dvhvscaval - Mathbox for Norm Megill

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Theorem dvhvscaval 41085

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 6864	. . 3 ⊢ (𝑡 = 𝑈 → (𝑡‘(1^st ‘𝑔)) = (𝑈‘(1^st ‘𝑔)))
2		coeq1 5829	. . 3 ⊢ (𝑡 = 𝑈 → (𝑡 ∘ (2^nd ‘𝑔)) = (𝑈 ∘ (2^nd ‘𝑔)))
3	1, 2	opeq12d 4853	. 2 ⊢ (𝑡 = 𝑈 → ⟨(𝑡‘(1^st ‘𝑔)), (𝑡 ∘ (2^nd ‘𝑔))⟩ = ⟨(𝑈‘(1^st ‘𝑔)), (𝑈 ∘ (2^nd ‘𝑔))⟩)
4		2fveq3 6870	. . 3 ⊢ (𝑔 = 𝐹 → (𝑈‘(1^st ‘𝑔)) = (𝑈‘(1^st ‘𝐹)))
5		fveq2 6865	. . . 4 ⊢ (𝑔 = 𝐹 → (2^nd ‘𝑔) = (2^nd ‘𝐹))
6	5	coeq2d 5834	. . 3 ⊢ (𝑔 = 𝐹 → (𝑈 ∘ (2^nd ‘𝑔)) = (𝑈 ∘ (2^nd ‘𝐹)))
7	4, 6	opeq12d 4853	. 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 41084	. 2 ⊢ · = (𝑡 ∈ 𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1^st ‘𝑔)), (𝑡 ∘ (2^nd ‘𝑔))⟩)
10		opex 5432	. 2 ⊢ ⟨(𝑈‘(1^st ‘𝐹)), (𝑈 ∘ (2^nd ‘𝐹))⟩ ∈ V
11	3, 7, 9, 10	ovmpo 7556	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 1540 ∈ wcel 2109 ⟨cop 4603 × cxp 5644 ∘ ccom 5650 ‘cfv 6519 (class class class)co 7394 ∈ cmpo 7396 1^st c1st 7975 2^nd c2nd 7976

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5259 ax-nul 5269 ax-pr 5395

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2880 df-ral 3047 df-rex 3056 df-rab 3412 df-v 3457 df-sbc 3762 df-dif 3925 df-un 3927 df-ss 3939 df-nul 4305 df-if 4497 df-sn 4598 df-pr 4600 df-op 4604 df-uni 4880 df-br 5116 df-opab 5178 df-id 5541 df-xp 5652 df-rel 5653 df-cnv 5654 df-co 5655 df-dm 5656 df-iota 6472 df-fun 6521 df-fv 6527 df-ov 7397 df-oprab 7398 df-mpo 7399

This theorem is referenced by: dvhvsca 41087 dvhopspN 41101

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