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Mirrors > Home > MPE Home > Th. List > Mathboxes > eulerpartlem1 | Structured version Visualization version GIF version |
Description: Lemma for eulerpart 33210. (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 12200 | . . . 4 ⊢ ℕ ∈ V | |
3 | 1, 2 | rabex2 5327 | . . 3 ⊢ 𝐽 ∈ V |
4 | nn0ex 12460 | . . 3 ⊢ ℕ0 ∈ V | |
5 | eqid 2731 | . . 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 31830 | . 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 4073 | . . . . . . 7 ⊢ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin} ⊆ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) | |
10 | 4 | pwex 5371 | . . . . . . . 8 ⊢ 𝒫 ℕ0 ∈ V |
11 | inss1 4224 | . . . . . . . 8 ⊢ (𝒫 ℕ0 ∩ Fin) ⊆ 𝒫 ℕ0 | |
12 | mapss 8866 | . . . . . . . 8 ⊢ ((𝒫 ℕ0 ∈ V ∧ (𝒫 ℕ0 ∩ Fin) ⊆ 𝒫 ℕ0) → ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ⊆ (𝒫 ℕ0 ↑m 𝐽)) | |
13 | 10, 11, 12 | mp2an 690 | . . . . . . 7 ⊢ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ⊆ (𝒫 ℕ0 ↑m 𝐽) |
14 | 9, 13 | sstri 3987 | . . . . . 6 ⊢ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑m 𝐽) ∣ (𝑟 supp ∅) ∈ Fin} ⊆ (𝒫 ℕ0 ↑m 𝐽) |
15 | 6, 14 | eqsstri 4012 | . . . . 5 ⊢ 𝐻 ⊆ (𝒫 ℕ0 ↑m 𝐽) |
16 | resmpt 6027 | . . . . 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 2762 | . . 3 ⊢ 𝑀 = ((𝑟 ∈ (𝒫 ℕ0 ↑m 𝐽) ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) ↾ 𝐻) |
19 | f1oeq1 6808 | . . 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 230 | 1 ⊢ 𝑀:𝐻–1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3060 {crab 3431 Vcvv 3473 ∩ cin 3943 ⊆ wss 3944 ∅c0 4318 𝒫 cpw 4596 class class class wbr 5141 {copab 5203 ↦ cmpt 5224 × cxp 5667 ◡ccnv 5668 ↾ cres 5671 “ cima 5672 –1-1-onto→wf1o 6531 ‘cfv 6532 (class class class)co 7393 ∈ cmpo 7395 supp csupp 8128 ↑m cmap 8803 Fincfn 8922 1c1 11093 · cmul 11097 ≤ cle 11231 ℕcn 12194 2c2 12249 ℕ0cn0 12454 ↑cexp 14009 Σcsu 15614 ∥ cdvds 16179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-ac2 10440 ax-cnex 11148 ax-1cn 11150 ax-addcl 11152 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-isom 6541 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-supp 8129 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-er 8686 df-map 8805 df-en 8923 df-dom 8924 df-fin 8926 df-card 9916 df-acn 9919 df-ac 10093 df-nn 12195 df-n0 12455 |
This theorem is referenced by: eulerpartgbij 33200 eulerpartlemgvv 33204 eulerpartlemgf 33207 |
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