| Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > tendoicbv | Structured version Visualization version GIF version | ||
| Description: Define inverse function for trace-preserving endomorphisms. Change bound variable to isolate it later. (Contributed by NM, 12-Jun-2013.) |
| Ref | Expression |
|---|---|
| tendoi.i | ⊢ 𝐼 = (𝑠 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ◡(𝑠‘𝑓))) |
| Ref | Expression |
|---|---|
| tendoicbv | ⊢ 𝐼 = (𝑢 ∈ 𝐸 ↦ (𝑔 ∈ 𝑇 ↦ ◡(𝑢‘𝑔))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | tendoi.i | . 2 ⊢ 𝐼 = (𝑠 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ◡(𝑠‘𝑓))) | |
| 2 | fveq1 6905 | . . . . . 6 ⊢ (𝑠 = 𝑢 → (𝑠‘𝑓) = (𝑢‘𝑓)) | |
| 3 | 2 | cnveqd 5886 | . . . . 5 ⊢ (𝑠 = 𝑢 → ◡(𝑠‘𝑓) = ◡(𝑢‘𝑓)) |
| 4 | 3 | mpteq2dv 5244 | . . . 4 ⊢ (𝑠 = 𝑢 → (𝑓 ∈ 𝑇 ↦ ◡(𝑠‘𝑓)) = (𝑓 ∈ 𝑇 ↦ ◡(𝑢‘𝑓))) |
| 5 | fveq2 6906 | . . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑢‘𝑓) = (𝑢‘𝑔)) | |
| 6 | 5 | cnveqd 5886 | . . . . 5 ⊢ (𝑓 = 𝑔 → ◡(𝑢‘𝑓) = ◡(𝑢‘𝑔)) |
| 7 | 6 | cbvmptv 5255 | . . . 4 ⊢ (𝑓 ∈ 𝑇 ↦ ◡(𝑢‘𝑓)) = (𝑔 ∈ 𝑇 ↦ ◡(𝑢‘𝑔)) |
| 8 | 4, 7 | eqtrdi 2793 | . . 3 ⊢ (𝑠 = 𝑢 → (𝑓 ∈ 𝑇 ↦ ◡(𝑠‘𝑓)) = (𝑔 ∈ 𝑇 ↦ ◡(𝑢‘𝑔))) |
| 9 | 8 | cbvmptv 5255 | . 2 ⊢ (𝑠 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ◡(𝑠‘𝑓))) = (𝑢 ∈ 𝐸 ↦ (𝑔 ∈ 𝑇 ↦ ◡(𝑢‘𝑔))) |
| 10 | 1, 9 | eqtri 2765 | 1 ⊢ 𝐼 = (𝑢 ∈ 𝐸 ↦ (𝑔 ∈ 𝑇 ↦ ◡(𝑢‘𝑔))) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ↦ cmpt 5225 ◡ccnv 5684 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-cnv 5693 df-iota 6514 df-fv 6569 |
| This theorem is referenced by: tendoi 40796 |
| Copyright terms: Public domain | W3C validator |