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Mirrors > Home > MPE Home > Th. List > Mathboxes > tendoplcbv | Structured version Visualization version GIF version |
Description: Define sum operation for trace-preserving endomorphisms. Change bound variables to isolate them later. (Contributed by NM, 11-Jun-2013.) |
Ref | Expression |
---|---|
tendoplcbv.p | ⊢ 𝑃 = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓)))) |
Ref | Expression |
---|---|
tendoplcbv | ⊢ 𝑃 = (𝑢 ∈ 𝐸, 𝑣 ∈ 𝐸 ↦ (𝑔 ∈ 𝑇 ↦ ((𝑢‘𝑔) ∘ (𝑣‘𝑔)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tendoplcbv.p | . 2 ⊢ 𝑃 = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓)))) | |
2 | fveq1 6891 | . . . . 5 ⊢ (𝑠 = 𝑢 → (𝑠‘𝑓) = (𝑢‘𝑓)) | |
3 | 2 | coeq1d 5862 | . . . 4 ⊢ (𝑠 = 𝑢 → ((𝑠‘𝑓) ∘ (𝑡‘𝑓)) = ((𝑢‘𝑓) ∘ (𝑡‘𝑓))) |
4 | 3 | mpteq2dv 5251 | . . 3 ⊢ (𝑠 = 𝑢 → (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))) = (𝑓 ∈ 𝑇 ↦ ((𝑢‘𝑓) ∘ (𝑡‘𝑓)))) |
5 | fveq1 6891 | . . . . . 6 ⊢ (𝑡 = 𝑣 → (𝑡‘𝑓) = (𝑣‘𝑓)) | |
6 | 5 | coeq2d 5863 | . . . . 5 ⊢ (𝑡 = 𝑣 → ((𝑢‘𝑓) ∘ (𝑡‘𝑓)) = ((𝑢‘𝑓) ∘ (𝑣‘𝑓))) |
7 | 6 | mpteq2dv 5251 | . . . 4 ⊢ (𝑡 = 𝑣 → (𝑓 ∈ 𝑇 ↦ ((𝑢‘𝑓) ∘ (𝑡‘𝑓))) = (𝑓 ∈ 𝑇 ↦ ((𝑢‘𝑓) ∘ (𝑣‘𝑓)))) |
8 | fveq2 6892 | . . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑢‘𝑓) = (𝑢‘𝑔)) | |
9 | fveq2 6892 | . . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑣‘𝑓) = (𝑣‘𝑔)) | |
10 | 8, 9 | coeq12d 5865 | . . . . 5 ⊢ (𝑓 = 𝑔 → ((𝑢‘𝑓) ∘ (𝑣‘𝑓)) = ((𝑢‘𝑔) ∘ (𝑣‘𝑔))) |
11 | 10 | cbvmptv 5262 | . . . 4 ⊢ (𝑓 ∈ 𝑇 ↦ ((𝑢‘𝑓) ∘ (𝑣‘𝑓))) = (𝑔 ∈ 𝑇 ↦ ((𝑢‘𝑔) ∘ (𝑣‘𝑔))) |
12 | 7, 11 | eqtrdi 2787 | . . 3 ⊢ (𝑡 = 𝑣 → (𝑓 ∈ 𝑇 ↦ ((𝑢‘𝑓) ∘ (𝑡‘𝑓))) = (𝑔 ∈ 𝑇 ↦ ((𝑢‘𝑔) ∘ (𝑣‘𝑔)))) |
13 | 4, 12 | cbvmpov 7507 | . 2 ⊢ (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓)))) = (𝑢 ∈ 𝐸, 𝑣 ∈ 𝐸 ↦ (𝑔 ∈ 𝑇 ↦ ((𝑢‘𝑔) ∘ (𝑣‘𝑔)))) |
14 | 1, 13 | eqtri 2759 | 1 ⊢ 𝑃 = (𝑢 ∈ 𝐸, 𝑣 ∈ 𝐸 ↦ (𝑔 ∈ 𝑇 ↦ ((𝑢‘𝑔) ∘ (𝑣‘𝑔)))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ↦ cmpt 5232 ∘ ccom 5681 ‘cfv 6544 ∈ cmpo 7414 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-rab 3432 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-co 5686 df-iota 6496 df-fv 6552 df-oprab 7416 df-mpo 7417 |
This theorem is referenced by: tendopl 39951 |
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