𝑡)‘𝑤)) ∧ ∀𝑚 ∈ (mVL‘𝑡)∀𝑒 ∈ (mEx‘𝑡)((mEval‘𝑡) “ {〈𝑚, 𝑒〉}) = (((mEval‘𝑡) “ {〈𝑚, ((mESyn‘𝑡)‘𝑒)〉}) ∩ ((mUV‘𝑡) “ {(1st
‘𝑒)})))} |
df-mitp 30774 | ⊢ mItp = (𝑡 ∈ V ↦ (𝑎 ∈ (mSA‘𝑡) ↦ (𝑔 ∈ X𝑖 ∈ ((mVars‘𝑡)‘𝑎)((mUV‘𝑡) “ {((mType‘𝑡)‘𝑖)}) ↦ (℩𝑥∃𝑚 ∈ (mVL‘𝑡)(𝑔 = (𝑚 ↾ ((mVars‘𝑡)‘𝑎)) ∧ 𝑥 = (𝑚(mEval‘𝑡)𝑎)))))) |
df-mfitp 30775 | ⊢ mFromItp = (𝑡 ∈ V ↦ (𝑓 ∈ X𝑎 ∈ (mSA‘𝑡)(((mUV‘𝑡) “ {((1st ‘𝑡)‘𝑎)}) ↑𝑚 X𝑖 ∈
((mVars‘𝑡)‘𝑎)((mUV‘𝑡) “ {((mType‘𝑡)‘𝑖)})) ↦ (℩𝑛 ∈ ((mUV‘𝑡) ↑pm
((mVL‘𝑡) ×
(mEx‘𝑡)))∀𝑚 ∈ (mVL‘𝑡)(∀𝑣 ∈ (mVR‘𝑡)〈𝑚, ((mVH‘𝑡)‘𝑣)〉𝑛(𝑚‘𝑣) ∧ ∀𝑒∀𝑎∀𝑔(𝑒(mST‘𝑡)〈𝑎, 𝑔〉 → 〈𝑚, 𝑒〉𝑛(𝑓‘(𝑖 ∈ ((mVars‘𝑡)‘𝑎) ↦ (𝑚𝑛(𝑔‘((mVH‘𝑡)‘𝑖)))))) ∧ ∀𝑒 ∈ (mEx‘𝑡)(𝑛 “ {〈𝑚, 𝑒〉}) = ((𝑛 “ {〈𝑚, ((mESyn‘𝑡)‘𝑒)〉}) ∩ ((mUV‘𝑡) “ {(1st
‘𝑒)})))))) |
citr 30776 | class IntgRing |
ccpms 30777 | class cplMetSp |
chlb 30778 | class HomLimB |
chlim 30779 | class HomLim |
cpfl 30780 | class polyFld |
csf1 30781 | class
splitFld1 |
csf 30782 | class splitFld |
cpsl 30783 | class polySplitLim |
df-irng 30784 | ⊢ IntgRing = (𝑟 ∈ V, 𝑠 ∈ V ↦ ∪ 𝑓 ∈ (Monic1p‘(𝑟 ↾s 𝑠))(◡𝑓 “ {(0g‘𝑟)})) |
df-cplmet 30785 | ⊢ cplMetSp = (𝑤 ∈ V ↦ ⦋((𝑤 ↑s
ℕ) ↾s (Cau‘(dist‘𝑤))) / 𝑟⦌⦋(Base‘𝑟) / 𝑣⦌⦋{〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ (𝑓 ↾
(ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶((𝑔‘𝑗)(ball‘(dist‘𝑤))𝑥))} / 𝑒⦌((𝑟 /s 𝑒) sSet {〈(dist‘ndx),
{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ∃𝑝 ∈ 𝑣 ∃𝑞 ∈ 𝑣 ((𝑥 = [𝑝]𝑒 ∧ 𝑦 = [𝑞]𝑒) ∧ (𝑝 ∘𝑓 (dist‘𝑟)𝑞) ⇝ 𝑧)}〉})) |
df-homlimb 30786 | ⊢ HomLimB = (𝑓 ∈ V ↦ ⦋∪ 𝑛 ∈ ℕ ({𝑛} × dom (𝑓‘𝑛)) / 𝑣⦌⦋∩ {𝑠
∣ (𝑠 Er 𝑣 ∧ (𝑥 ∈ 𝑣 ↦ 〈((1st ‘𝑥) + 1), ((𝑓‘(1st ‘𝑥))‘(2nd
‘𝑥))〉) ⊆
𝑠)} / 𝑒⦌〈(𝑣 / 𝑒), (𝑛 ∈ ℕ ↦ (𝑥 ∈ dom (𝑓‘𝑛) ↦ [〈𝑛, 𝑥〉]𝑒))〉) |
df-homlim 30787 | ⊢ HomLim = (𝑟 ∈ V, 𝑓 ∈ V ↦ ⦋( HomLimB
‘𝑓) / 𝑒⦌⦋(1st
‘𝑒) / 𝑣⦌⦋(2nd
‘𝑒) / 𝑔⦌({〈(Base‘ndx),
𝑣〉,
〈(+g‘ndx), ∪ 𝑛 ∈ ℕ ran (𝑥 ∈ dom (𝑔‘𝑛), 𝑦 ∈ dom (𝑔‘𝑛) ↦ 〈〈((𝑔‘𝑛)‘𝑥), ((𝑔‘𝑛)‘𝑦)〉, ((𝑔‘𝑛)‘(𝑥(+g‘(𝑟‘𝑛))𝑦))〉)〉,
〈(.r‘ndx), ∪ 𝑛 ∈ ℕ ran (𝑥 ∈ dom (𝑔‘𝑛), 𝑦 ∈ dom (𝑔‘𝑛) ↦ 〈〈((𝑔‘𝑛)‘𝑥), ((𝑔‘𝑛)‘𝑦)〉, ((𝑔‘𝑛)‘(𝑥(.r‘(𝑟‘𝑛))𝑦))〉)〉} ∪
{〈(TopOpen‘ndx), {𝑠 ∈ 𝒫 𝑣 ∣ ∀𝑛 ∈ ℕ (◡(𝑔‘𝑛) “ 𝑠) ∈ (TopOpen‘(𝑟‘𝑛))}〉, 〈(dist‘ndx), ∪ 𝑛 ∈ ℕ ran (𝑥 ∈ dom ((𝑔‘𝑛)‘𝑛), 𝑦 ∈ dom ((𝑔‘𝑛)‘𝑛) ↦ 〈〈((𝑔‘𝑛)‘𝑥), ((𝑔‘𝑛)‘𝑦)〉, (𝑥(dist‘(𝑟‘𝑛))𝑦)〉)〉, 〈(le‘ndx),
∪ 𝑛 ∈ ℕ (◡(𝑔‘𝑛) ∘ ((le‘(𝑟‘𝑛)) ∘ (𝑔‘𝑛)))〉})) |
df-plfl 30788 | ⊢ polyFld = (𝑟 ∈ V, 𝑝 ∈ V ↦
⦋(Poly1‘𝑟) / 𝑠⦌⦋((RSpan‘𝑠)‘{𝑝}) / 𝑖⦌⦋(𝑧 ∈ (Base‘𝑟) ↦ [(𝑧( ·𝑠
‘𝑠)(1r‘𝑠))](𝑠 ~QG 𝑖)) / 𝑓⦌〈⦋(𝑠 /s (𝑠 ~QG 𝑖)) / 𝑡⦌((𝑡 toNrmGrp (℩𝑛 ∈ (AbsVal‘𝑡)(𝑛 ∘ 𝑓) = (norm‘𝑟))) sSet 〈(le‘ndx),
⦋(𝑧 ∈
(Base‘𝑡) ↦
(℩𝑞 ∈ 𝑧 (𝑟 deg1 𝑞) < (𝑟 deg1 𝑝))) / 𝑔⦌(◡𝑔 ∘ ((le‘𝑠) ∘ 𝑔))〉), 𝑓〉) |
df-sfl1 30789 | ⊢ splitFld1 = (𝑟 ∈ V, 𝑗 ∈ V ↦ (𝑝 ∈ (Poly1‘𝑟) ↦ (rec((𝑠 ∈ V, 𝑓 ∈ V ↦ ⦋( mPoly
‘𝑠) / 𝑚⦌⦋{𝑔 ∈
((Monic1p‘𝑠) ∩ (Irred‘𝑚)) ∣ (𝑔(∥r‘𝑚)(𝑝 ∘ 𝑓) ∧ 1 < (𝑠 deg1 𝑔))} / 𝑏⦌if(((𝑝 ∘ 𝑓) = (0g‘𝑚) ∨ 𝑏 = ∅), 〈𝑠, 𝑓〉, ⦋(glb‘𝑏) / ℎ⦌⦋(𝑠 polyFld ℎ) / 𝑡⦌〈(1st
‘𝑡), (𝑓 ∘ (2nd
‘𝑡))〉)), 𝑗)‘(card‘(1...(𝑟 deg1 𝑝)))))) |
df-sfl 30790 | ⊢ splitFld = (𝑟 ∈ V, 𝑝 ∈ V ↦ (℩𝑥∃𝑓(𝑓 Isom < , (lt‘𝑟)((1...(#‘𝑝)), 𝑝) ∧ 𝑥 = (seq0((𝑒 ∈ V, 𝑔 ∈ V ↦ ((𝑟 splitFld1 𝑒)‘𝑔)), (𝑓 ∪ {〈0, 〈𝑟, ( I ↾ (Base‘𝑟))〉〉}))‘(#‘𝑝))))) |
df-psl 30791 | ⊢ polySplitLim = (𝑟 ∈ V, 𝑝 ∈ ((𝒫 (Base‘𝑟) ∩ Fin)
↑𝑚 ℕ) ↦ ⦋(1st
∘ seq0((𝑔 ∈ V,
𝑞 ∈ V ↦
⦋(1st ‘𝑔) / 𝑒⦌⦋(1st
‘𝑒) / 𝑠⦌⦋(𝑠 splitFld ran (𝑥 ∈ 𝑞 ↦ (𝑥 ∘ (2nd ‘𝑔)))) / 𝑓⦌〈𝑓, ((2nd ‘𝑔) ∘ (2nd ‘𝑓))〉), (𝑝 ∪ {〈0, 〈〈𝑟, ∅〉, ( I ↾
(Base‘𝑟))〉〉}))) / 𝑓⦌((1st ∘
(𝑓 shift 1)) HomLim
(2nd ∘ 𝑓))) |
czr 30792 | class ZRing |
cgf 30793 | class GF |
cgfo 30794 | class
GF∞ |
ceqp 30795 | class ~Qp |
crqp 30796 | class /Qp |
cqp 30797 | class Qp |
cqpOLD 30798 | class QpOLD |
czp 30799 | class Zp |
cqpa 30800 | class _Qp |
ccp 30801 | class Cp |
df-zrng 30802 | ⊢ ZRing = (𝑟 ∈ V ↦ (𝑟 IntgRing ran (ℤRHom‘𝑟))) |
df-gf 30803 | ⊢ GF = (𝑝 ∈ ℙ, 𝑛 ∈ ℕ ↦
⦋(ℤ/nℤ‘𝑝) / 𝑟⦌(1st ‘(𝑟 splitFld
{⦋(Poly1‘𝑟) / 𝑠⦌⦋(var1‘𝑟) / 𝑥⦌(((𝑝↑𝑛)(.g‘(mulGrp‘𝑠))𝑥)(-g‘𝑠)𝑥)}))) |
df-gfoo 30804 | ⊢ GF∞ = (𝑝 ∈ ℙ ↦
⦋(ℤ/nℤ‘𝑝) / 𝑟⦌(𝑟 polySplitLim (𝑛 ∈ ℕ ↦
{⦋(Poly1‘𝑟) / 𝑠⦌⦋(var1‘𝑟) / 𝑥⦌(((𝑝↑𝑛)(.g‘(mulGrp‘𝑠))𝑥)(-g‘𝑠)𝑥)}))) |
df-eqp 30805 | ⊢ ~Qp = (𝑝 ∈ ℙ ↦ {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ (ℤ ↑𝑚
ℤ) ∧ ∀𝑛
∈ ℤ Σ𝑘
∈ (ℤ≥‘-𝑛)(((𝑓‘-𝑘) − (𝑔‘-𝑘)) / (𝑝↑(𝑘 + (𝑛 + 1)))) ∈ ℤ)}) |
df-rqp 30806 | ⊢ /Qp = (𝑝 ∈ ℙ ↦ (~Qp ∩
⦋{𝑓 ∈
(ℤ ↑𝑚 ℤ) ∣ ∃𝑥 ∈ ran ℤ≥(◡𝑓 “ (ℤ ∖ {0})) ⊆ 𝑥} / 𝑦⦌(𝑦 × (𝑦 ∩ (ℤ ↑𝑚
(0...(𝑝 −
1))))))) |
df-qp 30807 | ⊢ Qp = (𝑝 ∈ ℙ ↦ ⦋{ℎ ∈ (ℤ
↑𝑚 (0...(𝑝 − 1))) ∣ ∃𝑥 ∈ ran
ℤ≥(◡ℎ “ (ℤ ∖ {0}))
⊆ 𝑥} / 𝑏⦌(({〈(Base‘ndx),
𝑏〉,
〈(+g‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ ((/Qp‘𝑝)‘(𝑓 ∘𝑓 + 𝑔)))〉,
〈(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ ((/Qp‘𝑝)‘(𝑛 ∈ ℤ ↦ Σ𝑘 ∈ ℤ ((𝑓‘𝑘) · (𝑔‘(𝑛 − 𝑘))))))〉} ∪ {〈(le‘ndx),
{〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ 𝑏 ∧ Σ𝑘 ∈ ℤ ((𝑓‘-𝑘) · ((𝑝 + 1)↑-𝑘)) < Σ𝑘 ∈ ℤ ((𝑔‘-𝑘) · ((𝑝 + 1)↑-𝑘)))}〉}) toNrmGrp (𝑓 ∈ 𝑏 ↦ if(𝑓 = (ℤ × {0}), 0, (𝑝↑-inf((◡𝑓 “ (ℤ ∖ {0})), ℝ,
< )))))) |
df-qpOLD 30808 | ⊢ QpOLD = (𝑝 ∈ ℙ ↦ ⦋{ℎ ∈ (ℤ
↑𝑚 (0...(𝑝 − 1))) ∣ ∃𝑥 ∈ ran
ℤ≥(◡ℎ “ (ℤ ∖ {0}))
⊆ 𝑥} / 𝑏⦌(({〈(Base‘ndx),
𝑏〉,
〈(+g‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ ((/Qp‘𝑝)‘(𝑓 ∘𝑓 + 𝑔)))〉,
〈(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ ((/Qp‘𝑝)‘(𝑛 ∈ ℤ ↦ Σ𝑘 ∈ ℤ ((𝑓‘𝑘) · (𝑔‘(𝑛 − 𝑘))))))〉} ∪ {〈(le‘ndx),
{〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ 𝑏 ∧ Σ𝑘 ∈ ℤ ((𝑓‘-𝑘) · ((𝑝 + 1)↑-𝑘)) < Σ𝑘 ∈ ℤ ((𝑔‘-𝑘) · ((𝑝 + 1)↑-𝑘)))}〉}) toNrmGrp (𝑓 ∈ 𝑏 ↦ if(𝑓 = (ℤ × {0}), 0, (𝑝↑-sup((◡𝑓 “ (ℤ ∖ {0})), ℝ,
◡ < )))))) |
df-zp 30809 | ⊢ Zp = (ZRing ∘
Qp) |
df-qpa 30810 | ⊢ _Qp = (𝑝 ∈ ℙ ↦
⦋(Qp‘𝑝)
/ 𝑟⦌(𝑟 polySplitLim (𝑛 ∈ ℕ ↦ {𝑓 ∈
(Poly1‘𝑟)
∣ ((𝑟 deg1
𝑓) ≤ 𝑛 ∧ ∀𝑑 ∈ ran (coe1‘𝑓)(◡𝑑 “ (ℤ ∖ {0})) ⊆
(0...𝑛))}))) |
df-cp 30811 | ⊢ Cp = ( cplMetSp ∘
_Qp) |
ctrpred 30961 | class TrPred(𝑅, 𝐴, 𝑋) |
df-trpred 30962 | ⊢ TrPred(𝑅, 𝐴, 𝑋) = ∪ ran
(rec((𝑎 ∈ V ↦
∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) |
cwsuc 30996 | class wsuc(𝑅, 𝐴, 𝑋) |
cwsucOLD 30997 | class wsucOLD(𝑅, 𝐴, 𝑋) |
cwlim 30998 | class WLim(𝑅, 𝐴) |
cwlimOLD 30999 | class WLimOLD(𝑅, 𝐴) |
df-wsuc 31000 | ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) |
df-wsucOLD 31001 | ⊢ wsucOLD(𝑅, 𝐴, 𝑋) = sup(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, ◡𝑅) |
df-wlim 31002 | ⊢ WLim(𝑅, 𝐴) = {𝑥 ∈ 𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))} |
df-wlimOLD 31003 | ⊢ WLimOLD(𝑅, 𝐴) = {𝑥 ∈ 𝐴 ∣ (𝑥 ≠ sup(𝐴, 𝐴, ◡𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))} |
csur 31037 | class No
|
cslt 31038 | class <s |
cbday 31039 | class bday
|
df-no 31040 | ⊢ No
= {𝑓 ∣
∃𝑎 ∈ On 𝑓:𝑎⟶{1𝑜,
2𝑜}} |
df-slt 31041 | ⊢ <s = {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ No
∧ 𝑔 ∈ No ) ∧ ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑔‘𝑦) ∧ (𝑓‘𝑥){〈1𝑜, ∅〉,
〈1𝑜, 2𝑜〉, 〈∅,
2𝑜〉} (𝑔‘𝑥)))} |
df-bday 31042 | ⊢ bday = (𝑥 ∈
No ↦ dom 𝑥) |
ctxp 31106 | class (𝐴 ⊗ 𝐵) |
cpprod 31107 | class pprod(𝑅, 𝑆) |
csset 31108 | class SSet
|
ctrans 31109 | class Trans
|
cbigcup 31110 | class Bigcup
|
cfix 31111 | class Fix
𝐴 |
climits 31112 | class Limits
|
cfuns 31113 | class Funs
|
csingle 31114 | class Singleton |
csingles 31115 | class
Singletons |
cimage 31116 | class Image𝐴 |
ccart 31117 | class Cart |
cimg 31118 | class Img |
cdomain 31119 | class Domain |
crange 31120 | class Range |
capply 31121 | class Apply |
ccup 31122 | class Cup |
ccap 31123 | class Cap |
csuccf 31124 | class Succ |
cfunpart 31125 | class Funpart𝐹 |
cfullfn 31126 | class FullFun𝐹 |
crestrict 31127 | class Restrict |
cub 31128 | class UB𝑅 |
clb 31129 | class LB𝑅 |
df-txp 31130 | ⊢ (𝐴 ⊗ 𝐵) = ((◡(1st ↾ (V × V))
∘ 𝐴) ∩ (◡(2nd ↾ (V × V))
∘ 𝐵)) |
df-pprod 31131 | ⊢ pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V ×
V))) ⊗ (𝐵 ∘
(2nd ↾ (V × V)))) |
df-sset 31132 | ⊢ SSet = ((V
× V) ∖ ran ( E ⊗ (V ∖ E ))) |
df-trans 31133 | ⊢ Trans = (V
∖ ran (( E ∘ E ) ∖ E )) |
df-bigcup 31134 | ⊢ Bigcup = ((V
× V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗
V))) |
df-fix 31135 | ⊢ Fix 𝐴 = dom (𝐴 ∩ I ) |
df-limits 31136 | ⊢ Limits = ((On
∩ Fix Bigcup )
∖ {∅}) |
df-funs 31137 | ⊢ Funs =
(𝒫 (V × V) ∖ Fix ( E ∘
((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘
◡ E ))) |
df-singleton 31138 | ⊢ Singleton = ((V × V) ∖ ran ((V
⊗ E ) △ ( I ⊗ V))) |
df-singles 31139 | ⊢ Singletons =
ran Singleton |
df-image 31140 | ⊢ Image𝐴 = ((V × V) ∖ ran ((V ⊗ E
) △ (( E ∘ ◡𝐴) ⊗ V))) |
df-cart 31141 | ⊢ Cart = (((V × V) × V) ∖ ran
((V ⊗ E ) △ (pprod( E , E ) ⊗ V))) |
df-img 31142 | ⊢ Img = (Image((2nd ∘
1st ) ↾ (1st ↾ (V × V))) ∘
Cart) |
df-domain 31143 | ⊢ Domain = Image(1st ↾ (V
× V)) |
df-range 31144 | ⊢ Range = Image(2nd ↾ (V
× V)) |
df-cup 31145 | ⊢ Cup = (((V × V) × V) ∖ ran
((V ⊗ E ) △ (((◡1st ∘ E ) ∪ (◡2nd ∘ E )) ⊗
V))) |
df-cap 31146 | ⊢ Cap = (((V × V) × V) ∖ ran
((V ⊗ E ) △ (((◡1st ∘ E ) ∩ (◡2nd ∘ E )) ⊗
V))) |
df-restrict 31147 | ⊢ Restrict = (Cap ∘ (1st
⊗ (Cart ∘ (2nd ⊗ (Range ∘ 1st
))))) |
df-succf 31148 | ⊢ Succ = (Cup ∘ ( I ⊗
Singleton)) |
df-apply 31149 | ⊢ Apply = (( Bigcup
∘ Bigcup ) ∘ (((V × V)
∖ ran ((V ⊗ E ) △ (( E ↾
Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘
pprod( I , Singleton)))) |
df-funpart 31150 | ⊢ Funpart𝐹 = (𝐹 ↾ dom ((Image𝐹 ∘ Singleton) ∩ (V × Singletons ))) |
df-fullfun 31151 | ⊢ FullFun𝐹 = (Funpart𝐹 ∪ ((V ∖ dom Funpart𝐹) ×
{∅})) |
df-ub 31152 | ⊢ UB𝑅 = ((V × V) ∖ ((V ∖ 𝑅) ∘ ◡ E )) |
df-lb 31153 | ⊢ LB𝑅 = UB◡𝑅 |
caltop 31233 | class ⟪𝐴, 𝐵⟫ |
caltxp 31234 | class (𝐴 ×× 𝐵) |
df-altop 31235 | ⊢ ⟪𝐴, 𝐵⟫ = {{𝐴}, {𝐴, {𝐵}}} |
df-altxp 31236 | ⊢ (𝐴 ×× 𝐵) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = ⟪𝑥, 𝑦⟫} |
cofs 31259 | class OuterFiveSeg |
df-ofs 31260 | ⊢ OuterFiveSeg = {〈𝑝, 𝑞〉 ∣ ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)∃𝑥 ∈ (𝔼‘𝑛)∃𝑦 ∈ (𝔼‘𝑛)∃𝑧 ∈ (𝔼‘𝑛)∃𝑤 ∈ (𝔼‘𝑛)(𝑝 = 〈〈𝑎, 𝑏〉, 〈𝑐, 𝑑〉〉 ∧ 𝑞 = 〈〈𝑥, 𝑦〉, 〈𝑧, 𝑤〉〉 ∧ ((𝑏 Btwn 〈𝑎, 𝑐〉 ∧ 𝑦 Btwn 〈𝑥, 𝑧〉) ∧ (〈𝑎, 𝑏〉Cgr〈𝑥, 𝑦〉 ∧ 〈𝑏, 𝑐〉Cgr〈𝑦, 𝑧〉) ∧ (〈𝑎, 𝑑〉Cgr〈𝑥, 𝑤〉 ∧ 〈𝑏, 𝑑〉Cgr〈𝑦, 𝑤〉)))} |
ctransport 31306 | class TransportTo |
df-transport 31307 | ⊢ TransportTo = {〈〈𝑝, 𝑞〉, 𝑥〉 ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd
‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn 〈(1st
‘𝑞), 𝑟〉 ∧
〈(2nd ‘𝑞), 𝑟〉Cgr𝑝)))} |
cifs 31312 | class InnerFiveSeg |
ccgr3 31313 | class Cgr3 |
ccolin 31314 | class Colinear |
cfs 31315 | class FiveSeg |
df-colinear 31316 | ⊢ Colinear = ◡{〈〈𝑏, 𝑐〉, 𝑎〉 ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑐 ∈ (𝔼‘𝑛)) ∧ (𝑎 Btwn 〈𝑏, 𝑐〉 ∨ 𝑏 Btwn 〈𝑐, 𝑎〉 ∨ 𝑐 Btwn 〈𝑎, 𝑏〉))} |
df-ifs 31317 | ⊢ InnerFiveSeg = {〈𝑝, 𝑞〉 ∣ ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)∃𝑥 ∈ (𝔼‘𝑛)∃𝑦 ∈ (𝔼‘𝑛)∃𝑧 ∈ (𝔼‘𝑛)∃𝑤 ∈ (𝔼‘𝑛)(𝑝 = 〈〈𝑎, 𝑏〉, 〈𝑐, 𝑑〉〉 ∧ 𝑞 = 〈〈𝑥, 𝑦〉, 〈𝑧, 𝑤〉〉 ∧ ((𝑏 Btwn 〈𝑎, 𝑐〉 ∧ 𝑦 Btwn 〈𝑥, 𝑧〉) ∧ (〈𝑎, 𝑐〉Cgr〈𝑥, 𝑧〉 ∧ 〈𝑏, 𝑐〉Cgr〈𝑦, 𝑧〉) ∧ (〈𝑎, 𝑑〉Cgr〈𝑥, 𝑤〉 ∧ 〈𝑐, 𝑑〉Cgr〈𝑧, 𝑤〉)))} |
df-cgr3 31318 | ⊢ Cgr3 = {〈𝑝, 𝑞〉 ∣ ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)∃𝑒 ∈ (𝔼‘𝑛)∃𝑓 ∈ (𝔼‘𝑛)(𝑝 = 〈𝑎, 〈𝑏, 𝑐〉〉 ∧ 𝑞 = 〈𝑑, 〈𝑒, 𝑓〉〉 ∧ (〈𝑎, 𝑏〉Cgr〈𝑑, 𝑒〉 ∧ 〈𝑎, 𝑐〉Cgr〈𝑑, 𝑓〉 ∧ 〈𝑏, 𝑐〉Cgr〈𝑒, 𝑓〉))} |
df-fs 31319 | ⊢ FiveSeg = {〈𝑝, 𝑞〉 ∣ ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)∃𝑥 ∈ (𝔼‘𝑛)∃𝑦 ∈ (𝔼‘𝑛)∃𝑧 ∈ (𝔼‘𝑛)∃𝑤 ∈ (𝔼‘𝑛)(𝑝 = 〈〈𝑎, 𝑏〉, 〈𝑐, 𝑑〉〉 ∧ 𝑞 = 〈〈𝑥, 𝑦〉, 〈𝑧, 𝑤〉〉 ∧ (𝑎 Colinear 〈𝑏, 𝑐〉 ∧ 〈𝑎, 〈𝑏, 𝑐〉〉Cgr3〈𝑥, 〈𝑦, 𝑧〉〉 ∧ (〈𝑎, 𝑑〉Cgr〈𝑥, 𝑤〉 ∧ 〈𝑏, 𝑑〉Cgr〈𝑦, 𝑤〉)))} |
csegle 31383 | class
Seg≤ |
df-segle 31384 | ⊢ Seg≤ = {〈𝑝, 𝑞〉 ∣ ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)(𝑝 = 〈𝑎, 𝑏〉 ∧ 𝑞 = 〈𝑐, 𝑑〉 ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn 〈𝑐, 𝑑〉 ∧ 〈𝑎, 𝑏〉Cgr〈𝑐, 𝑦〉))} |
coutsideof 31396 | class OutsideOf |
df-outsideof 31397 | ⊢ OutsideOf = ( Colinear ∖ Btwn
) |
cline2 31411 | class Line |
cray 31412 | class Ray |
clines2 31413 | class LinesEE |
df-line2 31414 | ⊢ Line = {〈〈𝑎, 𝑏〉, 𝑙〉 ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [〈𝑎, 𝑏〉]◡ Colinear )} |
df-ray 31415 | ⊢ Ray = {〈〈𝑝, 𝑎〉, 𝑟〉 ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf〈𝑎, 𝑥〉})} |
df-lines2 31416 | ⊢ LinesEE = ran Line |
cfwddif 31435 | class △ |
df-fwddif 31436 | ⊢ △ = (𝑓 ∈ (ℂ ↑pm
ℂ) ↦ (𝑥 ∈
{𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} ↦ ((𝑓‘(𝑥 + 1)) − (𝑓‘𝑥)))) |
cfwddifn 31437 | class
△n |
df-fwddifn 31438 | ⊢ △n = (𝑛 ∈ ℕ0,
𝑓 ∈ (ℂ
↑pm ℂ) ↦ (𝑥 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑛)(𝑦 + 𝑘) ∈ dom 𝑓} ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((-1↑(𝑛 − 𝑘)) · (𝑓‘(𝑥 + 𝑘)))))) |
chf 31449 | class Hf |
df-hf 31450 | ⊢ Hf = ∪ (𝑅1 “ ω) |
cfne 31501 | class Fne |
df-fne 31502 | ⊢ Fne = {〈𝑥, 𝑦〉 ∣ (∪
𝑥 = ∪ 𝑦
∧ ∀𝑧 ∈
𝑥 𝑧 ⊆ ∪ (𝑦 ∩ 𝒫 𝑧))} |
w3nand 31564 | wff (𝜑 ⊼ 𝜓 ⊼ 𝜒) |
df-3nand 31565 | ⊢ ((𝜑 ⊼ 𝜓 ⊼ 𝜒) ↔ (𝜑 → (𝜓 → ¬ 𝜒))) |
cgcdOLD 31628 | class gcdOLD (𝐴, 𝐵) |
df-gcdOLD 31629 | ⊢ gcdOLD (𝐴, 𝐵) = sup({𝑥 ∈ ℕ ∣ ((𝐴 / 𝑥) ∈ ℕ ∧ (𝐵 / 𝑥) ∈ ℕ)}, ℕ, <
) |
cprvb 31755 | wff Prv 𝜑 |
ax-prv1 31756 | ⊢ 𝜑 ⇒ ⊢ Prv 𝜑 |
ax-prv2 31757 | ⊢ (Prv (𝜑 → 𝜓) → (Prv 𝜑 → Prv 𝜓)) |
ax-prv3 31758 | ⊢ (Prv 𝜑 → Prv Prv 𝜑) |
wnff 31764 | wff ℲℲ𝑥𝜑 |
df-bj-nf 31765 | ⊢ (ℲℲ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) |
wssb 31808 | wff [𝑡/𝑥]b𝜑 |
df-ssb 31809 | ⊢ ([𝑡/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
wrnf 32122 | wff Ⅎ𝑥 ∈ 𝐴𝜑 |
df-bj-rnf 32123 | ⊢ (Ⅎ𝑥 ∈ 𝐴𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜑)) |
bj-csngl 32146 | class sngl 𝐴 |
df-bj-sngl 32147 | ⊢ sngl 𝐴 = {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = {𝑦}} |
bj-ctag 32155 | class tag 𝐴 |
df-bj-tag 32156 | ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅}) |
bj-cproj 32171 | class (𝐴 Proj 𝐵) |
df-bj-proj 32172 | ⊢ (𝐴 Proj 𝐵) = {𝑥 ∣ {𝑥} ∈ (𝐵 “ {𝐴})} |
bj-c1upl 32178 | class ⦅𝐴⦆ |
df-bj-1upl 32179 | ⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴) |
bj-cpr1 32181 | class pr1 𝐴 |
df-bj-pr1 32182 | ⊢ pr1 𝐴 = (∅ Proj 𝐴) |
bj-c2uple 32191 | class ⦅𝐴, 𝐵⦆ |
df-bj-2upl 32192 | ⊢ ⦅𝐴, 𝐵⦆ = (⦅𝐴⦆ ∪ ({1𝑜}
× tag 𝐵)) |
bj-cpr2 32195 | class pr2 𝐴 |
df-bj-pr2 32196 | ⊢ pr2 𝐴 = (1𝑜 Proj 𝐴) |
cmoo 32248 | class Moore |
df-bj-mre 32249 | ⊢ Moore = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(∪ 𝑥 ∩ ∩ 𝑦)
∈ 𝑥} |
cmpt3 32254 | class (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷) |
df-bj-mpt3 32255 | ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶 ↦ 𝐷) = {〈𝑠, 𝑡〉 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝑠 = 〈𝑥, 𝑦, 𝑧〉 ∧ 𝑡 = 𝐷)} |
cfset 32256 | class Set⟶ |
df-bj-fset 32257 | ⊢ Set⟶ = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∣ 𝑓:𝑥⟶𝑦}) |
ccur- 32258 | class curry_ |
cunc- 32259 | class uncurry_ |
df-bj-cur 32260 | ⊢ curry_ = (𝑥 ∈ V, 𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑓 ∈ ((𝑥 × 𝑦) Set⟶ 𝑧) ↦ (𝑎 ∈ 𝑥 ↦ (𝑏 ∈ 𝑦 ↦ (𝑓‘〈𝑎, 𝑏〉))))) |
df-bj-unc 32261 | ⊢ uncurry_ = (𝑥 ∈ V, 𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑓 ∈ (𝑥 Set⟶ (𝑦 Set⟶ 𝑧)) ↦ (𝑎 ∈ 𝑥, 𝑏 ∈ 𝑦 ↦ ((𝑓‘𝑎)‘𝑏)))) |
cdiag2 32265 | class Diag |
df-bj-diag 32266 | ⊢ Diag = (𝑥 ∈ V ↦ ( I ∩ (𝑥 × 𝑥))) |
cinftyexpi 32270 | class inftyexpi |
df-bj-inftyexpi 32271 | ⊢ inftyexpi = (𝑥 ∈ (-π(,]π) ↦ 〈𝑥,
ℂ〉) |
cccinfty 32275 | class
ℂ∞ |
df-bj-ccinfty 32276 | ⊢ ℂ∞ = ran
inftyexpi |
cccbar 32279 | class ℂ̅ |
df-bj-ccbar 32280 | ⊢ ℂ̅ = (ℂ ∪
ℂ∞) |
cpinfty 32283 | class +∞ |
df-bj-pinfty 32284 | ⊢ +∞ = (inftyexpi
‘0) |
cminfty 32287 | class -∞ |
df-bj-minfty 32288 | ⊢ -∞ = (inftyexpi
‘π) |
crrbar 32292 | class ℝ̅ |
df-bj-rrbar 32293 | ⊢ ℝ̅ = (ℝ ∪ {-∞,
+∞}) |
cinfty 32294 | class ∞ |
df-bj-infty 32295 | ⊢ ∞ = 𝒫 ∪ ℂ |
ccchat 32296 | class ℂ̂ |
df-bj-cchat 32297 | ⊢ ℂ̂ = (ℂ ∪
{∞}) |
crrhat 32298 | class ℝ̂ |
df-bj-rrhat 32299 | ⊢ ℝ̂ = (ℝ ∪
{∞}) |
caddcc 32301 | class
+ℂ̅ |
df-bj-addc 32302 | ⊢ +ℂ̅ = (𝑥 ∈ (((ℂ ×
ℂ̅) ∪ (ℂ̅ × ℂ)) ∪ ((ℂ̂
× ℂ̂) ∪ (Diag‘ℂ∞))) ↦
if(((1st ‘𝑥) = ∞ ∨ (2nd ‘𝑥) = ∞), ∞,
if((1st ‘𝑥) ∈ ℂ, if((2nd
‘𝑥) ∈ ℂ,
((1st ‘𝑥)
+ (2nd ‘𝑥)), (2nd ‘𝑥)), (1st ‘𝑥)))) |
coppcc 32303 | class
-ℂ̅ |
df-bj-oppc 32304 | ⊢ -ℂ̅ = (𝑥 ∈ (ℂ̅ ∪
ℂ̂) ↦ if(𝑥 = ∞, ∞, if(𝑥 ∈ ℂ, -𝑥, (inftyexpi ‘if(0 < (1st
‘𝑥), ((1st
‘𝑥) − π),
((1st ‘𝑥)
+ π)))))) |
cprcpal 32305 | class prcpal |
df-bj-prcpal 32306 | ⊢ prcpal = (𝑥 ∈ ℝ ↦ ((𝑥 mod (2 · π)) − if((𝑥 mod (2 · π)) ≤
π, 0, (2 · π)))) |
carg 32307 | class Arg |
df-bj-arg 32308 | ⊢ Arg = (𝑥 ∈ (ℂ̅ ∖ {0}) ↦
if(𝑥 ∈ ℂ,
(ℑ‘(log‘𝑥)), (1st ‘𝑥))) |
cmulc 32309 | class
·ℂ̅ |
df-bj-mulc 32310 | ⊢ ·ℂ̅ = (𝑥 ∈ ((ℂ̅ ×
ℂ̅) ∪ (ℂ̂ × ℂ̂)) ↦
if(((1st ‘𝑥) = 0 ∨ (2nd ‘𝑥) = 0), 0, if(((1st
‘𝑥) = ∞ ∨
(2nd ‘𝑥) =
∞), ∞, if(𝑥
∈ (ℂ × ℂ), ((1st ‘𝑥) · (2nd ‘𝑥)), (inftyexpi
‘(prcpal‘((Arg‘(1st ‘𝑥)) + (Arg‘(2nd ‘𝑥))))))))) |
cinvc 32311 | class
-1ℂ̅ |
df-bj-invc 32312 | ⊢ -1ℂ̅ =
(𝑥 ∈ (ℂ̅
∪ ℂ̂) ↦ if(𝑥 = 0, ∞, if(𝑥 ∈ ℂ, (1 / 𝑥), 0))) |
cfinsum 32322 | class FinSum |
df-bj-finsum 32323 | ⊢ FinSum = (𝑥 ∈ {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦))} ↦ (℩𝑠∃𝑚 ∈ ℕ0 ∃𝑓(𝑓:(1...𝑚)–1-1-onto→dom
(2nd ‘𝑥)
∧ 𝑠 =
(seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd
‘𝑥)‘(𝑓‘𝑛))))‘𝑚)))) |
crrvec 32325 | class ℝ-Vec |
df-bj-rrvec 32326 | ⊢ ℝ-Vec = {𝑥 ∈ LVec ∣ (Scalar‘𝑥) =
ℝfld} |
ctau 32340 | class τ |
df-tau 32341 | ⊢ τ = inf((ℝ+ ∩
(◡cos “ {1})), ℝ, <
) |
cfinxp 32396 | class (𝑈↑↑𝑁) |
df-finxp 32397 | ⊢ (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜 ∧
𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉))), 〈𝑁, 𝑦〉)‘𝑁))} |
ax-luk1 32417 | ⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒))) |
ax-luk2 32418 | ⊢ ((¬ 𝜑 → 𝜑) → 𝜑) |
ax-luk3 32419 | ⊢ (𝜑 → (¬ 𝜑 → 𝜓)) |
ax-wl-13v 32465 | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) |
ax-wl-11v 32540 | ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) |
ctotbnd 32735 | class TotBnd |
cbnd 32736 | class Bnd |
df-totbnd 32737 | ⊢ TotBnd = (𝑥 ∈ V ↦ {𝑚 ∈ (Met‘𝑥) ∣ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ Fin (∪ 𝑣 =
𝑥 ∧ ∀𝑏 ∈ 𝑣 ∃𝑦 ∈ 𝑥 𝑏 = (𝑦(ball‘𝑚)𝑑))}) |
df-bnd 32748 | ⊢ Bnd = (𝑥 ∈ V ↦ {𝑚 ∈ (Met‘𝑥) ∣ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑥 = (𝑦(ball‘𝑚)𝑟)}) |
cismty 32767 | class Ismty |
df-ismty 32768 | ⊢ Ismty = (𝑚 ∈ ∪ ran
∞Met, 𝑛 ∈ ∪ ran ∞Met ↦ {𝑓 ∣ (𝑓:dom dom 𝑚–1-1-onto→dom
dom 𝑛 ∧ ∀𝑥 ∈ dom dom 𝑚∀𝑦 ∈ dom dom 𝑚(𝑥𝑚𝑦) = ((𝑓‘𝑥)𝑛(𝑓‘𝑦)))}) |
crrn 32794 | class
ℝn |
df-rrn 32795 | ⊢ ℝn = (𝑖 ∈ Fin ↦ (𝑥 ∈ (ℝ
↑𝑚 𝑖), 𝑦 ∈ (ℝ ↑𝑚
𝑖) ↦
(√‘Σ𝑘
∈ 𝑖 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2)))) |
cass 32811 | class Ass |
df-ass 32812 | ⊢ Ass = {𝑔 ∣ ∀𝑥 ∈ dom dom 𝑔∀𝑦 ∈ dom dom 𝑔∀𝑧 ∈ dom dom 𝑔((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧))} |
cexid 32813 | class ExId |
df-exid 32814 | ⊢ ExId = {𝑔 ∣ ∃𝑥 ∈ dom dom 𝑔∀𝑦 ∈ dom dom 𝑔((𝑥𝑔𝑦) = 𝑦 ∧ (𝑦𝑔𝑥) = 𝑦)} |
cmagm 32817 | class Magma |
df-mgmOLD 32818 | ⊢ Magma = {𝑔 ∣ ∃𝑡 𝑔:(𝑡 × 𝑡)⟶𝑡} |
csem 32829 | class SemiGrp |
df-sgrOLD 32830 | ⊢ SemiGrp = (Magma ∩ Ass) |
cmndo 32835 | class MndOp |
df-mndo 32836 | ⊢ MndOp = (SemiGrp ∩ ExId
) |
cghomOLD 32852 | class GrpOpHom |
df-ghomOLD 32853 | ⊢ GrpOpHom = (𝑔 ∈ GrpOp, ℎ ∈ GrpOp ↦ {𝑓 ∣ (𝑓:ran 𝑔⟶ran ℎ ∧ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔((𝑓‘𝑥)ℎ(𝑓‘𝑦)) = (𝑓‘(𝑥𝑔𝑦)))}) |
crngo 32863 | class RingOps |
df-rngo 32864 | ⊢ RingOps = {〈𝑔, ℎ〉 ∣ ((𝑔 ∈ AbelOp ∧ ℎ:(ran 𝑔 × ran 𝑔)⟶ran 𝑔) ∧ (∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔∀𝑧 ∈ ran 𝑔(((𝑥ℎ𝑦)ℎ𝑧) = (𝑥ℎ(𝑦ℎ𝑧)) ∧ (𝑥ℎ(𝑦𝑔𝑧)) = ((𝑥ℎ𝑦)𝑔(𝑥ℎ𝑧)) ∧ ((𝑥𝑔𝑦)ℎ𝑧) = ((𝑥ℎ𝑧)𝑔(𝑦ℎ𝑧))) ∧ ∃𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔((𝑥ℎ𝑦) = 𝑦 ∧ (𝑦ℎ𝑥) = 𝑦)))} |
cdrng 32917 | class DivRingOps |
df-drngo 32918 | ⊢ DivRingOps = {〈𝑔, ℎ〉 ∣ (〈𝑔, ℎ〉 ∈ RingOps ∧ (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)} |
crnghom 32929 | class RngHom |
crngiso 32930 | class RngIso |
crisc 32931 | class
≃𝑟 |
df-rngohom 32932 | ⊢ RngHom = (𝑟 ∈ RingOps, 𝑠 ∈ RingOps ↦ {𝑓 ∈ (ran (1st ‘𝑠) ↑𝑚 ran
(1st ‘𝑟))
∣ ((𝑓‘(GId‘(2nd
‘𝑟))) =
(GId‘(2nd ‘𝑠)) ∧ ∀𝑥 ∈ ran (1st ‘𝑟)∀𝑦 ∈ ran (1st ‘𝑟)((𝑓‘(𝑥(1st ‘𝑟)𝑦)) = ((𝑓‘𝑥)(1st ‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(2nd ‘𝑟)𝑦)) = ((𝑓‘𝑥)(2nd ‘𝑠)(𝑓‘𝑦))))}) |
df-rngoiso 32945 | ⊢ RngIso = (𝑟 ∈ RingOps, 𝑠 ∈ RingOps ↦ {𝑓 ∈ (𝑟 RngHom 𝑠) ∣ 𝑓:ran (1st ‘𝑟)–1-1-onto→ran
(1st ‘𝑠)}) |
df-risc 32952 | ⊢ ≃𝑟 = {〈𝑟, 𝑠〉 ∣ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠))} |
ccm2 32958 | class Com2 |
df-com2 32959 | ⊢ Com2 = {〈𝑔, ℎ〉 ∣ ∀𝑎 ∈ ran 𝑔∀𝑏 ∈ ran 𝑔(𝑎ℎ𝑏) = (𝑏ℎ𝑎)} |
cfld 32960 | class Fld |
df-fld 32961 | ⊢ Fld = (DivRingOps ∩
Com2) |
ccring 32962 | class CRingOps |
df-crngo 32963 | ⊢ CRingOps = (RingOps ∩
Com2) |
cidl 32976 | class Idl |
cpridl 32977 | class PrIdl |
cmaxidl 32978 | class MaxIdl |
df-idl 32979 | ⊢ Idl = (𝑟 ∈ RingOps ↦ {𝑖 ∈ 𝒫 ran (1st
‘𝑟) ∣
((GId‘(1st ‘𝑟)) ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥(1st ‘𝑟)𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ ran (1st ‘𝑟)((𝑧(2nd ‘𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2nd ‘𝑟)𝑧) ∈ 𝑖)))}) |
df-pridl 32980 | ⊢ PrIdl = (𝑟 ∈ RingOps ↦ {𝑖 ∈ (Idl‘𝑟) ∣ (𝑖 ≠ ran (1st ‘𝑟) ∧ ∀𝑎 ∈ (Idl‘𝑟)∀𝑏 ∈ (Idl‘𝑟)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥(2nd ‘𝑟)𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖)))}) |
df-maxidl 32981 | ⊢ MaxIdl = (𝑟 ∈ RingOps ↦ {𝑖 ∈ (Idl‘𝑟) ∣ (𝑖 ≠ ran (1st ‘𝑟) ∧ ∀𝑗 ∈ (Idl‘𝑟)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = ran (1st ‘𝑟))))}) |
cprrng 33015 | class PrRing |
cdmn 33016 | class Dmn |
df-prrngo 33017 | ⊢ PrRing = {𝑟 ∈ RingOps ∣
{(GId‘(1st ‘𝑟))} ∈ (PrIdl‘𝑟)} |
df-dmn 33018 | ⊢ Dmn = (PrRing ∩ Com2) |
cigen 33028 | class IdlGen |
df-igen 33029 | ⊢ IdlGen = (𝑟 ∈ RingOps, 𝑠 ∈ 𝒫 ran (1st
‘𝑟) ↦ ∩ {𝑗
∈ (Idl‘𝑟)
∣ 𝑠 ⊆ 𝑗}) |
wprt 33174 | wff Prt 𝐴 |
df-prt 33175 | ⊢ (Prt 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)) |
ax-c5 33186 | ⊢ (∀𝑥𝜑 → 𝜑) |
ax-c4 33187 | ⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓)) |
ax-c7 33188 | ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑) |
ax-c10 33189 | ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑) |
ax-c11 33190 | ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑)) |
ax-c11n 33191 | ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) |
ax-c15 33192 | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) |
ax-c9 33193 | ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) |
ax-c14 33194 | ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦))) |
ax-c16 33195 | ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑)) |
ax-riotaBAD 33257 | ⊢ (℩𝑥 ∈ 𝐴 𝜑) = if(∃!𝑥 ∈ 𝐴 𝜑, (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)), (Undef‘{𝑥 ∣ 𝑥 ∈ 𝐴})) |
clsa 33279 | class LSAtoms |
clsh 33280 | class LSHyp |
df-lsatoms 33281 | ⊢ LSAtoms = (𝑤 ∈ V ↦ ran (𝑣 ∈ ((Base‘𝑤) ∖ {(0g‘𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣}))) |
df-lshyp 33282 | ⊢ LSHyp = (𝑤 ∈ V ↦ {𝑠 ∈ (LSubSp‘𝑤) ∣ (𝑠 ≠ (Base‘𝑤) ∧ ∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤))}) |
clcv 33323 | class
⋖L |
df-lcv 33324 | ⊢ ⋖L = (𝑤 ∈ V ↦ {〈𝑡, 𝑢〉 ∣ ((𝑡 ∈ (LSubSp‘𝑤) ∧ 𝑢 ∈ (LSubSp‘𝑤)) ∧ (𝑡 ⊊ 𝑢 ∧ ¬ ∃𝑠 ∈ (LSubSp‘𝑤)(𝑡 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑢)))}) |
clfn 33362 | class LFnl |
df-lfl 33363 | ⊢ LFnl = (𝑤 ∈ V ↦ {𝑓 ∈ ((Base‘(Scalar‘𝑤)) ↑𝑚
(Base‘𝑤)) ∣
∀𝑟 ∈
(Base‘(Scalar‘𝑤))∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑤)(𝑓‘((𝑟( ·𝑠
‘𝑤)𝑥)(+g‘𝑤)𝑦)) = ((𝑟(.r‘(Scalar‘𝑤))(𝑓‘𝑥))(+g‘(Scalar‘𝑤))(𝑓‘𝑦))}) |
clk 33390 | class LKer |
df-lkr 33391 | ⊢ LKer = (𝑤 ∈ V ↦ (𝑓 ∈ (LFnl‘𝑤) ↦ (◡𝑓 “
{(0g‘(Scalar‘𝑤))}))) |
cld 33428 | class LDual |
df-ldual 33429 | ⊢ LDual = (𝑣 ∈ V ↦ ({〈(Base‘ndx),
(LFnl‘𝑣)〉,
〈(+g‘ndx), ( ∘𝑓
(+g‘(Scalar‘𝑣)) ↾ ((LFnl‘𝑣) × (LFnl‘𝑣)))〉, 〈(Scalar‘ndx),
(oppr‘(Scalar‘𝑣))〉} ∪ {〈(
·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑣)), 𝑓 ∈ (LFnl‘𝑣) ↦ (𝑓 ∘𝑓
(.r‘(Scalar‘𝑣))((Base‘𝑣) × {𝑘})))〉})) |
cops 33477 | class OP |
ccmtN 33478 | class cm |
col 33479 | class OL |
coml 33480 | class OML |
df-oposet 33481 | ⊢ OP = {𝑝 ∈ Poset ∣ (((Base‘𝑝) ∈ dom (lub‘𝑝) ∧ (Base‘𝑝) ∈ dom (glb‘𝑝)) ∧ ∃𝑜(𝑜 = (oc‘𝑝) ∧ ∀𝑎 ∈ (Base‘𝑝)∀𝑏 ∈ (Base‘𝑝)(((𝑜‘𝑎) ∈ (Base‘𝑝) ∧ (𝑜‘(𝑜‘𝑎)) = 𝑎 ∧ (𝑎(le‘𝑝)𝑏 → (𝑜‘𝑏)(le‘𝑝)(𝑜‘𝑎))) ∧ (𝑎(join‘𝑝)(𝑜‘𝑎)) = (1.‘𝑝) ∧ (𝑎(meet‘𝑝)(𝑜‘𝑎)) = (0.‘𝑝))))} |
df-cmtN 33482 | ⊢ cm = (𝑝 ∈ V ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ (Base‘𝑝) ∧ 𝑦 ∈ (Base‘𝑝) ∧ 𝑥 = ((𝑥(meet‘𝑝)𝑦)(join‘𝑝)(𝑥(meet‘𝑝)((oc‘𝑝)‘𝑦))))}) |
df-ol 33483 | ⊢ OL = (Lat ∩
OP) |
df-oml 33484 | ⊢ OML = {𝑙 ∈ OL ∣ ∀𝑎 ∈ (Base‘𝑙)∀𝑏 ∈ (Base‘𝑙)(𝑎(le‘𝑙)𝑏 → 𝑏 = (𝑎(join‘𝑙)(𝑏(meet‘𝑙)((oc‘𝑙)‘𝑎))))} |
ccvr 33567 | class ⋖ |
catm 33568 | class Atoms |
cal 33569 | class AtLat |
clc 33570 | class CvLat |
df-covers 33571 | ⊢ ⋖ = (𝑝 ∈ V ↦ {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (Base‘𝑝) ∧ 𝑏 ∈ (Base‘𝑝)) ∧ 𝑎(lt‘𝑝)𝑏 ∧ ¬ ∃𝑧 ∈ (Base‘𝑝)(𝑎(lt‘𝑝)𝑧 ∧ 𝑧(lt‘𝑝)𝑏))}) |
df-ats 33572 | ⊢ Atoms = (𝑝 ∈ V ↦ {𝑎 ∈ (Base‘𝑝) ∣ (0.‘𝑝)( ⋖ ‘𝑝)𝑎}) |
df-atl 33603 | ⊢ AtLat = {𝑘 ∈ Lat ∣ ((Base‘𝑘) ∈ dom (glb‘𝑘) ∧ ∀𝑥 ∈ (Base‘𝑘)(𝑥 ≠ (0.‘𝑘) → ∃𝑝 ∈ (Atoms‘𝑘)𝑝(le‘𝑘)𝑥))} |
df-cvlat 33627 | ⊢ CvLat = {𝑘 ∈ AtLat ∣ ∀𝑎 ∈ (Atoms‘𝑘)∀𝑏 ∈ (Atoms‘𝑘)∀𝑐 ∈ (Base‘𝑘)((¬ 𝑎(le‘𝑘)𝑐 ∧ 𝑎(le‘𝑘)(𝑐(join‘𝑘)𝑏)) → 𝑏(le‘𝑘)(𝑐(join‘𝑘)𝑎))} |
chlt 33655 | class HL |
df-hlat 33656 | ⊢ HL = {𝑙 ∈ ((OML ∩ CLat) ∩ CvLat)
∣ (∀𝑎 ∈
(Atoms‘𝑙)∀𝑏 ∈ (Atoms‘𝑙)(𝑎 ≠ 𝑏 → ∃𝑐 ∈ (Atoms‘𝑙)(𝑐 ≠ 𝑎 ∧ 𝑐 ≠ 𝑏 ∧ 𝑐(le‘𝑙)(𝑎(join‘𝑙)𝑏))) ∧ ∃𝑎 ∈ (Base‘𝑙)∃𝑏 ∈ (Base‘𝑙)∃𝑐 ∈ (Base‘𝑙)(((0.‘𝑙)(lt‘𝑙)𝑎 ∧ 𝑎(lt‘𝑙)𝑏) ∧ (𝑏(lt‘𝑙)𝑐 ∧ 𝑐(lt‘𝑙)(1.‘𝑙))))} |
clln 33795 | class LLines |
clpl 33796 | class LPlanes |
clvol 33797 | class LVols |
clines 33798 | class Lines |
cpointsN 33799 | class Points |
cpsubsp 33800 | class PSubSp |
cpmap 33801 | class pmap |
df-llines 33802 | ⊢ LLines = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑝( ⋖ ‘𝑘)𝑥}) |
df-lplanes 33803 | ⊢ LPlanes = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (LLines‘𝑘)𝑝( ⋖ ‘𝑘)𝑥}) |
df-lvols 33804 | ⊢ LVols = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ ∃𝑝 ∈ (LPlanes‘𝑘)𝑝( ⋖ ‘𝑘)𝑥}) |
df-lines 33805 | ⊢ Lines = (𝑘 ∈ V ↦ {𝑠 ∣ ∃𝑞 ∈ (Atoms‘𝑘)∃𝑟 ∈ (Atoms‘𝑘)(𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)})}) |
df-pointsN 33806 | ⊢ Points = (𝑘 ∈ V ↦ {𝑞 ∣ ∃𝑝 ∈ (Atoms‘𝑘)𝑞 = {𝑝}}) |
df-psubsp 33807 | ⊢ PSubSp = (𝑘 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ (Atoms‘𝑘)(𝑟(le‘𝑘)(𝑝(join‘𝑘)𝑞) → 𝑟 ∈ 𝑠))}) |
df-pmap 33808 | ⊢ pmap = (𝑘 ∈ V ↦ (𝑎 ∈ (Base‘𝑘) ↦ {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)𝑎})) |
cpadd 34099 | class
+𝑃 |
df-padd 34100 | ⊢ +𝑃 = (𝑙 ∈ V ↦ (𝑚 ∈ 𝒫 (Atoms‘𝑙), 𝑛 ∈ 𝒫 (Atoms‘𝑙) ↦ ((𝑚 ∪ 𝑛) ∪ {𝑝 ∈ (Atoms‘𝑙) ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)}))) |
cpclN 34191 | class PCl |
df-pclN 34192 | ⊢ PCl = (𝑘 ∈ V ↦ (𝑥 ∈ 𝒫 (Atoms‘𝑘) ↦ ∩ {𝑦
∈ (PSubSp‘𝑘)
∣ 𝑥 ⊆ 𝑦})) |
cpolN 34206 | class
⊥𝑃 |
df-polarityN 34207 | ⊢ ⊥𝑃 = (𝑙 ∈ V ↦ (𝑚 ∈ 𝒫
(Atoms‘𝑙) ↦
((Atoms‘𝑙) ∩
∩ 𝑝 ∈ 𝑚 ((pmap‘𝑙)‘((oc‘𝑙)‘𝑝))))) |
cpscN 34238 | class PSubCl |
df-psubclN 34239 | ⊢ PSubCl = (𝑘 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧
((⊥𝑃‘𝑘)‘((⊥𝑃‘𝑘)‘𝑠)) = 𝑠)}) |
clh 34288 | class LHyp |
claut 34289 | class LAut |
cwpointsN 34290 | class WAtoms |
cpautN 34291 | class PAut |
df-lhyp 34292 | ⊢ LHyp = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ 𝑥( ⋖ ‘𝑘)(1.‘𝑘)}) |
df-laut 34293 | ⊢ LAut = (𝑘 ∈ V ↦ {𝑓 ∣ (𝑓:(Base‘𝑘)–1-1-onto→(Base‘𝑘) ∧ ∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 ↔ (𝑓‘𝑥)(le‘𝑘)(𝑓‘𝑦)))}) |
df-watsN 34294 | ⊢ WAtoms = (𝑘 ∈ V ↦ (𝑑 ∈ (Atoms‘𝑘) ↦ ((Atoms‘𝑘) ∖
((⊥𝑃‘𝑘)‘{𝑑})))) |
df-pautN 34295 | ⊢ PAut = (𝑘 ∈ V ↦ {𝑓 ∣ (𝑓:(PSubSp‘𝑘)–1-1-onto→(PSubSp‘𝑘) ∧ ∀𝑥 ∈ (PSubSp‘𝑘)∀𝑦 ∈ (PSubSp‘𝑘)(𝑥 ⊆ 𝑦 ↔ (𝑓‘𝑥) ⊆ (𝑓‘𝑦)))}) |
cldil 34404 | class LDil |
cltrn 34405 | class LTrn |
cdilN 34406 | class Dil |
ctrnN 34407 | class Trn |
df-ldil 34408 | ⊢ LDil = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑓 ∈ (LAut‘𝑘) ∣ ∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓‘𝑥) = 𝑥)})) |
df-ltrn 34409 | ⊢ LTrn = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑓 ∈ ((LDil‘𝑘)‘𝑤) ∣ ∀𝑝 ∈ (Atoms‘𝑘)∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑝(le‘𝑘)𝑤 ∧ ¬ 𝑞(le‘𝑘)𝑤) → ((𝑝(join‘𝑘)(𝑓‘𝑝))(meet‘𝑘)𝑤) = ((𝑞(join‘𝑘)(𝑓‘𝑞))(meet‘𝑘)𝑤))})) |
df-dilN 34410 | ⊢ Dil = (𝑘 ∈ V ↦ (𝑑 ∈ (Atoms‘𝑘) ↦ {𝑓 ∈ (PAut‘𝑘) ∣ ∀𝑥 ∈ (PSubSp‘𝑘)(𝑥 ⊆ ((WAtoms‘𝑘)‘𝑑) → (𝑓‘𝑥) = 𝑥)})) |
df-trnN 34411 | ⊢ Trn = (𝑘 ∈ V ↦ (𝑑 ∈ (Atoms‘𝑘) ↦ {𝑓 ∈ ((Dil‘𝑘)‘𝑑) ∣ ∀𝑞 ∈ ((WAtoms‘𝑘)‘𝑑)∀𝑟 ∈ ((WAtoms‘𝑘)‘𝑑)((𝑞(+𝑃‘𝑘)(𝑓‘𝑞)) ∩
((⊥𝑃‘𝑘)‘{𝑑})) = ((𝑟(+𝑃‘𝑘)(𝑓‘𝑟)) ∩
((⊥𝑃‘𝑘)‘{𝑑}))})) |
ctrl 34463 | class trL |
df-trl 34464 | ⊢ trL = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (℩𝑥 ∈ (Base‘𝑘)∀𝑝 ∈ (Atoms‘𝑘)(¬ 𝑝(le‘𝑘)𝑤 → 𝑥 = ((𝑝(join‘𝑘)(𝑓‘𝑝))(meet‘𝑘)𝑤)))))) |
ctgrp 35048 | class TGrp |
df-tgrp 35049 | ⊢ TGrp = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {〈(Base‘ndx),
((LTrn‘𝑘)‘𝑤)〉,
〈(+g‘ndx), (𝑓 ∈ ((LTrn‘𝑘)‘𝑤), 𝑔 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑓 ∘ 𝑔))〉})) |
ctendo 35058 | class TEndo |
cedring 35059 | class EDRing |
cedring-rN 35060 | class
EDRingR |
df-tendo 35061 | ⊢ TEndo = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑓 ∣ (𝑓:((LTrn‘𝑘)‘𝑤)⟶((LTrn‘𝑘)‘𝑤) ∧ ∀𝑥 ∈ ((LTrn‘𝑘)‘𝑤)∀𝑦 ∈ ((LTrn‘𝑘)‘𝑤)(𝑓‘(𝑥 ∘ 𝑦)) = ((𝑓‘𝑥) ∘ (𝑓‘𝑦)) ∧ ∀𝑥 ∈ ((LTrn‘𝑘)‘𝑤)(((trL‘𝑘)‘𝑤)‘(𝑓‘𝑥))(le‘𝑘)(((trL‘𝑘)‘𝑤)‘𝑥))})) |
df-edring-rN 35062 | ⊢ EDRingR = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {〈(Base‘ndx),
((TEndo‘𝑘)‘𝑤)〉, 〈(+g‘ndx),
(𝑠 ∈
((TEndo‘𝑘)‘𝑤), 𝑡 ∈ ((TEndo‘𝑘)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))〉, 〈(.r‘ndx),
(𝑠 ∈
((TEndo‘𝑘)‘𝑤), 𝑡 ∈ ((TEndo‘𝑘)‘𝑤) ↦ (𝑡 ∘ 𝑠))〉})) |
df-edring 35063 | ⊢ EDRing = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {〈(Base‘ndx),
((TEndo‘𝑘)‘𝑤)〉, 〈(+g‘ndx),
(𝑠 ∈
((TEndo‘𝑘)‘𝑤), 𝑡 ∈ ((TEndo‘𝑘)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))〉, 〈(.r‘ndx),
(𝑠 ∈
((TEndo‘𝑘)‘𝑤), 𝑡 ∈ ((TEndo‘𝑘)‘𝑤) ↦ (𝑠 ∘ 𝑡))〉})) |
cdveca 35308 | class DVecA |
df-dveca 35309 | ⊢ DVecA = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ ({〈(Base‘ndx),
((LTrn‘𝑘)‘𝑤)〉,
〈(+g‘ndx), (𝑓 ∈ ((LTrn‘𝑘)‘𝑤), 𝑔 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑓 ∘ 𝑔))〉, 〈(Scalar‘ndx),
((EDRing‘𝑘)‘𝑤)〉} ∪ {〈(
·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑠‘𝑓))〉}))) |
cdia 35335 | class DIsoA |
df-disoa 35336 | ⊢ DIsoA = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ {𝑦 ∈ (Base‘𝑘) ∣ 𝑦(le‘𝑘)𝑤} ↦ {𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ∣ (((trL‘𝑘)‘𝑤)‘𝑓)(le‘𝑘)𝑥}))) |
cdvh 35385 | class DVecH |
df-dvech 35386 | ⊢ DVecH = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ ({〈(Base‘ndx),
(((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤))〉, 〈(+g‘ndx),
(𝑓 ∈
(((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)), 𝑔 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)) ↦ 〈((1st
‘𝑓) ∘
(1st ‘𝑔)),
(ℎ ∈ ((LTrn‘𝑘)‘𝑤) ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))〉)〉, 〈(Scalar‘ndx),
((EDRing‘𝑘)‘𝑤)〉} ∪ {〈(
·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑓 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)) ↦ 〈(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))〉)〉}))) |
cocaN 35426 | class ocA |
df-docaN 35427 | ⊢ ocA = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤) ↦ (((DIsoA‘𝑘)‘𝑤)‘((((oc‘𝑘)‘(◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))(meet‘𝑘)𝑤))))) |
cdjaN 35438 | class vA |
df-djaN 35439 | ⊢ vA = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤), 𝑦 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤) ↦ (((ocA‘𝑘)‘𝑤)‘((((ocA‘𝑘)‘𝑤)‘𝑥) ∩ (((ocA‘𝑘)‘𝑤)‘𝑦)))))) |
cdib 35445 | class DIsoB |
df-dib 35446 | ⊢ DIsoB = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ dom ((DIsoA‘𝑘)‘𝑤) ↦ ((((DIsoA‘𝑘)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ( I ↾ (Base‘𝑘)))})))) |
cdic 35479 | class DIsoC |
df-dic 35480 | ⊢ DIsoC = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑞 ∈ {𝑟 ∈ (Atoms‘𝑘) ∣ ¬ 𝑟(le‘𝑘)𝑤} ↦ {〈𝑓, 𝑠〉 ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ ((LTrn‘𝑘)‘𝑤)(𝑔‘((oc‘𝑘)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝑘)‘𝑤))}))) |
cdih 35535 | class DIsoH |
df-dih 35536 | ⊢ DIsoH = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ (Base‘𝑘) ↦ if(𝑥(le‘𝑘)𝑤, (((DIsoB‘𝑘)‘𝑤)‘𝑥), (℩𝑢 ∈ (LSubSp‘((DVecH‘𝑘)‘𝑤))∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤))))))))) |
coch 35654 | class ocH |
df-doch 35655 | ⊢ ocH = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫
(Base‘((DVecH‘𝑘)‘𝑤)) ↦ (((DIsoH‘𝑘)‘𝑤)‘((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)})))))) |
cdjh 35701 | class joinH |
df-djh 35702 | ⊢ joinH = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫
(Base‘((DVecH‘𝑘)‘𝑤)), 𝑦 ∈ 𝒫
(Base‘((DVecH‘𝑘)‘𝑤)) ↦ (((ocH‘𝑘)‘𝑤)‘((((ocH‘𝑘)‘𝑤)‘𝑥) ∩ (((ocH‘𝑘)‘𝑤)‘𝑦)))))) |
clpoN 35787 | class LPol |
df-lpolN 35788 | ⊢ LPol = (𝑤 ∈ V ↦ {𝑜 ∈ ((LSubSp‘𝑤) ↑𝑚 𝒫
(Base‘𝑤)) ∣
((𝑜‘(Base‘𝑤)) = {(0g‘𝑤)} ∧ ∀𝑥∀𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑤)((𝑜‘𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥))}) |
clcd 35893 | class LCDual |
df-lcdual 35894 | ⊢ LCDual = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ ((LDual‘((DVecH‘𝑘)‘𝑤)) ↾s {𝑓 ∈ (LFnl‘((DVecH‘𝑘)‘𝑤)) ∣ (((ocH‘𝑘)‘𝑤)‘(((ocH‘𝑘)‘𝑤)‘((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓)}))) |
cmpd 35931 | class mapd |
df-mapd 35932 | ⊢ mapd = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑠 ∈ (LSubSp‘((DVecH‘𝑘)‘𝑤)) ↦ {𝑓 ∈ (LFnl‘((DVecH‘𝑘)‘𝑤)) ∣ ((((ocH‘𝑘)‘𝑤)‘(((ocH‘𝑘)‘𝑤)‘((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓) ∧ (((ocH‘𝑘)‘𝑤)‘((LKer‘((DVecH‘𝑘)‘𝑤))‘𝑓)) ⊆ 𝑠)}))) |
chvm 36063 | class HVMap |
df-hvmap 36064 | ⊢ HVMap = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ ((Base‘((DVecH‘𝑘)‘𝑤)) ∖
{(0g‘((DVecH‘𝑘)‘𝑤))}) ↦ (𝑣 ∈ (Base‘((DVecH‘𝑘)‘𝑤)) ↦ (℩𝑗 ∈
(Base‘(Scalar‘((DVecH‘𝑘)‘𝑤)))∃𝑡 ∈ (((ocH‘𝑘)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝑘)‘𝑤))(𝑗( ·𝑠
‘((DVecH‘𝑘)‘𝑤))𝑥))))))) |
chdma1 36099 | class HDMap1 |
chdma 36100 | class HDMap |
df-hdmap1 36101 | ⊢ HDMap1 = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑎 ∣ [((DVecH‘𝑘)‘𝑤) / 𝑢][(Base‘𝑢) / 𝑣][(LSpan‘𝑢) / 𝑛][((LCDual‘𝑘)‘𝑤) / 𝑐][(Base‘𝑐) / 𝑑][(LSpan‘𝑐) / 𝑗][((mapd‘𝑘)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ ((𝑣 × 𝑑) × 𝑣) ↦ if((2nd ‘𝑥) = (0g‘𝑢), (0g‘𝑐), (℩ℎ ∈ 𝑑 ((𝑚‘(𝑛‘{(2nd ‘𝑥)})) = (𝑗‘{ℎ}) ∧ (𝑚‘(𝑛‘{((1st
‘(1st ‘𝑥))(-g‘𝑢)(2nd ‘𝑥))})) = (𝑗‘{((2nd
‘(1st ‘𝑥))(-g‘𝑐)ℎ)})))))})) |
df-hdmap 36102 | ⊢ HDMap = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑎 ∣ [〈( I ↾
(Base‘𝑘)), ( I
↾ ((LTrn‘𝑘)‘𝑤))〉 / 𝑒][((DVecH‘𝑘)‘𝑤) / 𝑢][(Base‘𝑢) / 𝑣][((HDMap1‘𝑘)‘𝑤) / 𝑖]𝑎 ∈ (𝑡 ∈ 𝑣 ↦ (℩𝑦 ∈ (Base‘((LCDual‘𝑘)‘𝑤))∀𝑧 ∈ 𝑣 (¬ 𝑧 ∈ (((LSpan‘𝑢)‘{𝑒}) ∪ ((LSpan‘𝑢)‘{𝑡})) → 𝑦 = (𝑖‘〈𝑧, (𝑖‘〈𝑒, (((HVMap‘𝑘)‘𝑤)‘𝑒), 𝑧〉), 𝑡〉))))})) |
chg 36193 | class HGMap |
df-hgmap 36194 | ⊢ HGMap = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑎 ∣ [((DVecH‘𝑘)‘𝑤) / 𝑢][(Base‘(Scalar‘𝑢)) / 𝑏][((HDMap‘𝑘)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠
‘𝑢)𝑣)) = (𝑦( ·𝑠
‘((LCDual‘𝑘)‘𝑤))(𝑚‘𝑣))))})) |
chlh 36242 | class HLHil |
df-hlhil 36243 | ⊢ HLHil = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ ⦋((DVecH‘𝑘)‘𝑤) / 𝑢⦌⦋(Base‘𝑢) / 𝑣⦌({〈(Base‘ndx), 𝑣〉,
〈(+g‘ndx), (+g‘𝑢)〉, 〈(Scalar‘ndx),
(((EDRing‘𝑘)‘𝑤) sSet 〈(*𝑟‘ndx),
((HGMap‘𝑘)‘𝑤)〉)〉} ∪ {〈(
·𝑠 ‘ndx), (
·𝑠 ‘𝑢)〉,
〈(·𝑖‘ndx), (𝑥 ∈ 𝑣, 𝑦 ∈ 𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))〉}))) |
cnacs 36283 | class NoeACS |
df-nacs 36284 | ⊢ NoeACS = (𝑥 ∈ V ↦ {𝑐 ∈ (ACS‘𝑥) ∣ ∀𝑠 ∈ 𝑐 ∃𝑔 ∈ (𝒫 𝑥 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)}) |
cmzpcl 36302 | class mzPolyCld |
cmzp 36303 | class mzPoly |
df-mzpcl 36304 | ⊢ mzPolyCld = (𝑣 ∈ V ↦ {𝑝 ∈ 𝒫 (ℤ
↑𝑚 (ℤ ↑𝑚 𝑣)) ∣ ((∀𝑖 ∈ ℤ ((ℤ
↑𝑚 𝑣) × {𝑖}) ∈ 𝑝 ∧ ∀𝑗 ∈ 𝑣 (𝑥 ∈ (ℤ ↑𝑚
𝑣) ↦ (𝑥‘𝑗)) ∈ 𝑝) ∧ ∀𝑓 ∈ 𝑝 ∀𝑔 ∈ 𝑝 ((𝑓 ∘𝑓 + 𝑔) ∈ 𝑝 ∧ (𝑓 ∘𝑓 · 𝑔) ∈ 𝑝))}) |
df-mzp 36305 | ⊢ mzPoly = (𝑣 ∈ V ↦ ∩ (mzPolyCld‘𝑣)) |
cdioph 36336 | class Dioph |
df-dioph 36337 | ⊢ Dioph = (𝑛 ∈ ℕ0 ↦ ran
(𝑘 ∈
(ℤ≥‘𝑛), 𝑝 ∈ (mzPoly‘(1...𝑘)) ↦ {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑛)) ∧ (𝑝‘𝑢) = 0)})) |
csquarenn 36418 | class
◻NN |
cpell1qr 36419 | class Pell1QR |
cpell1234qr 36420 | class Pell1234QR |
cpell14qr 36421 | class Pell14QR |
cpellfund 36422 | class PellFund |
df-squarenn 36423 | ⊢ ◻NN = {𝑥 ∈ ℕ ∣
(√‘𝑥) ∈
ℚ} |
df-pell1qr 36424 | ⊢ Pell1QR = (𝑥 ∈ (ℕ ∖
◻NN) ↦ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0
∃𝑤 ∈
ℕ0 (𝑦 =
(𝑧 + ((√‘𝑥) · 𝑤)) ∧ ((𝑧↑2) − (𝑥 · (𝑤↑2))) = 1)}) |
df-pell14qr 36425 | ⊢ Pell14QR = (𝑥 ∈ (ℕ ∖
◻NN) ↦ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0
∃𝑤 ∈ ℤ
(𝑦 = (𝑧 + ((√‘𝑥) · 𝑤)) ∧ ((𝑧↑2) − (𝑥 · (𝑤↑2))) = 1)}) |
df-pell1234qr 36426 | ⊢ Pell1234QR = (𝑥 ∈ (ℕ ∖
◻NN) ↦ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝑥) · 𝑤)) ∧ ((𝑧↑2) − (𝑥 · (𝑤↑2))) = 1)}) |
df-pellfund 36427 | ⊢ PellFund = (𝑥 ∈ (ℕ ∖
◻NN) ↦ inf({𝑧 ∈ (Pell14QR‘𝑥) ∣ 1 < 𝑧}, ℝ, < )) |
crmx 36482 | class Xrm |
crmy 36483 | class Yrm |
df-rmx 36484 | ⊢ Xrm = (𝑎 ∈ (ℤ≥‘2),
𝑛 ∈ ℤ ↦
(1st ‘(◡(𝑏 ∈ (ℕ0
× ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd
‘𝑏))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛)))) |
df-rmy 36485 | ⊢ Yrm = (𝑎 ∈ (ℤ≥‘2),
𝑛 ∈ ℤ ↦
(2nd ‘(◡(𝑏 ∈ (ℕ0
× ℤ) ↦ ((1st ‘𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd
‘𝑏))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛)))) |
clfig 36655 | class LFinGen |
df-lfig 36656 | ⊢ LFinGen = {𝑤 ∈ LMod ∣ (Base‘𝑤) ∈ ((LSpan‘𝑤) “ (𝒫
(Base‘𝑤) ∩
Fin))} |
clnm 36663 | class LNoeM |
df-lnm 36664 | ⊢ LNoeM = {𝑤 ∈ LMod ∣ ∀𝑖 ∈ (LSubSp‘𝑤)(𝑤 ↾s 𝑖) ∈ LFinGen} |
clnr 36698 | class LNoeR |
df-lnr 36699 | ⊢ LNoeR = {𝑎 ∈ Ring ∣ (ringLMod‘𝑎) ∈ LNoeM} |
cldgis 36710 | class ldgIdlSeq |
df-ldgis 36711 | ⊢ ldgIdlSeq = (𝑟 ∈ V ↦ (𝑖 ∈
(LIdeal‘(Poly1‘𝑟)) ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((( deg1 ‘𝑟)‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))}))) |
cmnc 36720 | class Monic |
cplylt 36721 | class
Poly< |
df-mnc 36722 | ⊢ Monic = (𝑠 ∈ 𝒫 ℂ ↦ {𝑝 ∈ (Poly‘𝑠) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1}) |
df-plylt 36723 | ⊢ Poly< = (𝑠 ∈ 𝒫 ℂ, 𝑥 ∈ ℕ0 ↦ {𝑝 ∈ (Poly‘𝑠) ∣ (𝑝 = 0𝑝 ∨
(deg‘𝑝) < 𝑥)}) |
cdgraa 36729 | class
degAA |
cmpaa 36730 | class minPolyAA |
df-dgraa 36731 | ⊢ degAA = (𝑥 ∈ 𝔸 ↦ inf({𝑑 ∈ ℕ ∣
∃𝑝 ∈
((Poly‘ℚ) ∖ {0𝑝})((deg‘𝑝) = 𝑑 ∧ (𝑝‘𝑥) = 0)}, ℝ, < )) |
df-mpaa 36732 | ⊢ minPolyAA = (𝑥 ∈ 𝔸 ↦
(℩𝑝 ∈
(Poly‘ℚ)((deg‘𝑝) = (degAA‘𝑥) ∧ (𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝑥)) = 1))) |
citgo 36746 | class IntgOver |
cza 36747 | class ℤ |
df-itgo 36748 | ⊢ IntgOver = (𝑠 ∈ 𝒫 ℂ ↦ {𝑥 ∈ ℂ ∣
∃𝑝 ∈
(Poly‘𝑠)((𝑝‘𝑥) = 0 ∧ ((coeff‘𝑝)‘(deg‘𝑝)) = 1)}) |
df-za 36749 | ⊢ ℤ = (IntgOver‘ℤ) |
cmend 36764 | class MEndo |
df-mend 36765 | ⊢ MEndo = (𝑚 ∈ V ↦ ⦋(𝑚 LMHom 𝑚) / 𝑏⦌({〈(Base‘ndx),
𝑏〉,
〈(+g‘ndx), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 ∘𝑓
(+g‘𝑚)𝑦))〉, 〈(.r‘ndx),
(𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 ∘ 𝑦))〉} ∪ {〈(Scalar‘ndx),
(Scalar‘𝑚)〉,
〈( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘(Scalar‘𝑚)), 𝑦 ∈ 𝑏 ↦ (((Base‘𝑚) × {𝑥}) ∘𝑓 (
·𝑠 ‘𝑚)𝑦))〉})) |
csdrg 36784 | class SubDRing |
df-sdrg 36785 | ⊢ SubDRing = (𝑤 ∈ DivRing ↦ {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ DivRing}) |
ccytp 36799 | class CytP |
df-cytp 36800 | ⊢ CytP = (𝑛 ∈ ℕ ↦
((mulGrp‘(Poly1‘ℂfld))
Σg (𝑟 ∈ (◡(od‘((mulGrp‘ℂfld)
↾s (ℂ ∖ {0}))) “ {𝑛}) ↦
((var1‘ℂfld)(-g‘(Poly1‘ℂfld))((algSc‘(Poly1‘ℂfld))‘𝑟))))) |
ctopsep 36810 | class TopSep |
ctoplnd 36811 | class TopLnd |
df-topsep 36812 | ⊢ TopSep = {𝑗 ∈ Top ∣ ∃𝑥 ∈ 𝒫 ∪ 𝑗(𝑥 ≼ ω ∧ ((cls‘𝑗)‘𝑥) = ∪ 𝑗)} |
df-toplnd 36813 | ⊢ TopLnd = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥(∪
𝑥 = ∪ 𝑦
→ ∃𝑧 ∈
𝒫 𝑥(𝑧 ≼ ω ∧ ∪ 𝑥 =
∪ 𝑧))} |
crcl 36983 | class r* |
df-rcl 36984 | ⊢ r* = (𝑥 ∈ V ↦ ∩ {𝑧
∣ (𝑥 ⊆ 𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)}) |
whe 37086 | wff 𝑅 hereditary 𝐴 |
df-he 37087 | ⊢ (𝑅 hereditary 𝐴 ↔ (𝑅 “ 𝐴) ⊆ 𝐴) |
ax-frege1 37104 | ⊢ (𝜑 → (𝜓 → 𝜑)) |
ax-frege2 37105 | ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))) |
ax-frege8 37123 | ⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒))) |
ax-frege28 37144 | ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑)) |
ax-frege31 37148 | ⊢ (¬ ¬ 𝜑 → 𝜑) |
ax-frege41 37159 | ⊢ (𝜑 → ¬ ¬ 𝜑) |
ax-frege52a 37171 | ⊢ ((𝜑 ↔ 𝜓) → (if-(𝜑, 𝜃, 𝜒) → if-(𝜓, 𝜃, 𝜒))) |
ax-frege54a 37176 | ⊢ (𝜑 ↔ 𝜑) |
ax-frege58a 37189 | ⊢ ((𝜓 ∧ 𝜒) → if-(𝜑, 𝜓, 𝜒)) |
ax-frege52c 37202 | ⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 → [𝐵 / 𝑥]𝜑)) |
ax-frege54c 37206 | ⊢ 𝐴 = 𝐴 |
ax-frege58b 37215 | ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑) |
cbcc 37557 | class
C𝑐 |
df-bcc 37558 | ⊢ C𝑐 = (𝑐 ∈ ℂ, 𝑘 ∈ ℕ0 ↦ ((𝑐 FallFac 𝑘) / (!‘𝑘))) |
cplusr 37682 | class
+𝑟 |
cminusr 37683 | class
-𝑟 |
ctimesr 37684 | class
.𝑣 |
cptdfc 37685 | class PtDf(𝐴, 𝐵) |
crr3c 37686 | class RR3 |
cline3 37687 | class line3 |
df-addr 37688 | ⊢ +𝑟 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ ((𝑥‘𝑣) + (𝑦‘𝑣)))) |
df-subr 37689 | ⊢ -𝑟 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ ((𝑥‘𝑣) − (𝑦‘𝑣)))) |
df-mulv 37690 | ⊢ .𝑣 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ (𝑥 · (𝑦‘𝑣)))) |
df-ptdf 37701 | ⊢ PtDf(𝐴, 𝐵) = (𝑥 ∈ ℝ ↦ (((𝑥.𝑣(𝐵-𝑟𝐴)) +𝑣 𝐴) “ {1, 2, 3})) |
df-rr3 37702 | ⊢ RR3 = (ℝ ↑𝑚 {1,
2, 3}) |
df-line3 37703 | ⊢ line3 = {𝑥 ∈ 𝒫 RR3 ∣
(2𝑜 ≼ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑧 ≠ 𝑦 → ran PtDf(𝑦, 𝑧) = 𝑥))} |
wvd1 37806 | wff ( 𝜑 ▶ 𝜓 ) |
df-vd1 37807 | ⊢ (( 𝜑 ▶ 𝜓 ) ↔ (𝜑 → 𝜓)) |
wvd2 37814 | wff ( 𝜑 , 𝜓 ▶ 𝜒 ) |
df-vd2 37815 | ⊢ (( 𝜑 , 𝜓 ▶ 𝜒 ) ↔ ((𝜑 ∧ 𝜓) → 𝜒)) |
wvhc2 37817 | wff ( 𝜑 , 𝜓 ) |
df-vhc2 37818 | ⊢ (( 𝜑 , 𝜓 ) ↔ (𝜑 ∧ 𝜓)) |
wvd3 37824 | wff ( 𝜑 , 𝜓 , 𝜒 ▶ 𝜃 ) |
wvhc3 37825 | wff ( 𝜑 , 𝜓 , 𝜒 ) |
df-vhc3 37826 | ⊢ (( 𝜑 , 𝜓 , 𝜒 ) ↔ (𝜑 ∧ 𝜓 ∧ 𝜒)) |
df-vd3 37827 | ⊢ (( 𝜑 , 𝜓 , 𝜒 ▶ 𝜃 ) ↔ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜃)) |
wvhc4 37834 | wff ( 𝜑 , 𝜓 , 𝜒 , 𝜃 ) |
wvhc5 37835 | wff ( 𝜑 , 𝜓 , 𝜒 , 𝜃 , 𝜏 ) |
wvhc6 37836 | wff ( 𝜑 , 𝜓 , 𝜒 , 𝜃 , 𝜏 , 𝜂 ) |
wvhc7 37837 | wff ( 𝜑 , 𝜓 , 𝜒 , 𝜃 , 𝜏 , 𝜂 , 𝜁 ) |
wvhc8 37838 | wff ( 𝜑 , 𝜓 , 𝜒 , 𝜃 , 𝜏 , 𝜂 , 𝜁 , 𝜎 ) |
wvhc9 37839 | wff ( 𝜑 , 𝜓 , 𝜒 , 𝜃 , 𝜏 , 𝜂 , 𝜁 , 𝜎 , 𝜌 ) |
wvhc10 37840 | wff ( 𝜑 , 𝜓 , 𝜒 , 𝜃 , 𝜏 , 𝜂 , 𝜁 , 𝜎 , 𝜌 , 𝜇 ) |
wvhc11 37841 | wff ( 𝜑 , 𝜓 , 𝜒 , 𝜃 , 𝜏 , 𝜂 , 𝜁 , 𝜎 , 𝜌 , 𝜇 , 𝜆 ) |
wvhc12 37842 | wff ( 𝜑 , 𝜓 , 𝜒 , 𝜃 , 𝜏 , 𝜂 , 𝜁 , 𝜎 , 𝜌 , 𝜇 , 𝜆 , 𝜅 ) |
csalg 39204 | class SAlg |
df-salg 39205 | ⊢ SAlg = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (∪ 𝑥 ∖ 𝑦) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → ∪ 𝑦
∈ 𝑥))} |
csalon 39206 | class SalOn |
df-salon 39207 | ⊢ SalOn = (𝑥 ∈ V ↦ {𝑠 ∈ SAlg ∣ ∪ 𝑠 =
𝑥}) |
csalgen 39208 | class SalGen |
df-salgen 39209 | ⊢ SalGen = (𝑥 ∈ V ↦ ∩ {𝑠
∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑥 ∧ 𝑥 ⊆ 𝑠)}) |
csumge0 39255 | class
Σ^ |
df-sumge0 39256 | ⊢ Σ^ = (𝑥 ∈ V ↦ if(+∞
∈ ran 𝑥, +∞,
sup(ran (𝑦 ∈
(𝒫 dom 𝑥 ∩ Fin)
↦ Σ𝑤 ∈
𝑦 (𝑥‘𝑤)), ℝ*, <
))) |
cmea 39342 | class Meas |
df-mea 39343 | ⊢ Meas = {𝑥 ∣ (((𝑥:dom 𝑥⟶(0[,]+∞) ∧ dom 𝑥 ∈ SAlg) ∧ (𝑥‘∅) = 0) ∧
∀𝑦 ∈ 𝒫
dom 𝑥((𝑦 ≼ ω ∧ Disj 𝑤 ∈ 𝑦 𝑤) → (𝑥‘∪ 𝑦) =
(Σ^‘(𝑥 ↾ 𝑦))))} |
come 39379 | class OutMeas |
df-ome 39380 | ⊢ OutMeas = {𝑥 ∣ ((((𝑥:dom 𝑥⟶(0[,]+∞) ∧ dom 𝑥 = 𝒫 ∪ dom 𝑥) ∧ (𝑥‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑥∀𝑧 ∈ 𝒫 𝑦(𝑥‘𝑧) ≤ (𝑥‘𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≼ ω → (𝑥‘∪ 𝑦) ≤
(Σ^‘(𝑥 ↾ 𝑦))))} |
ccaragen 39381 | class CaraGen |
df-caragen 39382 | ⊢ CaraGen = (𝑜 ∈ OutMeas ↦ {𝑒 ∈ 𝒫 ∪ dom 𝑜 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑜((𝑜‘(𝑎 ∩ 𝑒)) +𝑒 (𝑜‘(𝑎 ∖ 𝑒))) = (𝑜‘𝑎)}) |
covoln 39426 | class voln* |
df-ovoln 39427 | ⊢ voln* = (𝑥 ∈ Fin ↦ (𝑦 ∈ 𝒫 (ℝ
↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, inf({𝑧 ∈ ℝ* ∣
∃𝑖 ∈ (((ℝ
× ℝ) ↑𝑚 𝑥) ↑𝑚 ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑥 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ*, <
)))) |
cvoln 39428 | class voln |
df-voln 39429 | ⊢ voln = (𝑥 ∈ Fin ↦ ((voln*‘𝑥) ↾
(CaraGen‘(voln*‘𝑥)))) |
csmblfn 39586 | class SMblFn |
df-smblfn 39587 | ⊢ SMblFn = (𝑠 ∈ SAlg ↦ {𝑓 ∈ (ℝ ↑pm
∪ 𝑠) ∣ ∀𝑎 ∈ ℝ (◡𝑓 “ (-∞(,)𝑎)) ∈ (𝑠 ↾t dom 𝑓)}) |
wdfat 39842 | wff 𝐹 defAt 𝐴 |
cafv 39843 | class (𝐹'''𝐴) |
caov 39844 | class ((𝐴𝐹𝐵)) |
df-dfat 39845 | ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) |
df-afv 39846 | ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), V) |
df-aov 39847 | ⊢ ((𝐴𝐹𝐵)) = (𝐹'''〈𝐴, 𝐵〉) |
ciccp 39951 | class RePart |
df-iccp 39952 | ⊢ RePart = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ*
↑𝑚 (0...𝑚)) ∣ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))}) |
cfmtno 39977 | class FermatNo |
df-fmtno 39978 | ⊢ FermatNo = (𝑛 ∈ ℕ0 ↦
((2↑(2↑𝑛)) +
1)) |
ceven 40075 | class Even |
codd 40076 | class Odd |
df-even 40077 | ⊢ Even = {𝑧 ∈ ℤ ∣ (𝑧 / 2) ∈ ℤ} |
df-odd 40078 | ⊢ Odd = {𝑧 ∈ ℤ ∣ ((𝑧 + 1) / 2) ∈ ℤ} |
cgbe 40167 | class GoldbachEven |
cgbo 40168 | class GoldbachOdd |
cgboa 40169 | class
GoldbachOddALTV |
df-gbe 40170 | ⊢ GoldbachEven = {𝑧 ∈ Even ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))} |
df-gbo 40171 | ⊢ GoldbachOdd = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)} |
df-gboa 40172 | ⊢ GoldbachOddALTV = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))} |
ax-bgbltosilva 40226 | ⊢ ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 ≤ (4 · (;10↑;18))) → 𝑁 ∈ GoldbachEven ) |
ax-hgprmladder 40228 | ⊢ ∃𝑑 ∈
(ℤ≥‘3)∃𝑓 ∈ (RePart‘𝑑)(((𝑓‘0) = 7 ∧ (𝑓‘1) = ;13 ∧ (𝑓‘𝑑) = (;89 · (;10↑;29))) ∧ ∀𝑖 ∈ (0..^𝑑)((𝑓‘𝑖) ∈ (ℙ ∖ {2}) ∧ ((𝑓‘(𝑖 + 1)) − (𝑓‘𝑖)) < ((4 · (;10↑;18)) − 4) ∧ 4 < ((𝑓‘(𝑖 + 1)) − (𝑓‘𝑖)))) |
ax-tgoldbachgt 40231 | ⊢ ∃𝑚 ∈ ℕ (𝑚 ≤ (;10↑;27) ∧ ∀𝑛 ∈ Odd (𝑚 < 𝑛 → 𝑛 ∈ GoldbachOddALTV )) |
ax-bgbltosilvaOLD 40233 | ⊢ ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 ≤ (4 · (10↑;18))) → 𝑁 ∈ GoldbachEven ) |
ax-hgprmladderOLD 40235 | ⊢ ∃𝑑 ∈
(ℤ≥‘3)∃𝑓 ∈ (RePart‘𝑑)(((𝑓‘0) = 7 ∧ (𝑓‘1) = ;13 ∧ (𝑓‘𝑑) = (;89 · (10↑;29))) ∧ ∀𝑖 ∈ (0..^𝑑)((𝑓‘𝑖) ∈ (ℙ ∖ {2}) ∧ ((𝑓‘(𝑖 + 1)) − (𝑓‘𝑖)) < ((4 · (10↑;18)) − 4) ∧ 4 < ((𝑓‘(𝑖 + 1)) − (𝑓‘𝑖)))) |
ax-tgoldbachgtOLD 40238 | ⊢ ∃𝑚 ∈ ℕ (𝑚 ≤ (10↑;27) ∧ ∀𝑛 ∈ Odd (𝑚 < 𝑛 → 𝑛 ∈ GoldbachOddALTV )) |
cpfx 40244 | class prefix |
df-pfx 40245 | ⊢ prefix = (𝑠 ∈ V, 𝑙 ∈ ℕ0 ↦ (𝑠 substr 〈0, 𝑙〉)) |
cuspgr 40378 | class USPGraph |
cusgr 40379 | class USGraph |
df-uspgr 40380 | ⊢ USPGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}} |
df-usgr 40381 | ⊢ USGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒–1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}} |
csubgr 40491 | class SubGraph |
df-subgr 40492 | ⊢ SubGraph = {〈𝑠, 𝑔〉 ∣ ((Vtx‘𝑠) ⊆ (Vtx‘𝑔) ∧ (iEdg‘𝑠) = ((iEdg‘𝑔) ↾ dom (iEdg‘𝑠)) ∧ (Edg‘𝑠) ⊆ 𝒫 (Vtx‘𝑠))} |
cfusgr 40535 | class FinUSGraph |
df-fusgr 40536 | ⊢ FinUSGraph = {𝑔 ∈ USGraph ∣ (Vtx‘𝑔) ∈ Fin} |
cnbgr 40550 | class NeighbVtx |
cuvtxa 40551 | class UnivVtx |
ccplgr 40552 | class ComplGraph |
ccusgr 40553 | class ComplUSGraph |
df-nbgr 40554 | ⊢ NeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒}) |
df-uvtxa 40556 | ⊢ UnivVtx = (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)}) |
df-cplgr 40557 | ⊢ ComplGraph = {𝑔 ∣ ∀𝑣 ∈ (Vtx‘𝑔)𝑣 ∈ (UnivVtx‘𝑔)} |
df-cusgr 40558 | ⊢ ComplUSGraph = {𝑔 ∈ USGraph ∣ 𝑔 ∈ ComplGraph} |
cvtxdg 40681 | class VtxDeg |
df-vtxdg 40682 | ⊢ VtxDeg = (𝑔 ∈ V ↦
⦋(Vtx‘𝑔) / 𝑣⦌⦋(iEdg‘𝑔) / 𝑒⦌(𝑢 ∈ 𝑣 ↦ ((#‘{𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}})))) |
crgr 40755 | class RegGraph |
crusgr 40756 | class RegUSGraph |
df-rgr 40757 | ⊢ RegGraph = {〈𝑔, 𝑘〉 ∣ (𝑘 ∈ ℕ0* ∧
∀𝑣 ∈
(Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘)} |
df-rusgr 40758 | ⊢ RegUSGraph = {〈𝑔, 𝑘〉 ∣ (𝑔 ∈ USGraph ∧ 𝑔 RegGraph 𝑘)} |
cewlks 40795 | class EdgWalks |
c1wlks 40796 | class 1Walks |
cwlks 40797 | class UPWalks |
cwlkson 40798 | class WalksOn |
df-ewlks 40799 | ⊢ EdgWalks = (𝑔 ∈ V, 𝑠 ∈ ℕ0*
↦ {𝑓 ∣
[(iEdg‘𝑔) /
𝑖](𝑓 ∈ Word dom 𝑖 ∧ ∀𝑘 ∈ (1..^(#‘𝑓))𝑠 ≤ (#‘((𝑖‘(𝑓‘(𝑘 − 1))) ∩ (𝑖‘(𝑓‘𝑘)))))}) |
df-1wlks 40800 | ⊢ 1Walks = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(#‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(#‘𝑓))if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓‘𝑘))))}) |
df-wlks 40801 | ⊢ UPWalks = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(#‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(#‘𝑓))((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})}) |
df-wlkson 40802 | ⊢ WalksOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(1Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏)})) |
ctrls 40899 | class TrailS |
ctrlson 40900 | class TrailsOn |
df-trls 40901 | ⊢ TrailS = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(1Walks‘𝑔)𝑝 ∧ Fun ◡𝑓)}) |
df-trlson 40902 | ⊢ TrailsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑎(WalksOn‘𝑔)𝑏)𝑝 ∧ 𝑓(TrailS‘𝑔)𝑝)})) |
cpths 40919 | class PathS |
cspths 40920 | class SPathS |
cpthson 40921 | class PathsOn |
cspthson 40922 | class SPathsOn |
df-pths 40923 | ⊢ PathS = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(TrailS‘𝑔)𝑝 ∧ Fun ◡(𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)}) |
df-spths 40924 | ⊢ SPathS = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(TrailS‘𝑔)𝑝 ∧ Fun ◡𝑝)}) |
df-pthson 40925 | ⊢ PathsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑎(TrailsOn‘𝑔)𝑏)𝑝 ∧ 𝑓(PathS‘𝑔)𝑝)})) |
df-spthson 40926 | ⊢ SPathsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑎(TrailsOn‘𝑔)𝑏)𝑝 ∧ 𝑓(SPathS‘𝑔)𝑝)})) |
cclwlks 40976 | class ClWalkS |
df-clwlks 40977 | ⊢ ClWalkS = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(1Walks‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))}) |
ccrcts 40990 | class CircuitS |
ccycls 40991 | class CycleS |
df-crcts 40992 | ⊢ CircuitS = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(TrailS‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))}) |
df-cycls 40993 | ⊢ CycleS = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(PathS‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))}) |
cwwlks 41028 | class WWalkS |
cwwlksn 41029 | class WWalkSN |
cwwlksnon 41030 | class WWalksNOn |
cwwspthsn 41031 | class WSPathsN |
cwwspthsnon 41032 | class WSPathsNOn |
df-wwlks 41033 | ⊢ WWalkS = (𝑔 ∈ V ↦ {𝑤 ∈ Word (Vtx‘𝑔) ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔))}) |
df-wwlksn 41034 | ⊢ WWalkSN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalkS‘𝑔) ∣ (#‘𝑤) = (𝑛 + 1)}) |
df-wwlksnon 41035 | ⊢ WWalksNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑛 WWalkSN 𝑔) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑛) = 𝑏)})) |
df-wspthsn 41036 | ⊢ WSPathsN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (𝑛 WWalkSN 𝑔) ∣ ∃𝑓 𝑓(SPathS‘𝑔)𝑤}) |
df-wspthsnon 41037 | ⊢ WSPathsNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑎(𝑛 WWalksNOn 𝑔)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤})) |
cclwwlks 41183 | class ClWWalkS |
cclwwlksn 41184 | class ClWWalkSN |
df-clwwlks 41185 | ⊢ ClWWalkS = (𝑔 ∈ V ↦ {𝑤 ∈ Word (Vtx‘𝑔) ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔) ∧ {( lastS ‘𝑤), (𝑤‘0)} ∈ (Edg‘𝑔))}) |
df-clwwlksn 41186 | ⊢ ClWWalkSN = (𝑛 ∈ ℕ, 𝑔 ∈ V ↦ {𝑤 ∈ (ClWWalkS‘𝑔) ∣ (#‘𝑤) = 𝑛}) |
cconngr 41353 | class ConnGraph |
df-conngr 41354 | ⊢ ConnGraph = {𝑔 ∣ [(Vtx‘𝑔) / 𝑣]∀𝑘 ∈ 𝑣 ∀𝑛 ∈ 𝑣 ∃𝑓∃𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝} |
ceupth 41364 | class EulerPaths |
df-eupth 41365 | ⊢ EulerPaths = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(TrailS‘𝑔)𝑝 ∧ 𝑓:(0..^(#‘𝑓))–onto→dom (iEdg‘𝑔))}) |
cfrgr 41428 | class FriendGraph |
df-frgr 41429 | ⊢ FriendGraph = {𝑔 ∣ (𝑔 ∈ USGraph ∧
[(Vtx‘𝑔) /
𝑣][(Edg‘𝑔) / 𝑒]∀𝑘 ∈ 𝑣 ∀𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥 ∈ 𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒)} |
cmgmhm 41567 | class MgmHom |
csubmgm 41568 | class SubMgm |
df-mgmhm 41569 | ⊢ MgmHom = (𝑠 ∈ Mgm, 𝑡 ∈ Mgm ↦ {𝑓 ∈ ((Base‘𝑡) ↑𝑚
(Base‘𝑠)) ∣
∀𝑥 ∈
(Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g‘𝑠)𝑦)) = ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦))}) |
df-submgm 41570 | ⊢ SubMgm = (𝑠 ∈ Mgm ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑠)𝑦) ∈ 𝑡}) |
ccllaw 41609 | class clLaw |
casslaw 41610 | class assLaw |
ccomlaw 41611 | class comLaw |
df-cllaw 41612 | ⊢ clLaw = {〈𝑜, 𝑚〉 ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) ∈ 𝑚} |
df-comlaw 41613 | ⊢ comLaw = {〈𝑜, 𝑚〉 ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) = (𝑦𝑜𝑥)} |
df-asslaw 41614 | ⊢ assLaw = {〈𝑜, 𝑚〉 ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 ∀𝑧 ∈ 𝑚 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))} |
cintop 41622 | class intOp |
cclintop 41623 | class clIntOp |
cassintop 41624 | class assIntOp |
df-intop 41625 | ⊢ intOp = (𝑚 ∈ V, 𝑛 ∈ V ↦ (𝑛 ↑𝑚 (𝑚 × 𝑚))) |
df-clintop 41626 | ⊢ clIntOp = (𝑚 ∈ V ↦ (𝑚 intOp 𝑚)) |
df-assintop 41627 | ⊢ assIntOp = (𝑚 ∈ V ↦ {𝑜 ∈ ( clIntOp ‘𝑚) ∣ 𝑜 assLaw 𝑚}) |
cmgm2 41641 | class MgmALT |
ccmgm2 41642 | class CMgmALT |
csgrp2 41643 | class SGrpALT |
ccsgrp2 41644 | class CSGrpALT |
df-mgm2 41645 | ⊢ MgmALT = {𝑚 ∣ (+g‘𝑚) clLaw (Base‘𝑚)} |
df-cmgm2 41646 | ⊢ CMgmALT = {𝑚 ∈ MgmALT ∣
(+g‘𝑚)
comLaw (Base‘𝑚)} |
df-sgrp2 41647 | ⊢ SGrpALT = {𝑔 ∈ MgmALT ∣
(+g‘𝑔)
assLaw (Base‘𝑔)} |
df-csgrp2 41648 | ⊢ CSGrpALT = {𝑔 ∈ SGrpALT ∣
(+g‘𝑔)
comLaw (Base‘𝑔)} |
crng 41664 | class Rng |
df-rng0 41665 | ⊢ Rng = {𝑓 ∈ Abel ∣ ((mulGrp‘𝑓) ∈ SGrp ∧
[(Base‘𝑓) /
𝑏][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))))} |
crngh 41675 | class RngHomo |
crngs 41676 | class RngIsom |
df-rnghomo 41677 | ⊢ RngHomo = (𝑟 ∈ Rng, 𝑠 ∈ Rng ↦
⦋(Base‘𝑟) / 𝑣⦌⦋(Base‘𝑠) / 𝑤⦌{𝑓 ∈ (𝑤 ↑𝑚 𝑣) ∣ ∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦)))}) |
df-rngisom 41678 | ⊢ RngIsom = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RngHomo 𝑠) ∣ ◡𝑓 ∈ (𝑠 RngHomo 𝑟)}) |
crngc 41749 | class RngCat |
crngcALTV 41750 | class RngCatALTV |
df-rngc 41751 | ⊢ RngCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat (
RngHomo ↾ ((𝑢 ∩
Rng) × (𝑢 ∩
Rng))))) |
df-rngcALTV 41752 | ⊢ RngCatALTV = (𝑢 ∈ V ↦ ⦋(𝑢 ∩ Rng) / 𝑏⦌{〈(Base‘ndx), 𝑏〉, 〈(Hom ‘ndx),
(𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 RngHomo 𝑦))〉, 〈(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHomo (2nd
‘𝑣)) ↦ (𝑔 ∘ 𝑓)))〉}) |
cringc 41795 | class RingCat |
cringcALTV 41796 | class RingCatALTV |
df-ringc 41797 | ⊢ RingCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat (
RingHom ↾ ((𝑢 ∩
Ring) × (𝑢 ∩
Ring))))) |
df-ringcALTV 41798 | ⊢ RingCatALTV = (𝑢 ∈ V ↦ ⦋(𝑢 ∩ Ring) / 𝑏⦌{〈(Base‘ndx), 𝑏〉, 〈(Hom ‘ndx),
(𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 RingHom 𝑦))〉, 〈(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RingHom (2nd
‘𝑣)) ↦ (𝑔 ∘ 𝑓)))〉}) |
cdmatalt 41979 | class DMatALT |
cscmatalt 41980 | class ScMatALT |
df-dmatalt 41981 | ⊢ DMatALT = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ⦋(𝑛 Mat 𝑟) / 𝑎⦌(𝑎 ↾s {𝑚 ∈ (Base‘𝑎) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = (0g‘𝑟))})) |
df-scmatalt 41982 | ⊢ ScMatALT = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ⦋(𝑛 Mat 𝑟) / 𝑎⦌(𝑎 ↾s {𝑚 ∈ (Base‘𝑎) ∣ ∃𝑐 ∈ (Base‘𝑟)∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑟))})) |
clinc 41987 | class linC |
clinco 41988 | class LinCo |
df-linc 41989 | ⊢ linC = (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚
𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑥 ∈ 𝑣 ↦ ((𝑠‘𝑥)( ·𝑠
‘𝑚)𝑥))))) |
df-lco 41990 | ⊢ LinCo = (𝑚 ∈ V, 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ {𝑐 ∈ (Base‘𝑚) ∣ ∃𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚
𝑣)(𝑠 finSupp
(0g‘(Scalar‘𝑚)) ∧ 𝑐 = (𝑠( linC ‘𝑚)𝑣))}) |
clininds 42023 | class linIndS |
clindeps 42024 | class linDepS |
df-lininds 42025 | ⊢ linIndS = {〈𝑠, 𝑚〉 ∣ (𝑠 ∈ 𝒫 (Base‘𝑚) ∧ ∀𝑓 ∈
((Base‘(Scalar‘𝑚)) ↑𝑚 𝑠)((𝑓 finSupp
(0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)𝑠) = (0g‘𝑚)) → ∀𝑥 ∈ 𝑠 (𝑓‘𝑥) = (0g‘(Scalar‘𝑚))))} |
df-lindeps 42027 | ⊢ linDepS = {〈𝑠, 𝑚〉 ∣ ¬ 𝑠 linIndS 𝑚} |
cfdiv 42129 | class /f |
df-fdiv 42130 | ⊢ /f = (𝑓 ∈ V, 𝑔 ∈ V ↦ ((𝑓 ∘𝑓 / 𝑔) ↾ (𝑔 supp 0))) |
cbigo 42139 | class Ο |
df-bigo 42140 | ⊢ Ο = (𝑔 ∈ (ℝ ↑pm
ℝ) ↦ {𝑓 ∈
(ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝑔‘𝑦))}) |
cblen 42161 | class #b |
df-blen 42162 | ⊢ #b = (𝑛 ∈ V ↦ if(𝑛 = 0, 1, ((⌊‘(2 logb
(abs‘𝑛))) +
1))) |
cdig 42187 | class digit |
df-dig 42188 | ⊢ digit = (𝑏 ∈ ℕ ↦ (𝑘 ∈ ℤ, 𝑟 ∈ (0[,)+∞) ↦
((⌊‘((𝑏↑-𝑘) · 𝑟)) mod 𝑏))) |
csetrecs 42229 | class setrecs(𝐹) |
df-setrecs 42230 | ⊢ setrecs(𝐹) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} |
cpg 42251 | class Pg |
df-pg 42252 | ⊢ Pg = setrecs((𝑥 ∈ V ↦ (𝒫 𝑥 × 𝒫 𝑥))) |
cge-real 42260 | class ≥ |
cgt 42261 | class > |
df-gte 42262 | ⊢ ≥ = ◡ ≤ |
df-gt 42263 | ⊢ > = ◡ < |
csinh 42270 | class sinh |
ccosh 42271 | class cosh |
ctanh 42272 | class tanh |
df-sinh 42273 | ⊢ sinh = (𝑥 ∈ ℂ ↦ ((sin‘(i
· 𝑥)) /
i)) |
df-cosh 42274 | ⊢ cosh = (𝑥 ∈ ℂ ↦ (cos‘(i
· 𝑥))) |
df-tanh 42275 | ⊢ tanh = (𝑥 ∈ (◡cosh “ (ℂ ∖ {0})) ↦
((tan‘(i · 𝑥))
/ i)) |
csec 42281 | class sec |
ccsc 42282 | class csc |
ccot 42283 | class cot |
df-sec 42284 | ⊢ sec = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (cos‘𝑦) ≠ 0} ↦ (1 /
(cos‘𝑥))) |
df-csc 42285 | ⊢ csc = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ (1 /
(sin‘𝑥))) |
df-cot 42286 | ⊢ cot = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦
((cos‘𝑥) /
(sin‘𝑥))) |
cdp2 42304 | class _𝐴𝐵 |
cdp 42305 | class . |
df-dp2 42306 | ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10)) |
df-dp 42308 | ⊢ . = (𝑥 ∈ ℕ0, 𝑦 ∈ ℝ ↦ _𝑥𝑦) |
clog- 42315 | class log_ |
df-logbALT 42316 | ⊢ log_ = (𝑏 ∈ (ℂ ∖ {0, 1}) ↦
(𝑥 ∈ (ℂ ∖
{0}) ↦ ((log‘𝑥)
/ (log‘𝑏)))) |
wreflexive 42317 | wff 𝑅Reflexive𝐴 |
df-reflexive 42318 | ⊢ (𝑅Reflexive𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥 ∈ 𝐴 𝑥𝑅𝑥)) |
wirreflexive 42319 | wff 𝑅Irreflexive𝐴 |
df-irreflexive 42320 | ⊢ (𝑅Irreflexive𝐴 ↔ (𝑅 ⊆ (𝐴 × 𝐴) ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥)) |
walsi 42341 | wff ∀!𝑥(𝜑 → 𝜓) |
walsc 42342 | wff ∀!𝑥 ∈ 𝐴𝜑 |
df-alsi 42343 | ⊢ (∀!𝑥(𝜑 → 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥𝜑)) |
df-alsc 42344 | ⊢ (∀!𝑥 ∈ 𝐴𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 𝑥 ∈ 𝐴)) |