dvhvaddcbv - Mathbox for Norm Megill

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Theorem dvhvaddcbv 41904

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 6888	. . . . 5 ⊢ (𝑓 = ℎ → (1^st ‘𝑓) = (1^st ‘ℎ))
3	2	coeq1d 5852	. . . 4 ⊢ (𝑓 = ℎ → ((1^st ‘𝑓) ∘ (1^st ‘𝑔)) = ((1^st ‘ℎ) ∘ (1^st ‘𝑔)))
4		fveq2 6888	. . . . 5 ⊢ (𝑓 = ℎ → (2^nd ‘𝑓) = (2^nd ‘ℎ))
5	4	oveq1d 7438	. . . 4 ⊢ (𝑓 = ℎ → ((2^nd ‘𝑓) ⨣ (2^nd ‘𝑔)) = ((2^nd ‘ℎ) ⨣ (2^nd ‘𝑔)))
6	3, 5	opeq12d 4851	. . 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 6888	. . . . 5 ⊢ (𝑔 = 𝑖 → (1^st ‘𝑔) = (1^st ‘𝑖))
8	7	coeq2d 5853	. . . 4 ⊢ (𝑔 = 𝑖 → ((1^st ‘ℎ) ∘ (1^st ‘𝑔)) = ((1^st ‘ℎ) ∘ (1^st ‘𝑖)))
9		fveq2 6888	. . . . 5 ⊢ (𝑔 = 𝑖 → (2^nd ‘𝑔) = (2^nd ‘𝑖))
10	9	oveq2d 7439	. . . 4 ⊢ (𝑔 = 𝑖 → ((2^nd ‘ℎ) ⨣ (2^nd ‘𝑔)) = ((2^nd ‘ℎ) ⨣ (2^nd ‘𝑖)))
11	8, 10	opeq12d 4851	. . 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 7518	. 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 2789	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

This proof depends on syntax axioms: = wceq 1570 ⟨cop 4600 × cxp 5664 ∘ ccom 5670 ‘cfv 6543 (class class class)co 7423 ∈ cmpo 7425 1^st c1st 7993 2^nd c2nd 7994

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 2148 ax-9 2156 ax-ext 2738

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-sb 2100 df-clab 2745 df-cleq 2758 df-clel 2841 df-rab 3420 df-v 3460 df-dif 3911 df-un 3913 df-ss 3925 df-nul 4290 df-if 4493 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-br 5115 df-opab 5179 df-co 5675 df-iota 6499 df-fv 6551 df-ov 7426 df-oprab 7427 df-mpo 7428

This theorem is used by: dvhvaddval 41905

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