![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > eulerpartlem1 | Structured version Visualization version GIF version |
Description: Lemma for eulerpart 31318. (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 11445 | . . . 4 ⊢ ℕ ∈ V | |
3 | 1, 2 | rabex2 5090 | . . 3 ⊢ 𝐽 ∈ V |
4 | nn0ex 11713 | . . 3 ⊢ ℕ0 ∈ V | |
5 | eqid 2773 | . . 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 30247 | . 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 3941 | . . . . . . 7 ⊢ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin} ⊆ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) | |
10 | 4 | pwex 5131 | . . . . . . . 8 ⊢ 𝒫 ℕ0 ∈ V |
11 | inss1 4087 | . . . . . . . 8 ⊢ (𝒫 ℕ0 ∩ Fin) ⊆ 𝒫 ℕ0 | |
12 | mapss 8250 | . . . . . . . 8 ⊢ ((𝒫 ℕ0 ∈ V ∧ (𝒫 ℕ0 ∩ Fin) ⊆ 𝒫 ℕ0) → ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ⊆ (𝒫 ℕ0 ↑𝑚 𝐽)) | |
13 | 10, 11, 12 | mp2an 680 | . . . . . . 7 ⊢ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ⊆ (𝒫 ℕ0 ↑𝑚 𝐽) |
14 | 9, 13 | sstri 3862 | . . . . . 6 ⊢ {𝑟 ∈ ((𝒫 ℕ0 ∩ Fin) ↑𝑚 𝐽) ∣ (𝑟 supp ∅) ∈ Fin} ⊆ (𝒫 ℕ0 ↑𝑚 𝐽) |
15 | 6, 14 | eqsstri 3886 | . . . . 5 ⊢ 𝐻 ⊆ (𝒫 ℕ0 ↑𝑚 𝐽) |
16 | resmpt 5748 | . . . . 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 2800 | . . 3 ⊢ 𝑀 = ((𝑟 ∈ (𝒫 ℕ0 ↑𝑚 𝐽) ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ (𝑟‘𝑥))}) ↾ 𝐻) |
19 | f1oeq1 6431 | . . 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 223 | 1 ⊢ 𝑀:𝐻–1-1-onto→(𝒫 (𝐽 × ℕ0) ∩ Fin) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 198 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ∀wral 3083 {crab 3087 Vcvv 3410 ∩ cin 3823 ⊆ wss 3824 ∅c0 4173 𝒫 cpw 4417 class class class wbr 4926 {copab 4988 ↦ cmpt 5005 × cxp 5402 ◡ccnv 5403 ↾ cres 5406 “ cima 5407 –1-1-onto→wf1o 6185 ‘cfv 6186 (class class class)co 6975 ∈ cmpo 6977 supp csupp 7632 ↑𝑚 cmap 8205 Fincfn 8305 1c1 10335 · cmul 10339 ≤ cle 10474 ℕcn 11438 2c2 11494 ℕ0cn0 11706 ↑cexp 13243 Σcsu 14902 ∥ cdvds 15466 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-rep 5046 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-ac2 9682 ax-cnex 10390 ax-1cn 10392 ax-addcl 10394 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-int 4747 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-se 5364 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-isom 6195 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-om 7396 df-1st 7500 df-2nd 7501 df-supp 7633 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-1o 7904 df-oadd 7908 df-er 8088 df-map 8207 df-en 8306 df-dom 8307 df-fin 8309 df-card 9161 df-acn 9164 df-ac 9335 df-nn 11439 df-n0 11707 |
This theorem is referenced by: eulerpartgbij 31308 eulerpartlemgvv 31312 eulerpartlemgf 31315 |
Copyright terms: Public domain | W3C validator |