dvhvaddval - Mathbox for Norm Megill

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Theorem dvhvaddval 41051

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 6886	. . . 4 ⊢ (ℎ = 𝐹 → (1^st ‘ℎ) = (1^st ‘𝐹))
2	1	coeq1d 5852	. . 3 ⊢ (ℎ = 𝐹 → ((1^st ‘ℎ) ∘ (1^st ‘𝑖)) = ((1^st ‘𝐹) ∘ (1^st ‘𝑖)))
3		fveq2 6886	. . . 4 ⊢ (ℎ = 𝐹 → (2^nd ‘ℎ) = (2^nd ‘𝐹))
4	3	oveq1d 7428	. . 3 ⊢ (ℎ = 𝐹 → ((2^nd ‘ℎ) ⨣ (2^nd ‘𝑖)) = ((2^nd ‘𝐹) ⨣ (2^nd ‘𝑖)))
5	2, 4	opeq12d 4861	. 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 6886	. . . 4 ⊢ (𝑖 = 𝐺 → (1^st ‘𝑖) = (1^st ‘𝐺))
7	6	coeq2d 5853	. . 3 ⊢ (𝑖 = 𝐺 → ((1^st ‘𝐹) ∘ (1^st ‘𝑖)) = ((1^st ‘𝐹) ∘ (1^st ‘𝐺)))
8		fveq2 6886	. . . 4 ⊢ (𝑖 = 𝐺 → (2^nd ‘𝑖) = (2^nd ‘𝐺))
9	8	oveq2d 7429	. . 3 ⊢ (𝑖 = 𝐺 → ((2^nd ‘𝐹) ⨣ (2^nd ‘𝑖)) = ((2^nd ‘𝐹) ⨣ (2^nd ‘𝐺)))
10	7, 9	opeq12d 4861	. 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 41050	. 2 ⊢ + = (ℎ ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ ⟨((1^st ‘ℎ) ∘ (1^st ‘𝑖)), ((2^nd ‘ℎ) ⨣ (2^nd ‘𝑖))⟩)
13		opex 5449	. 2 ⊢ ⟨((1^st ‘𝐹) ∘ (1^st ‘𝐺)), ((2^nd ‘𝐹) ⨣ (2^nd ‘𝐺))⟩ ∈ V
14	5, 10, 12, 13	ovmpo 7575	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 395 = wceq 1539 ∈ wcel 2107 ⟨cop 4612 × cxp 5663 ∘ ccom 5669 ‘cfv 6541 (class class class)co 7413 ∈ cmpo 7415 1^st c1st 7994 2^nd c2nd 7995

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-sep 5276 ax-nul 5286 ax-pr 5412

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ral 3051 df-rex 3060 df-rab 3420 df-v 3465 df-sbc 3771 df-dif 3934 df-un 3936 df-ss 3948 df-nul 4314 df-if 4506 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4888 df-br 5124 df-opab 5186 df-id 5558 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-iota 6494 df-fun 6543 df-fv 6549 df-ov 7416 df-oprab 7417 df-mpo 7418

This theorem is referenced by: dvhvadd 41053 dvhopaddN 41075

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