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Mirrors > Home > MPE Home > Th. List > Mathboxes > oddpwdcv | Structured version Visualization version GIF version |
Description: Lemma for eulerpart 29771: 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 7096 | . . 3 ⊢ (𝑊 ∈ (𝐽 × ℕ0) → 𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉) | |
2 | 1 | fveq2d 6107 | . 2 ⊢ (𝑊 ∈ (𝐽 × ℕ0) → (𝐹‘𝑊) = (𝐹‘〈(1st ‘𝑊), (2nd ‘𝑊)〉)) |
3 | df-ov 6552 | . . 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 7091 | . . . 4 ⊢ (𝑊 ∈ (𝐽 × ℕ0) ↔ (𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉 ∧ ((1st ‘𝑊) ∈ 𝐽 ∧ (2nd ‘𝑊) ∈ ℕ0))) | |
6 | 5 | simprbi 479 | . . 3 ⊢ (𝑊 ∈ (𝐽 × ℕ0) → ((1st ‘𝑊) ∈ 𝐽 ∧ (2nd ‘𝑊) ∈ ℕ0)) |
7 | oveq2 6557 | . . . 4 ⊢ (𝑥 = (1st ‘𝑊) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · (1st ‘𝑊))) | |
8 | oveq2 6557 | . . . . 5 ⊢ (𝑦 = (2nd ‘𝑊) → (2↑𝑦) = (2↑(2nd ‘𝑊))) | |
9 | 8 | oveq1d 6564 | . . . 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 6577 | . . . 4 ⊢ ((2↑(2nd ‘𝑊)) · (1st ‘𝑊)) ∈ V | |
12 | 7, 9, 10, 11 | ovmpt2 6694 | . . 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 2650 | 1 ⊢ (𝑊 ∈ (𝐽 × ℕ0) → (𝐹‘𝑊) = ((2↑(2nd ‘𝑊)) · (1st ‘𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 〈cop 4131 class class class wbr 4583 × cxp 5036 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 1st c1st 7057 2nd c2nd 7058 · cmul 9820 ℕcn 10897 2c2 10947 ℕ0cn0 11169 ↑cexp 12722 ∥ cdvds 14821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 |
This theorem is referenced by: eulerpartlemgvv 29765 eulerpartlemgh 29767 eulerpartlemgs2 29769 |
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