dvhvaddcbv - Mathbox for Norm Megill

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Theorem dvhvaddcbv 41725

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 6871	. . . . 5 ⊢ (𝑓 = ℎ → (1^st ‘𝑓) = (1^st ‘ℎ))
3	2	coeq1d 5838	. . . 4 ⊢ (𝑓 = ℎ → ((1^st ‘𝑓) ∘ (1^st ‘𝑔)) = ((1^st ‘ℎ) ∘ (1^st ‘𝑔)))
4		fveq2 6871	. . . . 5 ⊢ (𝑓 = ℎ → (2^nd ‘𝑓) = (2^nd ‘ℎ))
5	4	oveq1d 7415	. . . 4 ⊢ (𝑓 = ℎ → ((2^nd ‘𝑓) ⨣ (2^nd ‘𝑔)) = ((2^nd ‘ℎ) ⨣ (2^nd ‘𝑔)))
6	3, 5	opeq12d 4842	. . 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 6871	. . . . 5 ⊢ (𝑔 = 𝑖 → (1^st ‘𝑔) = (1^st ‘𝑖))
8	7	coeq2d 5839	. . . 4 ⊢ (𝑔 = 𝑖 → ((1^st ‘ℎ) ∘ (1^st ‘𝑔)) = ((1^st ‘ℎ) ∘ (1^st ‘𝑖)))
9		fveq2 6871	. . . . 5 ⊢ (𝑔 = 𝑖 → (2^nd ‘𝑔) = (2^nd ‘𝑖))
10	9	oveq2d 7416	. . . 4 ⊢ (𝑔 = 𝑖 → ((2^nd ‘ℎ) ⨣ (2^nd ‘𝑔)) = ((2^nd ‘ℎ) ⨣ (2^nd ‘𝑖)))
11	8, 10	opeq12d 4842	. . 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 7495	. 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 2788	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 1563 ⟨cop 4591 × cxp 5650 ∘ ccom 5656 ‘cfv 6525 (class class class)co 7400 ∈ cmpo 7402 1^st c1st 7972 2^nd c2nd 7973

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-ext 2737

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-sb 2094 df-clab 2744 df-cleq 2757 df-clel 2840 df-rab 3418 df-v 3459 df-dif 3910 df-un 3912 df-ss 3924 df-nul 4289 df-if 4484 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4869 df-br 5106 df-opab 5168 df-co 5661 df-iota 6481 df-fv 6533 df-ov 7403 df-oprab 7404 df-mpo 7405

This theorem is referenced by: dvhvaddval 41726

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