| Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > oddpwdcv | Structured version Visualization version GIF version | ||
| Description: Lemma for eulerpart 34384: value of the 𝐹 function. (Contributed by Thierry Arnoux, 9-Sep-2017.) |
| Ref | Expression |
|---|---|
| oddpwdc.j | ⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧} |
| oddpwdc.f | ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) |
| Ref | Expression |
|---|---|
| oddpwdcv | ⊢ (𝑊 ∈ (𝐽 × ℕ0) → (𝐹‘𝑊) = ((2↑(2nd ‘𝑊)) · (1st ‘𝑊))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 1st2nd2 8053 | . . 3 ⊢ (𝑊 ∈ (𝐽 × ℕ0) → 𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉) | |
| 2 | 1 | fveq2d 6910 | . 2 ⊢ (𝑊 ∈ (𝐽 × ℕ0) → (𝐹‘𝑊) = (𝐹‘〈(1st ‘𝑊), (2nd ‘𝑊)〉)) |
| 3 | df-ov 7434 | . . 3 ⊢ ((1st ‘𝑊)𝐹(2nd ‘𝑊)) = (𝐹‘〈(1st ‘𝑊), (2nd ‘𝑊)〉) | |
| 4 | 3 | a1i 11 | . 2 ⊢ (𝑊 ∈ (𝐽 × ℕ0) → ((1st ‘𝑊)𝐹(2nd ‘𝑊)) = (𝐹‘〈(1st ‘𝑊), (2nd ‘𝑊)〉)) |
| 5 | elxp6 8048 | . . . 4 ⊢ (𝑊 ∈ (𝐽 × ℕ0) ↔ (𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉 ∧ ((1st ‘𝑊) ∈ 𝐽 ∧ (2nd ‘𝑊) ∈ ℕ0))) | |
| 6 | 5 | simprbi 496 | . . 3 ⊢ (𝑊 ∈ (𝐽 × ℕ0) → ((1st ‘𝑊) ∈ 𝐽 ∧ (2nd ‘𝑊) ∈ ℕ0)) |
| 7 | oveq2 7439 | . . . 4 ⊢ (𝑥 = (1st ‘𝑊) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · (1st ‘𝑊))) | |
| 8 | oveq2 7439 | . . . . 5 ⊢ (𝑦 = (2nd ‘𝑊) → (2↑𝑦) = (2↑(2nd ‘𝑊))) | |
| 9 | 8 | oveq1d 7446 | . . . 4 ⊢ (𝑦 = (2nd ‘𝑊) → ((2↑𝑦) · (1st ‘𝑊)) = ((2↑(2nd ‘𝑊)) · (1st ‘𝑊))) |
| 10 | oddpwdc.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ0 ↦ ((2↑𝑦) · 𝑥)) | |
| 11 | ovex 7464 | . . . 4 ⊢ ((2↑(2nd ‘𝑊)) · (1st ‘𝑊)) ∈ V | |
| 12 | 7, 9, 10, 11 | ovmpo 7593 | . . 3 ⊢ (((1st ‘𝑊) ∈ 𝐽 ∧ (2nd ‘𝑊) ∈ ℕ0) → ((1st ‘𝑊)𝐹(2nd ‘𝑊)) = ((2↑(2nd ‘𝑊)) · (1st ‘𝑊))) |
| 13 | 6, 12 | syl 17 | . 2 ⊢ (𝑊 ∈ (𝐽 × ℕ0) → ((1st ‘𝑊)𝐹(2nd ‘𝑊)) = ((2↑(2nd ‘𝑊)) · (1st ‘𝑊))) |
| 14 | 2, 4, 13 | 3eqtr2d 2783 | 1 ⊢ (𝑊 ∈ (𝐽 × ℕ0) → (𝐹‘𝑊) = ((2↑(2nd ‘𝑊)) · (1st ‘𝑊))) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 〈cop 4632 class class class wbr 5143 × cxp 5683 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 1st c1st 8012 2nd c2nd 8013 · cmul 11160 ℕcn 12266 2c2 12321 ℕ0cn0 12526 ↑cexp 14102 ∥ cdvds 16290 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 |
| This theorem is referenced by: eulerpartlemgvv 34378 eulerpartlemgh 34380 eulerpartlemgs2 34382 |
| Copyright terms: Public domain | W3C validator |