dvhvaddval - Mathbox for Norm Megill

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Theorem dvhvaddval 40619

Description: The vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013.)

**Hypothesis**
Ref	Expression
dvhvaddval.a	⊢ + = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1^st ‘𝑓) ∘ (1^st ‘𝑔)), ((2^nd ‘𝑓) ⨣ (2^nd ‘𝑔))⟩)

**Assertion**
Ref	Expression
dvhvaddval	⊢ ((𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸)) → (𝐹 + 𝐺) = ⟨((1^st ‘𝐹) ∘ (1^st ‘𝐺)), ((2^nd ‘𝐹) ⨣ (2^nd ‘𝐺))⟩)

Distinct variable groups: 𝑓,𝑔,𝐸 ⨣ ,𝑓,𝑔 𝑇,𝑓,𝑔 Allowed substitution hints: + (𝑓,𝑔) 𝐹(𝑓,𝑔) 𝐺(𝑓,𝑔)

Proof of Theorem dvhvaddval Dummy variables ℎ 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6892	. . . 4 ⊢ (ℎ = 𝐹 → (1^st ‘ℎ) = (1^st ‘𝐹))
2	1	coeq1d 5858	. . 3 ⊢ (ℎ = 𝐹 → ((1^st ‘ℎ) ∘ (1^st ‘𝑖)) = ((1^st ‘𝐹) ∘ (1^st ‘𝑖)))
3		fveq2 6892	. . . 4 ⊢ (ℎ = 𝐹 → (2^nd ‘ℎ) = (2^nd ‘𝐹))
4	3	oveq1d 7431	. . 3 ⊢ (ℎ = 𝐹 → ((2^nd ‘ℎ) ⨣ (2^nd ‘𝑖)) = ((2^nd ‘𝐹) ⨣ (2^nd ‘𝑖)))
5	2, 4	opeq12d 4877	. 2 ⊢ (ℎ = 𝐹 → ⟨((1^st ‘ℎ) ∘ (1^st ‘𝑖)), ((2^nd ‘ℎ) ⨣ (2^nd ‘𝑖))⟩ = ⟨((1^st ‘𝐹) ∘ (1^st ‘𝑖)), ((2^nd ‘𝐹) ⨣ (2^nd ‘𝑖))⟩)
6		fveq2 6892	. . . 4 ⊢ (𝑖 = 𝐺 → (1^st ‘𝑖) = (1^st ‘𝐺))
7	6	coeq2d 5859	. . 3 ⊢ (𝑖 = 𝐺 → ((1^st ‘𝐹) ∘ (1^st ‘𝑖)) = ((1^st ‘𝐹) ∘ (1^st ‘𝐺)))
8		fveq2 6892	. . . 4 ⊢ (𝑖 = 𝐺 → (2^nd ‘𝑖) = (2^nd ‘𝐺))
9	8	oveq2d 7432	. . 3 ⊢ (𝑖 = 𝐺 → ((2^nd ‘𝐹) ⨣ (2^nd ‘𝑖)) = ((2^nd ‘𝐹) ⨣ (2^nd ‘𝐺)))
10	7, 9	opeq12d 4877	. 2 ⊢ (𝑖 = 𝐺 → ⟨((1^st ‘𝐹) ∘ (1^st ‘𝑖)), ((2^nd ‘𝐹) ⨣ (2^nd ‘𝑖))⟩ = ⟨((1^st ‘𝐹) ∘ (1^st ‘𝐺)), ((2^nd ‘𝐹) ⨣ (2^nd ‘𝐺))⟩)
11		dvhvaddval.a	. . 3 ⊢ + = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1^st ‘𝑓) ∘ (1^st ‘𝑔)), ((2^nd ‘𝑓) ⨣ (2^nd ‘𝑔))⟩)
12	11	dvhvaddcbv 40618	. 2 ⊢ + = (ℎ ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ ⟨((1^st ‘ℎ) ∘ (1^st ‘𝑖)), ((2^nd ‘ℎ) ⨣ (2^nd ‘𝑖))⟩)
13		opex 5460	. 2 ⊢ ⟨((1^st ‘𝐹) ∘ (1^st ‘𝐺)), ((2^nd ‘𝐹) ⨣ (2^nd ‘𝐺))⟩ ∈ V
14	5, 10, 12, 13	ovmpo 7578	1 ⊢ ((𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸)) → (𝐹 + 𝐺) = ⟨((1^st ‘𝐹) ∘ (1^st ‘𝐺)), ((2^nd ‘𝐹) ⨣ (2^nd ‘𝐺))⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ⟨cop 4630 × cxp 5670 ∘ ccom 5676 ‘cfv 6543 (class class class)co 7416 ∈ cmpo 7418 1^st c1st 7989 2^nd c2nd 7990

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3769 df-dif 3942 df-un 3944 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7419 df-oprab 7420 df-mpo 7421

This theorem is referenced by: dvhvadd 40621 dvhopaddN 40643

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