![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
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 6897 | . . . . 5 ⊢ (𝑓 = ℎ → (1st ‘𝑓) = (1st ‘ℎ)) | |
3 | 2 | coeq1d 5864 | . . . 4 ⊢ (𝑓 = ℎ → ((1st ‘𝑓) ∘ (1st ‘𝑔)) = ((1st ‘ℎ) ∘ (1st ‘𝑔))) |
4 | fveq2 6897 | . . . . 5 ⊢ (𝑓 = ℎ → (2nd ‘𝑓) = (2nd ‘ℎ)) | |
5 | 4 | oveq1d 7435 | . . . 4 ⊢ (𝑓 = ℎ → ((2nd ‘𝑓) ⨣ (2nd ‘𝑔)) = ((2nd ‘ℎ) ⨣ (2nd ‘𝑔))) |
6 | 3, 5 | opeq12d 4882 | . . 3 ⊢ (𝑓 = ℎ → 〈((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓) ⨣ (2nd ‘𝑔))〉 = 〈((1st ‘ℎ) ∘ (1st ‘𝑔)), ((2nd ‘ℎ) ⨣ (2nd ‘𝑔))〉) |
7 | fveq2 6897 | . . . . 5 ⊢ (𝑔 = 𝑖 → (1st ‘𝑔) = (1st ‘𝑖)) | |
8 | 7 | coeq2d 5865 | . . . 4 ⊢ (𝑔 = 𝑖 → ((1st ‘ℎ) ∘ (1st ‘𝑔)) = ((1st ‘ℎ) ∘ (1st ‘𝑖))) |
9 | fveq2 6897 | . . . . 5 ⊢ (𝑔 = 𝑖 → (2nd ‘𝑔) = (2nd ‘𝑖)) | |
10 | 9 | oveq2d 7436 | . . . 4 ⊢ (𝑔 = 𝑖 → ((2nd ‘ℎ) ⨣ (2nd ‘𝑔)) = ((2nd ‘ℎ) ⨣ (2nd ‘𝑖))) |
11 | 8, 10 | opeq12d 4882 | . . 3 ⊢ (𝑔 = 𝑖 → 〈((1st ‘ℎ) ∘ (1st ‘𝑔)), ((2nd ‘ℎ) ⨣ (2nd ‘𝑔))〉 = 〈((1st ‘ℎ) ∘ (1st ‘𝑖)), ((2nd ‘ℎ) ⨣ (2nd ‘𝑖))〉) |
12 | 6, 11 | cbvmpov 7515 | . 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 2756 | 1 ⊢ + = (ℎ ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ 〈((1st ‘ℎ) ∘ (1st ‘𝑖)), ((2nd ‘ℎ) ⨣ (2nd ‘𝑖))〉) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 〈cop 4635 × cxp 5676 ∘ ccom 5682 ‘cfv 6548 (class class class)co 7420 ∈ cmpo 7422 1st c1st 7991 2nd c2nd 7992 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-co 5687 df-iota 6500 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 |
This theorem is referenced by: dvhvaddval 40563 |
Copyright terms: Public domain | W3C validator |