dvhvaddval - Mathbox for Norm Megill

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Theorem dvhvaddval 41963

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 6878	. . . 4 ⊢ (ℎ = 𝐹 → (1^st ‘ℎ) = (1^st ‘𝐹))
2	1	coeq1d 5841	. . 3 ⊢ (ℎ = 𝐹 → ((1^st ‘ℎ) ∘ (1^st ‘𝑖)) = ((1^st ‘𝐹) ∘ (1^st ‘𝑖)))
3		fveq2 6878	. . . 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 4841	. 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 6878	. . . 4 ⊢ (𝑖 = 𝐺 → (1^st ‘𝑖) = (1^st ‘𝐺))
7	6	coeq2d 5842	. . 3 ⊢ (𝑖 = 𝐺 → ((1^st ‘𝐹) ∘ (1^st ‘𝑖)) = ((1^st ‘𝐹) ∘ (1^st ‘𝐺)))
8		fveq2 6878	. . . 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 4841	. 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 41962	. 2 ⊢ + = (ℎ ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ ⟨((1^st ‘ℎ) ∘ (1^st ‘𝑖)), ((2^nd ‘ℎ) ⨣ (2^nd ‘𝑖))⟩)
13		opex 5439	. 2 ⊢ ⟨((1^st ‘𝐹) ∘ (1^st ‘𝐺)), ((2^nd ‘𝐹) ⨣ (2^nd ‘𝐺))⟩ ∈ V
14	5, 10, 12, 13	ovmpo 7573	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

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ⟨cop 4590 × cxp 5653 ∘ ccom 5659 ‘cfv 6533 (class class class)co 7413 ∈ cmpo 7415 1^st c1st 7984 2^nd c2nd 7985

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-pr 5398

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3740 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-iota 6489 df-fun 6535 df-fv 6541 df-ov 7416 df-oprab 7417 df-mpo 7418

This theorem is used by: dvhvadd 41965 dvhopaddN 41987

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