dvhvscaval - Mathbox for Norm Megill

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Theorem dvhvscaval 41058

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 6921	. . 3 ⊢ (𝑡 = 𝑈 → (𝑡‘(1^st ‘𝑔)) = (𝑈‘(1^st ‘𝑔)))
2		coeq1 5882	. . 3 ⊢ (𝑡 = 𝑈 → (𝑡 ∘ (2^nd ‘𝑔)) = (𝑈 ∘ (2^nd ‘𝑔)))
3	1, 2	opeq12d 4905	. 2 ⊢ (𝑡 = 𝑈 → ⟨(𝑡‘(1^st ‘𝑔)), (𝑡 ∘ (2^nd ‘𝑔))⟩ = ⟨(𝑈‘(1^st ‘𝑔)), (𝑈 ∘ (2^nd ‘𝑔))⟩)
4		2fveq3 6927	. . 3 ⊢ (𝑔 = 𝐹 → (𝑈‘(1^st ‘𝑔)) = (𝑈‘(1^st ‘𝐹)))
5		fveq2 6922	. . . 4 ⊢ (𝑔 = 𝐹 → (2^nd ‘𝑔) = (2^nd ‘𝐹))
6	5	coeq2d 5887	. . 3 ⊢ (𝑔 = 𝐹 → (𝑈 ∘ (2^nd ‘𝑔)) = (𝑈 ∘ (2^nd ‘𝐹)))
7	4, 6	opeq12d 4905	. 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 41057	. 2 ⊢ · = (𝑡 ∈ 𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1^st ‘𝑔)), (𝑡 ∘ (2^nd ‘𝑔))⟩)
10		opex 5484	. 2 ⊢ ⟨(𝑈‘(1^st ‘𝐹)), (𝑈 ∘ (2^nd ‘𝐹))⟩ ∈ V
11	3, 7, 9, 10	ovmpo 7612	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 395 = wceq 1537 ∈ wcel 2108 ⟨cop 4654 × cxp 5698 ∘ ccom 5704 ‘cfv 6575 (class class class)co 7450 ∈ cmpo 7452 1^st c1st 8030 2^nd c2nd 8031

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-dif 3979 df-un 3981 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-iota 6527 df-fun 6577 df-fv 6583 df-ov 7453 df-oprab 7454 df-mpo 7455

This theorem is referenced by: dvhvsca 41060 dvhopspN 41074

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