dvhvscaval - Mathbox for Norm Megill

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Theorem dvhvscaval 39113

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 6773	. . 3 ⊢ (𝑡 = 𝑈 → (𝑡‘(1^st ‘𝑔)) = (𝑈‘(1^st ‘𝑔)))
2		coeq1 5766	. . 3 ⊢ (𝑡 = 𝑈 → (𝑡 ∘ (2^nd ‘𝑔)) = (𝑈 ∘ (2^nd ‘𝑔)))
3	1, 2	opeq12d 4812	. 2 ⊢ (𝑡 = 𝑈 → ⟨(𝑡‘(1^st ‘𝑔)), (𝑡 ∘ (2^nd ‘𝑔))⟩ = ⟨(𝑈‘(1^st ‘𝑔)), (𝑈 ∘ (2^nd ‘𝑔))⟩)
4		2fveq3 6779	. . 3 ⊢ (𝑔 = 𝐹 → (𝑈‘(1^st ‘𝑔)) = (𝑈‘(1^st ‘𝐹)))
5		fveq2 6774	. . . 4 ⊢ (𝑔 = 𝐹 → (2^nd ‘𝑔) = (2^nd ‘𝐹))
6	5	coeq2d 5771	. . 3 ⊢ (𝑔 = 𝐹 → (𝑈 ∘ (2^nd ‘𝑔)) = (𝑈 ∘ (2^nd ‘𝐹)))
7	4, 6	opeq12d 4812	. 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 39112	. 2 ⊢ · = (𝑡 ∈ 𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1^st ‘𝑔)), (𝑡 ∘ (2^nd ‘𝑔))⟩)
10		opex 5379	. 2 ⊢ ⟨(𝑈‘(1^st ‘𝐹)), (𝑈 ∘ (2^nd ‘𝐹))⟩ ∈ V
11	3, 7, 9, 10	ovmpo 7433	1 ⊢ ((𝑈 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸)) → (𝑈 · 𝐹) = ⟨(𝑈‘(1^st ‘𝐹)), (𝑈 ∘ (2^nd ‘𝐹))⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ⟨cop 4567 × cxp 5587 ∘ ccom 5593 ‘cfv 6433 (class class class)co 7275 ∈ cmpo 7277 1^st c1st 7829 2^nd c2nd 7830

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

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-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-iota 6391 df-fun 6435 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280

This theorem is referenced by: dvhvsca 39115 dvhopspN 39129

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