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| Mirrors > Home > MPE Home > Th. List > Mathboxes > oddpwdcv | Structured version Visualization version GIF version | ||
| Description: Lemma for eulerpart 34539: 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 7972 | . . 3 ⊢ (𝑊 ∈ (𝐽 × ℕ0) → 𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉) | |
| 2 | 1 | fveq2d 6838 | . 2 ⊢ (𝑊 ∈ (𝐽 × ℕ0) → (𝐹‘𝑊) = (𝐹‘〈(1st ‘𝑊), (2nd ‘𝑊)〉)) |
| 3 | df-ov 7361 | . . 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 7967 | . . . 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 7366 | . . . 4 ⊢ (𝑥 = (1st ‘𝑊) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · (1st ‘𝑊))) | |
| 8 | oveq2 7366 | . . . . 5 ⊢ (𝑦 = (2nd ‘𝑊) → (2↑𝑦) = (2↑(2nd ‘𝑊))) | |
| 9 | 8 | oveq1d 7373 | . . . 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 7391 | . . . 4 ⊢ ((2↑(2nd ‘𝑊)) · (1st ‘𝑊)) ∈ V | |
| 12 | 7, 9, 10, 11 | ovmpo 7518 | . . 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 2777 | 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 1541 ∈ wcel 2113 {crab 3399 〈cop 4586 class class class wbr 5098 × cxp 5622 ‘cfv 6492 (class class class)co 7358 ∈ cmpo 7360 1st c1st 7931 2nd c2nd 7932 · cmul 11031 ℕcn 12145 2c2 12200 ℕ0cn0 12401 ↑cexp 13984 ∥ cdvds 16179 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pr 5377 ax-un 7680 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rab 3400 df-v 3442 df-sbc 3741 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-br 5099 df-opab 5161 df-mpt 5180 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-iota 6448 df-fun 6494 df-fv 6500 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7933 df-2nd 7934 |
| This theorem is referenced by: eulerpartlemgvv 34533 eulerpartlemgh 34535 eulerpartlemgs2 34537 |
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