dvhvaddval - Mathbox for Norm Megill

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Theorem dvhvaddval 41753

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 6882	. . . 4 ⊢ (ℎ = 𝐹 → (1^st ‘ℎ) = (1^st ‘𝐹))
2	1	coeq1d 5848	. . 3 ⊢ (ℎ = 𝐹 → ((1^st ‘ℎ) ∘ (1^st ‘𝑖)) = ((1^st ‘𝐹) ∘ (1^st ‘𝑖)))
3		fveq2 6882	. . . 4 ⊢ (ℎ = 𝐹 → (2^nd ‘ℎ) = (2^nd ‘𝐹))
4	3	oveq1d 7426	. . 3 ⊢ (ℎ = 𝐹 → ((2^nd ‘ℎ) ⨣ (2^nd ‘𝑖)) = ((2^nd ‘𝐹) ⨣ (2^nd ‘𝑖)))
5	2, 4	opeq12d 4850	. 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 6882	. . . 4 ⊢ (𝑖 = 𝐺 → (1^st ‘𝑖) = (1^st ‘𝐺))
7	6	coeq2d 5849	. . 3 ⊢ (𝑖 = 𝐺 → ((1^st ‘𝐹) ∘ (1^st ‘𝑖)) = ((1^st ‘𝐹) ∘ (1^st ‘𝐺)))
8		fveq2 6882	. . . 4 ⊢ (𝑖 = 𝐺 → (2^nd ‘𝑖) = (2^nd ‘𝐺))
9	8	oveq2d 7427	. . 3 ⊢ (𝑖 = 𝐺 → ((2^nd ‘𝐹) ⨣ (2^nd ‘𝑖)) = ((2^nd ‘𝐹) ⨣ (2^nd ‘𝐺)))
10	7, 9	opeq12d 4850	. 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 41752	. 2 ⊢ + = (ℎ ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ ⟨((1^st ‘ℎ) ∘ (1^st ‘𝑖)), ((2^nd ‘ℎ) ⨣ (2^nd ‘𝑖))⟩)
13		opex 5446	. 2 ⊢ ⟨((1^st ‘𝐹) ∘ (1^st ‘𝐺)), ((2^nd ‘𝐹) ⨣ (2^nd ‘𝐺))⟩ ∈ V
14	5, 10, 12, 13	ovmpo 7571	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 400 = wceq 1567 ∈ wcel 2149 ⟨cop 4600 × cxp 5660 ∘ ccom 5666 ‘cfv 6537 (class class class)co 7411 ∈ cmpo 7413 1^st c1st 7983 2^nd c2nd 7984

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5261 ax-pr 5405

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ral 3086 df-rex 3096 df-rab 3424 df-v 3465 df-sbc 3754 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4295 df-if 4493 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-br 5114 df-opab 5178 df-id 5557 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-iota 6493 df-fun 6539 df-fv 6545 df-ov 7414 df-oprab 7415 df-mpo 7416

This theorem is referenced by: dvhvadd 41755 dvhopaddN 41777

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