dvhvaddcbv - Mathbox for Norm Megill

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Theorem dvhvaddcbv 40618

Description: Change bound variables to isolate them later. (Contributed by NM, 3-Nov-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
dvhvaddcbv	⊢ + = (ℎ ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ ⟨((1^st ‘ℎ) ∘ (1^st ‘𝑖)), ((2^nd ‘ℎ) ⨣ (2^nd ‘𝑖))⟩)

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

**Proof of Theorem dvhvaddcbv**
Step	Hyp	Ref	Expression
1		dvhvaddval.a	. 2 ⊢ + = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1^st ‘𝑓) ∘ (1^st ‘𝑔)), ((2^nd ‘𝑓) ⨣ (2^nd ‘𝑔))⟩)
2		fveq2 6892	. . . . 5 ⊢ (𝑓 = ℎ → (1^st ‘𝑓) = (1^st ‘ℎ))
3	2	coeq1d 5858	. . . 4 ⊢ (𝑓 = ℎ → ((1^st ‘𝑓) ∘ (1^st ‘𝑔)) = ((1^st ‘ℎ) ∘ (1^st ‘𝑔)))
4		fveq2 6892	. . . . 5 ⊢ (𝑓 = ℎ → (2^nd ‘𝑓) = (2^nd ‘ℎ))
5	4	oveq1d 7431	. . . 4 ⊢ (𝑓 = ℎ → ((2^nd ‘𝑓) ⨣ (2^nd ‘𝑔)) = ((2^nd ‘ℎ) ⨣ (2^nd ‘𝑔)))
6	3, 5	opeq12d 4877	. . 3 ⊢ (𝑓 = ℎ → ⟨((1^st ‘𝑓) ∘ (1^st ‘𝑔)), ((2^nd ‘𝑓) ⨣ (2^nd ‘𝑔))⟩ = ⟨((1^st ‘ℎ) ∘ (1^st ‘𝑔)), ((2^nd ‘ℎ) ⨣ (2^nd ‘𝑔))⟩)
7		fveq2 6892	. . . . 5 ⊢ (𝑔 = 𝑖 → (1^st ‘𝑔) = (1^st ‘𝑖))
8	7	coeq2d 5859	. . . 4 ⊢ (𝑔 = 𝑖 → ((1^st ‘ℎ) ∘ (1^st ‘𝑔)) = ((1^st ‘ℎ) ∘ (1^st ‘𝑖)))
9		fveq2 6892	. . . . 5 ⊢ (𝑔 = 𝑖 → (2^nd ‘𝑔) = (2^nd ‘𝑖))
10	9	oveq2d 7432	. . . 4 ⊢ (𝑔 = 𝑖 → ((2^nd ‘ℎ) ⨣ (2^nd ‘𝑔)) = ((2^nd ‘ℎ) ⨣ (2^nd ‘𝑖)))
11	8, 10	opeq12d 4877	. . 3 ⊢ (𝑔 = 𝑖 → ⟨((1^st ‘ℎ) ∘ (1^st ‘𝑔)), ((2^nd ‘ℎ) ⨣ (2^nd ‘𝑔))⟩ = ⟨((1^st ‘ℎ) ∘ (1^st ‘𝑖)), ((2^nd ‘ℎ) ⨣ (2^nd ‘𝑖))⟩)
12	6, 11	cbvmpov 7512	. 2 ⊢ (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1^st ‘𝑓) ∘ (1^st ‘𝑔)), ((2^nd ‘𝑓) ⨣ (2^nd ‘𝑔))⟩) = (ℎ ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ ⟨((1^st ‘ℎ) ∘ (1^st ‘𝑖)), ((2^nd ‘ℎ) ⨣ (2^nd ‘𝑖))⟩)
13	1, 12	eqtri 2753	1 ⊢ + = (ℎ ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ ⟨((1^st ‘ℎ) ∘ (1^st ‘𝑖)), ((2^nd ‘ℎ) ⨣ (2^nd ‘𝑖))⟩)

Colors of variables: wff setvar class

Syntax hints: = wceq 1533 ⟨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-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-rab 3420 df-v 3465 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-co 5681 df-iota 6495 df-fv 6551 df-ov 7419 df-oprab 7420 df-mpo 7421

This theorem is referenced by: dvhvaddval 40619

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