oddpwdcv - Mathbox for Thierry Arnoux

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Theorem oddpwdcv 33884

Description: Lemma for eulerpart 33911: value of the 𝐹 function. (Contributed by Thierry Arnoux, 9-Sep-2017.)

**Hypotheses**
Ref	Expression
oddpwdc.j	⊢ 𝐽 = {𝑧 ∈ ℕ ∣ ¬ 2 ∥ 𝑧}
oddpwdc.f	⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ₀ ↦ ((2↑𝑦) · 𝑥))

**Assertion**
Ref	Expression
oddpwdcv	⊢ (𝑊 ∈ (𝐽 × ℕ₀) → (𝐹‘𝑊) = ((2↑(2^nd ‘𝑊)) · (1^st ‘𝑊)))

Distinct variable groups: 𝑥,𝑦,𝑧 𝑥,𝐽,𝑦 𝑥,𝑊,𝑦 Allowed substitution hints: 𝐹(𝑥,𝑦,𝑧) 𝐽(𝑧) 𝑊(𝑧)

**Proof of Theorem oddpwdcv**
Step	Hyp	Ref	Expression
1		1st2nd2 8013	. . 3 ⊢ (𝑊 ∈ (𝐽 × ℕ₀) → 𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩)
2	1	fveq2d 6889	. 2 ⊢ (𝑊 ∈ (𝐽 × ℕ₀) → (𝐹‘𝑊) = (𝐹‘⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩))
3		df-ov 7408	. . 3 ⊢ ((1^st ‘𝑊)𝐹(2^nd ‘𝑊)) = (𝐹‘⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩)
4	3	a1i 11	. 2 ⊢ (𝑊 ∈ (𝐽 × ℕ₀) → ((1^st ‘𝑊)𝐹(2^nd ‘𝑊)) = (𝐹‘⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩))
5		elxp6 8008	. . . 4 ⊢ (𝑊 ∈ (𝐽 × ℕ₀) ↔ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∧ ((1^st ‘𝑊) ∈ 𝐽 ∧ (2^nd ‘𝑊) ∈ ℕ₀)))
6	5	simprbi 496	. . 3 ⊢ (𝑊 ∈ (𝐽 × ℕ₀) → ((1^st ‘𝑊) ∈ 𝐽 ∧ (2^nd ‘𝑊) ∈ ℕ₀))
7		oveq2 7413	. . . 4 ⊢ (𝑥 = (1^st ‘𝑊) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · (1^st ‘𝑊)))
8		oveq2 7413	. . . . 5 ⊢ (𝑦 = (2^nd ‘𝑊) → (2↑𝑦) = (2↑(2^nd ‘𝑊)))
9	8	oveq1d 7420	. . . 4 ⊢ (𝑦 = (2^nd ‘𝑊) → ((2↑𝑦) · (1^st ‘𝑊)) = ((2↑(2^nd ‘𝑊)) · (1^st ‘𝑊)))
10		oddpwdc.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐽, 𝑦 ∈ ℕ₀ ↦ ((2↑𝑦) · 𝑥))
11		ovex 7438	. . . 4 ⊢ ((2↑(2^nd ‘𝑊)) · (1^st ‘𝑊)) ∈ V
12	7, 9, 10, 11	ovmpo 7564	. . 3 ⊢ (((1^st ‘𝑊) ∈ 𝐽 ∧ (2^nd ‘𝑊) ∈ ℕ₀) → ((1^st ‘𝑊)𝐹(2^nd ‘𝑊)) = ((2↑(2^nd ‘𝑊)) · (1^st ‘𝑊)))
13	6, 12	syl 17	. 2 ⊢ (𝑊 ∈ (𝐽 × ℕ₀) → ((1^st ‘𝑊)𝐹(2^nd ‘𝑊)) = ((2↑(2^nd ‘𝑊)) · (1^st ‘𝑊)))
14	2, 4, 13	3eqtr2d 2772	1 ⊢ (𝑊 ∈ (𝐽 × ℕ₀) → (𝐹‘𝑊) = ((2↑(2^nd ‘𝑊)) · (1^st ‘𝑊)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {crab 3426 ⟨cop 4629 class class class wbr 5141 × cxp 5667 ‘cfv 6537 (class class class)co 7405 ∈ cmpo 7407 1^st c1st 7972 2^nd c2nd 7973 · cmul 11117 ℕcn 12216 2c2 12271 ℕ₀cn0 12476 ↑cexp 14032 ∥ cdvds 16204

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6489 df-fun 6539 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975

This theorem is referenced by: eulerpartlemgvv 33905 eulerpartlemgh 33907 eulerpartlemgs2 33909

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