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Mirrors > Home > MPE Home > Th. List > Mathboxes > eulerpartlem1 | Structured version Visualization version GIF version |
Description: Lemma for eulerpart 29771. (Contributed by Thierry Arnoux, 27-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.) |
Ref | Expression |
---|---|
eulerpart.p | ⊢ 𝑃 = {𝑓 ∈ (ℕ0 ↑𝑚 ℕ) ∣ ((◡𝑓 “ ℕ) ∈ Fin ∧ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘) = 𝑁)} |
eulerpart.o | ⊢ 𝑂 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ (◡𝑔 “ ℕ) ¬ 2 ∥ 𝑛} |
eulerpart.d | ⊢ 𝐷 = {𝑔 ∈ 𝑃 ∣ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ≤ 1} |
eulerpart.j | ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} |
eulerpart.f | ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) |
eulerpart.h | ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin} |
eulerpart.m | ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) |
Ref | Expression |
---|---|
eulerpartlem1 | ⊢ 𝑀:𝐻–1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eulerpart.j | . . . 4 ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} | |
2 | nnex 10903 | . . . 4 ⊢ ℕ ∈ V | |
3 | 1, 2 | rabex2 4742 | . . 3 ⊢ 𝐽 ∈ V |
4 | nn0ex 11175 | . . 3 ⊢ ℕ0 ∈ V | |
5 | eqid 2610 | . . 3 ⊢ (𝑟 ∈ (𝒫 ℕ0 ↑𝑚 𝐽) ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) = (𝑟 ∈ (𝒫 ℕ0 ↑𝑚 𝐽) ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) | |
6 | eulerpart.h | . . 3 ⊢ 𝐻 = {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin} | |
7 | 3, 4, 5, 6 | fpwrelmapffs 28897 | . 2 ⊢ ((𝑟 ∈ (𝒫 ℕ0 ↑𝑚 𝐽) ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) ↾ 𝐻):𝐻–1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin) |
8 | eulerpart.m | . . . 4 ⊢ 𝑀 = (𝑟 ∈ 𝐻 ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) | |
9 | ssrab2 3650 | . . . . . . 7 ⊢ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin} ⊆ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) | |
10 | 4 | pwex 4774 | . . . . . . . 8 ⊢ 𝒫 ℕ0 ∈ V |
11 | inss1 3795 | . . . . . . . 8 ⊢ (𝒫 ℕ0 ∩ Fin) ⊆ 𝒫 ℕ0 | |
12 | mapss 7786 | . . . . . . . 8 ⊢ ((𝒫 ℕ0 ∈ V ∧ (𝒫 ℕ0 ∩ Fin) ⊆ 𝒫 ℕ0) → ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ⊆ (𝒫 ℕ0 ↑𝑚 𝐽)) | |
13 | 10, 11, 12 | mp2an 704 | . . . . . . 7 ⊢ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ⊆ (𝒫 ℕ0 ↑𝑚 𝐽) |
14 | 9, 13 | sstri 3577 | . . . . . 6 ⊢ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin} ⊆ (𝒫 ℕ0 ↑𝑚 𝐽) |
15 | 6, 14 | eqsstri 3598 | . . . . 5 ⊢ 𝐻 ⊆ (𝒫 ℕ0 ↑𝑚 𝐽) |
16 | resmpt 5369 | . . . . 5 ⊢ (𝐻 ⊆ (𝒫 ℕ0 ↑𝑚 𝐽) → ((𝑟 ∈ (𝒫 ℕ0 ↑𝑚 𝐽) ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) ↾ 𝐻) = (𝑟 ∈ 𝐻 ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))})) | |
17 | 15, 16 | ax-mp 5 | . . . 4 ⊢ ((𝑟 ∈ (𝒫 ℕ0 ↑𝑚 𝐽) ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) ↾ 𝐻) = (𝑟 ∈ 𝐻 ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) |
18 | 8, 17 | eqtr4i 2635 | . . 3 ⊢ 𝑀 = ((𝑟 ∈ (𝒫 ℕ0 ↑𝑚 𝐽) ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) ↾ 𝐻) |
19 | f1oeq1 6040 | . . 3 ⊢ (𝑀 = ((𝑟 ∈ (𝒫 ℕ0 ↑𝑚 𝐽) ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) ↾ 𝐻) → (𝑀:𝐻–1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin) ↔ ((𝑟 ∈ (𝒫 ℕ0 ↑𝑚 𝐽) ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) ↾ 𝐻):𝐻–1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin))) | |
20 | 18, 19 | ax-mp 5 | . 2 ⊢ (𝑀:𝐻–1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin) ↔ ((𝑟 ∈ (𝒫 ℕ0 ↑𝑚 𝐽) ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) ↾ 𝐻):𝐻–1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin)) |
21 | 7, 20 | mpbir 220 | 1 ⊢ 𝑀:𝐻–1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 class class class wbr 4583 {copab 4642 ↦ cmpt 4643 × cxp 5036 ◡ccnv 5037 ↾ cres 5040 “ cima 5041 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 supp csupp 7182 ↑𝑚 cmap 7744 Fincfn 7841 1c1 9816 · cmul 9820 ≤ cle 9954 ℕcn 10897 2c2 10947 ℕ0cn0 11169 ↑cexp 12722 Σcsu 14264 ∥ cdvds 14821 |
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-ac2 9168 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-i2m1 9883 ax-1ne0 9884 ax-rrecex 9887 ax-cnre 9888 |
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-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-int 4411 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-se 4998 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-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-fin 7845 df-card 8648 df-acn 8651 df-ac 8822 df-nn 10898 df-n0 11170 |
This theorem is referenced by: eulerpartgbij 29761 eulerpartlemgvv 29765 eulerpartlemgf 29768 |
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