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| Mirrors > Home > MPE Home > Th. List > Mathboxes > oddpwdcv | Structured version Visualization version GIF version | ||
| Description: Lemma for eulerpart 34395: 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 7960 | . . 3 ⊢ (𝑊 ∈ (𝐽 × ℕ0) → 𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉) | |
| 2 | 1 | fveq2d 6826 | . 2 ⊢ (𝑊 ∈ (𝐽 × ℕ0) → (𝐹‘𝑊) = (𝐹‘〈(1st ‘𝑊), (2nd ‘𝑊)〉)) |
| 3 | df-ov 7349 | . . 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 7955 | . . . 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 7354 | . . . 4 ⊢ (𝑥 = (1st ‘𝑊) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · (1st ‘𝑊))) | |
| 8 | oveq2 7354 | . . . . 5 ⊢ (𝑦 = (2nd ‘𝑊) → (2↑𝑦) = (2↑(2nd ‘𝑊))) | |
| 9 | 8 | oveq1d 7361 | . . . 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 7379 | . . . 4 ⊢ ((2↑(2nd ‘𝑊)) · (1st ‘𝑊)) ∈ V | |
| 12 | 7, 9, 10, 11 | ovmpo 7506 | . . 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 2772 | 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 2111 {crab 3395 〈cop 4579 class class class wbr 5089 × cxp 5612 ‘cfv 6481 (class class class)co 7346 ∈ cmpo 7348 1st c1st 7919 2nd c2nd 7920 · cmul 11011 ℕcn 12125 2c2 12180 ℕ0cn0 12381 ↑cexp 13968 ∥ cdvds 16163 |
| 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 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5232 ax-nul 5242 ax-pr 5368 ax-un 7668 |
| 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 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-sbc 3737 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4281 df-if 4473 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-iota 6437 df-fun 6483 df-fv 6489 df-ov 7349 df-oprab 7350 df-mpo 7351 df-1st 7921 df-2nd 7922 |
| This theorem is referenced by: eulerpartlemgvv 34389 eulerpartlemgh 34391 eulerpartlemgs2 34393 |
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