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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvhvaddcbv | Structured version Visualization version GIF version |
Description: Change bound variables to isolate them later. (Contributed by NM, 3-Nov-2013.) |
Ref | Expression |
---|---|
dvhvaddval.a | ⊢ + = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1^{st} ‘𝑓) ∘ (1^{st} ‘𝑔)), ((2^{nd} ‘𝑓) ⨣ (2^{nd} ‘𝑔))⟩) |
Ref | Expression |
---|---|
dvhvaddcbv | ⊢ + = (ℎ ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ ⟨((1^{st} ‘ℎ) ∘ (1^{st} ‘𝑖)), ((2^{nd} ‘ℎ) ⨣ (2^{nd} ‘𝑖))⟩) |
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 6663 | . . . . 5 ⊢ (𝑓 = ℎ → (1^{st} ‘𝑓) = (1^{st} ‘ℎ)) | |
3 | 2 | coeq1d 5707 | . . . 4 ⊢ (𝑓 = ℎ → ((1^{st} ‘𝑓) ∘ (1^{st} ‘𝑔)) = ((1^{st} ‘ℎ) ∘ (1^{st} ‘𝑔))) |
4 | fveq2 6663 | . . . . 5 ⊢ (𝑓 = ℎ → (2^{nd} ‘𝑓) = (2^{nd} ‘ℎ)) | |
5 | 4 | oveq1d 7171 | . . . 4 ⊢ (𝑓 = ℎ → ((2^{nd} ‘𝑓) ⨣ (2^{nd} ‘𝑔)) = ((2^{nd} ‘ℎ) ⨣ (2^{nd} ‘𝑔))) |
6 | 3, 5 | opeq12d 4774 | . . 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 6663 | . . . . 5 ⊢ (𝑔 = 𝑖 → (1^{st} ‘𝑔) = (1^{st} ‘𝑖)) | |
8 | 7 | coeq2d 5708 | . . . 4 ⊢ (𝑔 = 𝑖 → ((1^{st} ‘ℎ) ∘ (1^{st} ‘𝑔)) = ((1^{st} ‘ℎ) ∘ (1^{st} ‘𝑖))) |
9 | fveq2 6663 | . . . . 5 ⊢ (𝑔 = 𝑖 → (2^{nd} ‘𝑔) = (2^{nd} ‘𝑖)) | |
10 | 9 | oveq2d 7172 | . . . 4 ⊢ (𝑔 = 𝑖 → ((2^{nd} ‘ℎ) ⨣ (2^{nd} ‘𝑔)) = ((2^{nd} ‘ℎ) ⨣ (2^{nd} ‘𝑖))) |
11 | 8, 10 | opeq12d 4774 | . . 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 7249 | . 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 2781 | 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 1538 ⟨cop 4531 × cxp 5526 ∘ ccom 5532 ‘cfv 6340 (class class class)co 7156 ∈ cmpo 7158 1^{st} c1st 7697 2^{nd} c2nd 7698 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-sep 5173 ax-nul 5180 ax-pr 5302 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-v 3411 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-nul 4228 df-if 4424 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4802 df-br 5037 df-opab 5099 df-co 5537 df-iota 6299 df-fv 6348 df-ov 7159 df-oprab 7160 df-mpo 7161 |
This theorem is referenced by: dvhvaddval 38701 |
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