dvhvscaval - Mathbox for Norm Megill

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Theorem dvhvscaval 41972

Description: The scalar product operation for the constructed full vector space H. (Contributed by NM, 20-Nov-2013.)

**Hypothesis**
Ref	Expression
dvhvscaval.s	⊢ · = (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩)

**Assertion**
Ref	Expression
dvhvscaval	⊢ ((𝑈 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸)) → (𝑈 · 𝐹) = ⟨(𝑈‘(1^st ‘𝐹)), (𝑈 ∘ (2^nd ‘𝐹))⟩)

Distinct variable groups: 𝑓,𝑠,𝐸 𝑇,𝑠,𝑓 Allowed substitution hints: · (𝑓, 𝑠) 𝑈(𝑓, 𝑠) 𝐹(𝑓, 𝑠)

Proof of Theorem dvhvscaval Dummy variables 𝑡 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 6877	. . 3 ⊢ (𝑡 = 𝑈 → (𝑡‘(1^st ‘𝑔)) = (𝑈‘(1^st ‘𝑔)))
2		coeq1 5837	. . 3 ⊢ (𝑡 = 𝑈 → (𝑡 ∘ (2^nd ‘𝑔)) = (𝑈 ∘ (2^nd ‘𝑔)))
3	1, 2	opeq12d 4841	. 2 ⊢ (𝑡 = 𝑈 → ⟨(𝑡‘(1^st ‘𝑔)), (𝑡 ∘ (2^nd ‘𝑔))⟩ = ⟨(𝑈‘(1^st ‘𝑔)), (𝑈 ∘ (2^nd ‘𝑔))⟩)
4		2fveq3 6883	. . 3 ⊢ (𝑔 = 𝐹 → (𝑈‘(1^st ‘𝑔)) = (𝑈‘(1^st ‘𝐹)))
5		fveq2 6878	. . . 4 ⊢ (𝑔 = 𝐹 → (2^nd ‘𝑔) = (2^nd ‘𝐹))
6	5	coeq2d 5842	. . 3 ⊢ (𝑔 = 𝐹 → (𝑈 ∘ (2^nd ‘𝑔)) = (𝑈 ∘ (2^nd ‘𝐹)))
7	4, 6	opeq12d 4841	. 2 ⊢ (𝑔 = 𝐹 → ⟨(𝑈‘(1^st ‘𝑔)), (𝑈 ∘ (2^nd ‘𝑔))⟩ = ⟨(𝑈‘(1^st ‘𝐹)), (𝑈 ∘ (2^nd ‘𝐹))⟩)
8		dvhvscaval.s	. . 3 ⊢ · = (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1^st ‘𝑓)), (𝑠 ∘ (2^nd ‘𝑓))⟩)
9	8	dvhvscacbv 41971	. 2 ⊢ · = (𝑡 ∈ 𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1^st ‘𝑔)), (𝑡 ∘ (2^nd ‘𝑔))⟩)
10		opex 5439	. 2 ⊢ ⟨(𝑈‘(1^st ‘𝐹)), (𝑈 ∘ (2^nd ‘𝐹))⟩ ∈ V
11	3, 7, 9, 10	ovmpo 7573	1 ⊢ ((𝑈 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸)) → (𝑈 · 𝐹) = ⟨(𝑈‘(1^st ‘𝐹)), (𝑈 ∘ (2^nd ‘𝐹))⟩)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ⟨cop 4590 × cxp 5653 ∘ ccom 5659 ‘cfv 6533 (class class class)co 7413 ∈ cmpo 7415 1^st c1st 7984 2^nd c2nd 7985

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-pr 5398

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3740 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-iota 6489 df-fun 6535 df-fv 6541 df-ov 7416 df-oprab 7417 df-mpo 7418

This theorem is used by: dvhvsca 41974 dvhopspN 41988

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