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Mirrors > Home > MPE Home > Th. List > psrbagev1 | Structured version Visualization version GIF version |
Description: A bag of multipliers provides the conditions for a valid sum. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 18-Jul-2019.) |
Ref | Expression |
---|---|
psrbagev1.d | ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} |
psrbagev1.c | ⊢ 𝐶 = (Base‘𝑇) |
psrbagev1.x | ⊢ · = (.g‘𝑇) |
psrbagev1.z | ⊢ 0 = (0g‘𝑇) |
psrbagev1.t | ⊢ (𝜑 → 𝑇 ∈ CMnd) |
psrbagev1.b | ⊢ (𝜑 → 𝐵 ∈ 𝐷) |
psrbagev1.g | ⊢ (𝜑 → 𝐺:𝐼⟶𝐶) |
psrbagev1.i | ⊢ (𝜑 → 𝐼 ∈ V) |
Ref | Expression |
---|---|
psrbagev1 | ⊢ (𝜑 → ((𝐵 ∘𝑓 · 𝐺):𝐼⟶𝐶 ∧ (𝐵 ∘𝑓 · 𝐺) finSupp 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psrbagev1.t | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ CMnd) | |
2 | cmnmnd 18031 | . . . . 5 ⊢ (𝑇 ∈ CMnd → 𝑇 ∈ Mnd) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑇 ∈ Mnd) |
4 | psrbagev1.c | . . . . . 6 ⊢ 𝐶 = (Base‘𝑇) | |
5 | psrbagev1.x | . . . . . 6 ⊢ · = (.g‘𝑇) | |
6 | 4, 5 | mulgnn0cl 17381 | . . . . 5 ⊢ ((𝑇 ∈ Mnd ∧ 𝑦 ∈ ℕ0 ∧ 𝑧 ∈ 𝐶) → (𝑦 · 𝑧) ∈ 𝐶) |
7 | 6 | 3expb 1258 | . . . 4 ⊢ ((𝑇 ∈ Mnd ∧ (𝑦 ∈ ℕ0 ∧ 𝑧 ∈ 𝐶)) → (𝑦 · 𝑧) ∈ 𝐶) |
8 | 3, 7 | sylan 487 | . . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ ℕ0 ∧ 𝑧 ∈ 𝐶)) → (𝑦 · 𝑧) ∈ 𝐶) |
9 | psrbagev1.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ V) | |
10 | psrbagev1.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
11 | psrbagev1.d | . . . . 5 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
12 | 11 | psrbagf 19186 | . . . 4 ⊢ ((𝐼 ∈ V ∧ 𝐵 ∈ 𝐷) → 𝐵:𝐼⟶ℕ0) |
13 | 9, 10, 12 | syl2anc 691 | . . 3 ⊢ (𝜑 → 𝐵:𝐼⟶ℕ0) |
14 | psrbagev1.g | . . 3 ⊢ (𝜑 → 𝐺:𝐼⟶𝐶) | |
15 | inidm 3784 | . . 3 ⊢ (𝐼 ∩ 𝐼) = 𝐼 | |
16 | 8, 13, 14, 9, 9, 15 | off 6810 | . 2 ⊢ (𝜑 → (𝐵 ∘𝑓 · 𝐺):𝐼⟶𝐶) |
17 | ovex 6577 | . . . 4 ⊢ (𝐵 ∘𝑓 · 𝐺) ∈ V | |
18 | 17 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐵 ∘𝑓 · 𝐺) ∈ V) |
19 | ffn 5958 | . . . . . 6 ⊢ (𝐵:𝐼⟶ℕ0 → 𝐵 Fn 𝐼) | |
20 | 13, 19 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐵 Fn 𝐼) |
21 | ffn 5958 | . . . . . 6 ⊢ (𝐺:𝐼⟶𝐶 → 𝐺 Fn 𝐼) | |
22 | 14, 21 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐺 Fn 𝐼) |
23 | 20, 22, 9, 9, 15 | offn 6806 | . . . 4 ⊢ (𝜑 → (𝐵 ∘𝑓 · 𝐺) Fn 𝐼) |
24 | fnfun 5902 | . . . 4 ⊢ ((𝐵 ∘𝑓 · 𝐺) Fn 𝐼 → Fun (𝐵 ∘𝑓 · 𝐺)) | |
25 | 23, 24 | syl 17 | . . 3 ⊢ (𝜑 → Fun (𝐵 ∘𝑓 · 𝐺)) |
26 | psrbagev1.z | . . . . 5 ⊢ 0 = (0g‘𝑇) | |
27 | fvex 6113 | . . . . 5 ⊢ (0g‘𝑇) ∈ V | |
28 | 26, 27 | eqeltri 2684 | . . . 4 ⊢ 0 ∈ V |
29 | 28 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ V) |
30 | 11 | psrbagfsupp 19330 | . . . . 5 ⊢ ((𝐵 ∈ 𝐷 ∧ 𝐼 ∈ V) → 𝐵 finSupp 0) |
31 | 10, 9, 30 | syl2anc 691 | . . . 4 ⊢ (𝜑 → 𝐵 finSupp 0) |
32 | 31 | fsuppimpd 8165 | . . 3 ⊢ (𝜑 → (𝐵 supp 0) ∈ Fin) |
33 | ssid 3587 | . . . . 5 ⊢ (𝐵 supp 0) ⊆ (𝐵 supp 0) | |
34 | 33 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐵 supp 0) ⊆ (𝐵 supp 0)) |
35 | 4, 26, 5 | mulg0 17369 | . . . . 5 ⊢ (𝑧 ∈ 𝐶 → (0 · 𝑧) = 0 ) |
36 | 35 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (0 · 𝑧) = 0 ) |
37 | c0ex 9913 | . . . . 5 ⊢ 0 ∈ V | |
38 | 37 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ V) |
39 | 34, 36, 13, 14, 9, 38 | suppssof1 7215 | . . 3 ⊢ (𝜑 → ((𝐵 ∘𝑓 · 𝐺) supp 0 ) ⊆ (𝐵 supp 0)) |
40 | suppssfifsupp 8173 | . . 3 ⊢ ((((𝐵 ∘𝑓 · 𝐺) ∈ V ∧ Fun (𝐵 ∘𝑓 · 𝐺) ∧ 0 ∈ V) ∧ ((𝐵 supp 0) ∈ Fin ∧ ((𝐵 ∘𝑓 · 𝐺) supp 0 ) ⊆ (𝐵 supp 0))) → (𝐵 ∘𝑓 · 𝐺) finSupp 0 ) | |
41 | 18, 25, 29, 32, 39, 40 | syl32anc 1326 | . 2 ⊢ (𝜑 → (𝐵 ∘𝑓 · 𝐺) finSupp 0 ) |
42 | 16, 41 | jca 553 | 1 ⊢ (𝜑 → ((𝐵 ∘𝑓 · 𝐺):𝐼⟶𝐶 ∧ (𝐵 ∘𝑓 · 𝐺) finSupp 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 ⊆ wss 3540 class class class wbr 4583 ◡ccnv 5037 “ cima 5041 Fun wfun 5798 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ∘𝑓 cof 6793 supp csupp 7182 ↑𝑚 cmap 7744 Fincfn 7841 finSupp cfsupp 8158 0cc0 9815 ℕcn 10897 ℕ0cn0 11169 Basecbs 15695 0gc0g 15923 Mndcmnd 17117 .gcmg 17363 CMndccmn 18016 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-seq 12664 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mulg 17364 df-cmn 18018 |
This theorem is referenced by: psrbagev2 19332 evlslem1 19336 |
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