dvhvaddval - Mathbox for Norm Megill

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Theorem dvhvaddval 35397

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 6103	. . . 4 ⊢ (ℎ = 𝐹 → (1^st ‘ℎ) = (1^st ‘𝐹))
2	1	coeq1d 5205	. . 3 ⊢ (ℎ = 𝐹 → ((1^st ‘ℎ) ∘ (1^st ‘𝑖)) = ((1^st ‘𝐹) ∘ (1^st ‘𝑖)))
3		fveq2 6103	. . . 4 ⊢ (ℎ = 𝐹 → (2^nd ‘ℎ) = (2^nd ‘𝐹))
4	3	oveq1d 6564	. . 3 ⊢ (ℎ = 𝐹 → ((2^nd ‘ℎ) ⨣ (2^nd ‘𝑖)) = ((2^nd ‘𝐹) ⨣ (2^nd ‘𝑖)))
5	2, 4	opeq12d 4348	. 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 6103	. . . 4 ⊢ (𝑖 = 𝐺 → (1^st ‘𝑖) = (1^st ‘𝐺))
7	6	coeq2d 5206	. . 3 ⊢ (𝑖 = 𝐺 → ((1^st ‘𝐹) ∘ (1^st ‘𝑖)) = ((1^st ‘𝐹) ∘ (1^st ‘𝐺)))
8		fveq2 6103	. . . 4 ⊢ (𝑖 = 𝐺 → (2^nd ‘𝑖) = (2^nd ‘𝐺))
9	8	oveq2d 6565	. . 3 ⊢ (𝑖 = 𝐺 → ((2^nd ‘𝐹) ⨣ (2^nd ‘𝑖)) = ((2^nd ‘𝐹) ⨣ (2^nd ‘𝐺)))
10	7, 9	opeq12d 4348	. 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 35396	. 2 ⊢ + = (ℎ ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ ⟨((1^st ‘ℎ) ∘ (1^st ‘𝑖)), ((2^nd ‘ℎ) ⨣ (2^nd ‘𝑖))⟩)
13		opex 4859	. 2 ⊢ ⟨((1^st ‘𝐹) ∘ (1^st ‘𝐺)), ((2^nd ‘𝐹) ⨣ (2^nd ‘𝐺))⟩ ∈ V
14	5, 10, 12, 13	ovmpt2 6694	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 383 = wceq 1475 ∈ wcel 1977 ⟨cop 4131 × cxp 5036 ∘ ccom 5042 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 1^st c1st 7057 2^nd c2nd 7058

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554

This theorem is referenced by: dvhvadd 35399 dvhopaddN 35421

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