dvhvscaval - Mathbox for Norm Megill

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Theorem dvhvscaval 41146

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 6821	. . 3 ⊢ (𝑡 = 𝑈 → (𝑡‘(1^st ‘𝑔)) = (𝑈‘(1^st ‘𝑔)))
2		coeq1 5796	. . 3 ⊢ (𝑡 = 𝑈 → (𝑡 ∘ (2^nd ‘𝑔)) = (𝑈 ∘ (2^nd ‘𝑔)))
3	1, 2	opeq12d 4830	. 2 ⊢ (𝑡 = 𝑈 → ⟨(𝑡‘(1^st ‘𝑔)), (𝑡 ∘ (2^nd ‘𝑔))⟩ = ⟨(𝑈‘(1^st ‘𝑔)), (𝑈 ∘ (2^nd ‘𝑔))⟩)
4		2fveq3 6827	. . 3 ⊢ (𝑔 = 𝐹 → (𝑈‘(1^st ‘𝑔)) = (𝑈‘(1^st ‘𝐹)))
5		fveq2 6822	. . . 4 ⊢ (𝑔 = 𝐹 → (2^nd ‘𝑔) = (2^nd ‘𝐹))
6	5	coeq2d 5801	. . 3 ⊢ (𝑔 = 𝐹 → (𝑈 ∘ (2^nd ‘𝑔)) = (𝑈 ∘ (2^nd ‘𝐹)))
7	4, 6	opeq12d 4830	. 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 41145	. 2 ⊢ · = (𝑡 ∈ 𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1^st ‘𝑔)), (𝑡 ∘ (2^nd ‘𝑔))⟩)
10		opex 5402	. 2 ⊢ ⟨(𝑈‘(1^st ‘𝐹)), (𝑈 ∘ (2^nd ‘𝐹))⟩ ∈ V
11	3, 7, 9, 10	ovmpo 7506	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 1541 ∈ wcel 2111 ⟨cop 4579 × cxp 5612 ∘ ccom 5618 ‘cfv 6481 (class class class)co 7346 ∈ cmpo 7348 1^st c1st 7919 2^nd c2nd 7920

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5232 ax-nul 5242 ax-pr 5368

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-sbc 3737 df-dif 3900 df-un 3902 df-ss 3914 df-nul 4281 df-if 4473 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-br 5090 df-opab 5152 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-iota 6437 df-fun 6483 df-fv 6489 df-ov 7349 df-oprab 7350 df-mpo 7351

This theorem is referenced by: dvhvsca 41148 dvhopspN 41162

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