oddpwdcv - Mathbox for Thierry Arnoux

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Theorem oddpwdcv 34320

Description: Lemma for eulerpart 34347: 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 8069	. . 3 ⊢ (𝑊 ∈ (𝐽 × ℕ₀) → 𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩)
2	1	fveq2d 6924	. 2 ⊢ (𝑊 ∈ (𝐽 × ℕ₀) → (𝐹‘𝑊) = (𝐹‘⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩))
3		df-ov 7451	. . 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 8064	. . . 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 7456	. . . 4 ⊢ (𝑥 = (1^st ‘𝑊) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · (1^st ‘𝑊)))
8		oveq2 7456	. . . . 5 ⊢ (𝑦 = (2^nd ‘𝑊) → (2↑𝑦) = (2↑(2^nd ‘𝑊)))
9	8	oveq1d 7463	. . . 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 7481	. . . 4 ⊢ ((2↑(2^nd ‘𝑊)) · (1^st ‘𝑊)) ∈ V
12	7, 9, 10, 11	ovmpo 7610	. . 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 2786	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 1537 ∈ wcel 2108 {crab 3443 ⟨cop 4654 class class class wbr 5166 × cxp 5698 ‘cfv 6573 (class class class)co 7448 ∈ cmpo 7450 1^st c1st 8028 2^nd c2nd 8029 · cmul 11189 ℕcn 12293 2c2 12348 ℕ₀cn0 12553 ↑cexp 14112 ∥ cdvds 16302

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-iota 6525 df-fun 6575 df-fv 6581 df-ov 7451 df-oprab 7452 df-mpo 7453 df-1st 8030 df-2nd 8031

This theorem is referenced by: eulerpartlemgvv 34341 eulerpartlemgh 34343 eulerpartlemgs2 34345

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