oddpwdcv - Mathbox for Thierry Arnoux

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Theorem oddpwdcv 34758

Description: Lemma for eulerpart 34785: 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 8027	. . 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 7419	. . 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 8022	. . . 4 ⊢ (𝑊 ∈ (𝐽 × ℕ₀) ↔ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∧ ((1^st ‘𝑊) ∈ 𝐽 ∧ (2^nd ‘𝑊) ∈ ℕ₀)))
6	5	simprbi 503	. . 3 ⊢ (𝑊 ∈ (𝐽 × ℕ₀) → ((1^st ‘𝑊) ∈ 𝐽 ∧ (2^nd ‘𝑊) ∈ ℕ₀))
7		oveq2 7424	. . . 4 ⊢ (𝑥 = (1^st ‘𝑊) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · (1^st ‘𝑊)))
8		oveq2 7424	. . . . 5 ⊢ (𝑦 = (2^nd ‘𝑊) → (2↑𝑦) = (2↑(2^nd ‘𝑊)))
9	8	oveq1d 7431	. . . 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 7449	. . . 4 ⊢ ((2↑(2^nd ‘𝑊)) · (1^st ‘𝑊)) ∈ V
12	7, 9, 10, 11	ovmpo 7576	. . 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 18	. 2 ⊢ (𝑊 ∈ (𝐽 × ℕ₀) → ((1^st ‘𝑊)𝐹(2^nd ‘𝑊)) = ((2↑(2^nd ‘𝑊)) · (1^st ‘𝑊)))
14	2, 4, 13	3eqtr2d 2807	1 ⊢ (𝑊 ∈ (𝐽 × ℕ₀) → (𝐹‘𝑊) = ((2↑(2^nd ‘𝑊)) · (1^st ‘𝑊)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 {crab 3419 ⟨cop 4598 class class class wbr 5112 × cxp 5662 ‘cfv 6540 (class class class)co 7416 ∈ cmpo 7418 1^st c1st 7986 2^nd c2nd 7987 · cmul 11115 ℕcn 12243 2c2 12305 ℕ₀cn0 12514 ↑cexp 14108 ∥ cdvds 16320

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-sep 5260 ax-nul 5272 ax-pr 5407 ax-un 7738

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-ral 3083 df-rex 3093 df-rab 3420 df-v 3460 df-sbc 3748 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4290 df-if 4491 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4876 df-br 5113 df-opab 5177 df-mpt 5196 df-id 5559 df-xp 5670 df-rel 5671 df-cnv 5672 df-co 5673 df-dm 5674 df-rn 5675 df-iota 6496 df-fun 6542 df-fv 6548 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7988 df-2nd 7989

This theorem is used by: eulerpartlemgvv 34779 eulerpartlemgh 34781 eulerpartlemgs2 34783

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