oddpwdcv - Mathbox for Thierry Arnoux

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Theorem oddpwdcv 30545

Description: Lemma for eulerpart 30572: 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 7249	. . 3 ⊢ (𝑊 ∈ (𝐽 × ℕ₀) → 𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩)
2	1	fveq2d 6233	. 2 ⊢ (𝑊 ∈ (𝐽 × ℕ₀) → (𝐹‘𝑊) = (𝐹‘⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩))
3		df-ov 6693	. . 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 7244	. . . 4 ⊢ (𝑊 ∈ (𝐽 × ℕ₀) ↔ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∧ ((1^st ‘𝑊) ∈ 𝐽 ∧ (2^nd ‘𝑊) ∈ ℕ₀)))
6	5	simprbi 479	. . 3 ⊢ (𝑊 ∈ (𝐽 × ℕ₀) → ((1^st ‘𝑊) ∈ 𝐽 ∧ (2^nd ‘𝑊) ∈ ℕ₀))
7		oveq2 6698	. . . 4 ⊢ (𝑥 = (1^st ‘𝑊) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · (1^st ‘𝑊)))
8		oveq2 6698	. . . . 5 ⊢ (𝑦 = (2^nd ‘𝑊) → (2↑𝑦) = (2↑(2^nd ‘𝑊)))
9	8	oveq1d 6705	. . . 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 6718	. . . 4 ⊢ ((2↑(2^nd ‘𝑊)) · (1^st ‘𝑊)) ∈ V
12	7, 9, 10, 11	ovmpt2 6838	. . 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 2691	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 383 = wceq 1523 ∈ wcel 2030 {crab 2945 ⟨cop 4216 class class class wbr 4685 × cxp 5141 ‘cfv 5926 (class class class)co 6690 ↦ cmpt2 6692 1^st c1st 7208 2^nd c2nd 7209 · cmul 9979 ℕcn 11058 2c2 11108 ℕ₀cn0 11330 ↑cexp 12900 ∥ cdvds 15027

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-iota 5889 df-fun 5928 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-1st 7210 df-2nd 7211

This theorem is referenced by: eulerpartlemgvv 30566 eulerpartlemgh 30568 eulerpartlemgs2 30570

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