oddpwdcv - Mathbox for Thierry Arnoux

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Theorem oddpwdcv 32322

Description: Lemma for eulerpart 32349: 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 7870	. . 3 ⊢ (𝑊 ∈ (𝐽 × ℕ₀) → 𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩)
2	1	fveq2d 6778	. 2 ⊢ (𝑊 ∈ (𝐽 × ℕ₀) → (𝐹‘𝑊) = (𝐹‘⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩))
3		df-ov 7278	. . 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 7865	. . . 4 ⊢ (𝑊 ∈ (𝐽 × ℕ₀) ↔ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∧ ((1^st ‘𝑊) ∈ 𝐽 ∧ (2^nd ‘𝑊) ∈ ℕ₀)))
6	5	simprbi 497	. . 3 ⊢ (𝑊 ∈ (𝐽 × ℕ₀) → ((1^st ‘𝑊) ∈ 𝐽 ∧ (2^nd ‘𝑊) ∈ ℕ₀))
7		oveq2 7283	. . . 4 ⊢ (𝑥 = (1^st ‘𝑊) → ((2↑𝑦) · 𝑥) = ((2↑𝑦) · (1^st ‘𝑊)))
8		oveq2 7283	. . . . 5 ⊢ (𝑦 = (2^nd ‘𝑊) → (2↑𝑦) = (2↑(2^nd ‘𝑊)))
9	8	oveq1d 7290	. . . 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 7308	. . . 4 ⊢ ((2↑(2^nd ‘𝑊)) · (1^st ‘𝑊)) ∈ V
12	7, 9, 10, 11	ovmpo 7433	. . 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 2784	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 396 = wceq 1539 ∈ wcel 2106 {crab 3068 ⟨cop 4567 class class class wbr 5074 × cxp 5587 ‘cfv 6433 (class class class)co 7275 ∈ cmpo 7277 1^st c1st 7829 2^nd c2nd 7830 · cmul 10876 ℕcn 11973 2c2 12028 ℕ₀cn0 12233 ↑cexp 13782 ∥ cdvds 15963

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-iota 6391 df-fun 6435 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832

This theorem is referenced by: eulerpartlemgvv 32343 eulerpartlemgh 32345 eulerpartlemgs2 32347

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