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Theorem tendoplcbv 39950
Description: Define sum operation for trace-preserving endomorphisms. Change bound variables to isolate them later. (Contributed by NM, 11-Jun-2013.)
Hypothesis
Ref Expression
tendoplcbv.p 𝑃 = (𝑠𝐸, 𝑡𝐸 ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))
Assertion
Ref Expression
tendoplcbv 𝑃 = (𝑢𝐸, 𝑣𝐸 ↦ (𝑔𝑇 ↦ ((𝑢𝑔) ∘ (𝑣𝑔))))
Distinct variable groups:   𝑡,𝑠,𝑢,𝑣,𝐸   𝑓,𝑔,𝑠,𝑡,𝑢,𝑣,𝑇
Allowed substitution hints:   𝑃(𝑣,𝑢,𝑡,𝑓,𝑔,𝑠)   𝐸(𝑓,𝑔)

Proof of Theorem tendoplcbv
StepHypRef Expression
1 tendoplcbv.p . 2 𝑃 = (𝑠𝐸, 𝑡𝐸 ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))
2 fveq1 6891 . . . . 5 (𝑠 = 𝑢 → (𝑠𝑓) = (𝑢𝑓))
32coeq1d 5862 . . . 4 (𝑠 = 𝑢 → ((𝑠𝑓) ∘ (𝑡𝑓)) = ((𝑢𝑓) ∘ (𝑡𝑓)))
43mpteq2dv 5251 . . 3 (𝑠 = 𝑢 → (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))) = (𝑓𝑇 ↦ ((𝑢𝑓) ∘ (𝑡𝑓))))
5 fveq1 6891 . . . . . 6 (𝑡 = 𝑣 → (𝑡𝑓) = (𝑣𝑓))
65coeq2d 5863 . . . . 5 (𝑡 = 𝑣 → ((𝑢𝑓) ∘ (𝑡𝑓)) = ((𝑢𝑓) ∘ (𝑣𝑓)))
76mpteq2dv 5251 . . . 4 (𝑡 = 𝑣 → (𝑓𝑇 ↦ ((𝑢𝑓) ∘ (𝑡𝑓))) = (𝑓𝑇 ↦ ((𝑢𝑓) ∘ (𝑣𝑓))))
8 fveq2 6892 . . . . . 6 (𝑓 = 𝑔 → (𝑢𝑓) = (𝑢𝑔))
9 fveq2 6892 . . . . . 6 (𝑓 = 𝑔 → (𝑣𝑓) = (𝑣𝑔))
108, 9coeq12d 5865 . . . . 5 (𝑓 = 𝑔 → ((𝑢𝑓) ∘ (𝑣𝑓)) = ((𝑢𝑔) ∘ (𝑣𝑔)))
1110cbvmptv 5262 . . . 4 (𝑓𝑇 ↦ ((𝑢𝑓) ∘ (𝑣𝑓))) = (𝑔𝑇 ↦ ((𝑢𝑔) ∘ (𝑣𝑔)))
127, 11eqtrdi 2787 . . 3 (𝑡 = 𝑣 → (𝑓𝑇 ↦ ((𝑢𝑓) ∘ (𝑡𝑓))) = (𝑔𝑇 ↦ ((𝑢𝑔) ∘ (𝑣𝑔))))
134, 12cbvmpov 7507 . 2 (𝑠𝐸, 𝑡𝐸 ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓)))) = (𝑢𝐸, 𝑣𝐸 ↦ (𝑔𝑇 ↦ ((𝑢𝑔) ∘ (𝑣𝑔))))
141, 13eqtri 2759 1 𝑃 = (𝑢𝐸, 𝑣𝐸 ↦ (𝑔𝑇 ↦ ((𝑢𝑔) ∘ (𝑣𝑔))))
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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