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Mirrors > Home > MPE Home > Th. List > Mathboxes > oddpwdcv | Structured version Visualization version GIF version |
Description: Lemma for eulerpart 34364: 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 8052 | . . 3 ⊢ (𝑊 ∈ (𝐽 × ℕ0) → 𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉) | |
2 | 1 | fveq2d 6911 | . 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 8047 | . . . 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 2781 | 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 1537 ∈ wcel 2106 {crab 3433 〈cop 4637 class class class wbr 5148 × cxp 5687 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 1st c1st 8011 2nd c2nd 8012 · cmul 11158 ℕcn 12264 2c2 12319 ℕ0cn0 12524 ↑cexp 14099 ∥ cdvds 16287 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 |
This theorem is referenced by: eulerpartlemgvv 34358 eulerpartlemgh 34360 eulerpartlemgs2 34362 |
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