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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) ↦ 〈((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓) ⨣ (2nd ‘𝑔))〉) |
Ref | Expression |
---|---|
dvhvaddcbv | ⊢ + = (ℎ ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ 〈((1st ‘ℎ) ∘ (1st ‘𝑖)), ((2nd ‘ℎ) ⨣ (2nd ‘𝑖))〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvhvaddval.a | . 2 ⊢ + = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓) ⨣ (2nd ‘𝑔))〉) | |
2 | fveq2 6672 | . . . . 5 ⊢ (𝑓 = ℎ → (1st ‘𝑓) = (1st ‘ℎ)) | |
3 | 2 | coeq1d 5734 | . . . 4 ⊢ (𝑓 = ℎ → ((1st ‘𝑓) ∘ (1st ‘𝑔)) = ((1st ‘ℎ) ∘ (1st ‘𝑔))) |
4 | fveq2 6672 | . . . . 5 ⊢ (𝑓 = ℎ → (2nd ‘𝑓) = (2nd ‘ℎ)) | |
5 | 4 | oveq1d 7173 | . . . 4 ⊢ (𝑓 = ℎ → ((2nd ‘𝑓) ⨣ (2nd ‘𝑔)) = ((2nd ‘ℎ) ⨣ (2nd ‘𝑔))) |
6 | 3, 5 | opeq12d 4813 | . . 3 ⊢ (𝑓 = ℎ → 〈((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓) ⨣ (2nd ‘𝑔))〉 = 〈((1st ‘ℎ) ∘ (1st ‘𝑔)), ((2nd ‘ℎ) ⨣ (2nd ‘𝑔))〉) |
7 | fveq2 6672 | . . . . 5 ⊢ (𝑔 = 𝑖 → (1st ‘𝑔) = (1st ‘𝑖)) | |
8 | 7 | coeq2d 5735 | . . . 4 ⊢ (𝑔 = 𝑖 → ((1st ‘ℎ) ∘ (1st ‘𝑔)) = ((1st ‘ℎ) ∘ (1st ‘𝑖))) |
9 | fveq2 6672 | . . . . 5 ⊢ (𝑔 = 𝑖 → (2nd ‘𝑔) = (2nd ‘𝑖)) | |
10 | 9 | oveq2d 7174 | . . . 4 ⊢ (𝑔 = 𝑖 → ((2nd ‘ℎ) ⨣ (2nd ‘𝑔)) = ((2nd ‘ℎ) ⨣ (2nd ‘𝑖))) |
11 | 8, 10 | opeq12d 4813 | . . 3 ⊢ (𝑔 = 𝑖 → 〈((1st ‘ℎ) ∘ (1st ‘𝑔)), ((2nd ‘ℎ) ⨣ (2nd ‘𝑔))〉 = 〈((1st ‘ℎ) ∘ (1st ‘𝑖)), ((2nd ‘ℎ) ⨣ (2nd ‘𝑖))〉) |
12 | 6, 11 | cbvmpov 7251 | . 2 ⊢ (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓) ⨣ (2nd ‘𝑔))〉) = (ℎ ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ 〈((1st ‘ℎ) ∘ (1st ‘𝑖)), ((2nd ‘ℎ) ⨣ (2nd ‘𝑖))〉) |
13 | 1, 12 | eqtri 2846 | 1 ⊢ + = (ℎ ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ 〈((1st ‘ℎ) ∘ (1st ‘𝑖)), ((2nd ‘ℎ) ⨣ (2nd ‘𝑖))〉) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 〈cop 4575 × cxp 5555 ∘ ccom 5561 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 1st c1st 7689 2nd c2nd 7690 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-co 5566 df-iota 6316 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 |
This theorem is referenced by: dvhvaddval 38228 |
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