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Theorem tendoplcbv 39738
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 6890 . . . . 5 (𝑠 = 𝑢 → (𝑠𝑓) = (𝑢𝑓))
32coeq1d 5861 . . . 4 (𝑠 = 𝑢 → ((𝑠𝑓) ∘ (𝑡𝑓)) = ((𝑢𝑓) ∘ (𝑡𝑓)))
43mpteq2dv 5250 . . 3 (𝑠 = 𝑢 → (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))) = (𝑓𝑇 ↦ ((𝑢𝑓) ∘ (𝑡𝑓))))
5 fveq1 6890 . . . . . 6 (𝑡 = 𝑣 → (𝑡𝑓) = (𝑣𝑓))
65coeq2d 5862 . . . . 5 (𝑡 = 𝑣 → ((𝑢𝑓) ∘ (𝑡𝑓)) = ((𝑢𝑓) ∘ (𝑣𝑓)))
76mpteq2dv 5250 . . . 4 (𝑡 = 𝑣 → (𝑓𝑇 ↦ ((𝑢𝑓) ∘ (𝑡𝑓))) = (𝑓𝑇 ↦ ((𝑢𝑓) ∘ (𝑣𝑓))))
8 fveq2 6891 . . . . . 6 (𝑓 = 𝑔 → (𝑢𝑓) = (𝑢𝑔))
9 fveq2 6891 . . . . . 6 (𝑓 = 𝑔 → (𝑣𝑓) = (𝑣𝑔))
108, 9coeq12d 5864 . . . . 5 (𝑓 = 𝑔 → ((𝑢𝑓) ∘ (𝑣𝑓)) = ((𝑢𝑔) ∘ (𝑣𝑔)))
1110cbvmptv 5261 . . . 4 (𝑓𝑇 ↦ ((𝑢𝑓) ∘ (𝑣𝑓))) = (𝑔𝑇 ↦ ((𝑢𝑔) ∘ (𝑣𝑔)))
127, 11eqtrdi 2788 . . 3 (𝑡 = 𝑣 → (𝑓𝑇 ↦ ((𝑢𝑓) ∘ (𝑡𝑓))) = (𝑔𝑇 ↦ ((𝑢𝑔) ∘ (𝑣𝑔))))
134, 12cbvmpov 7506 . 2 (𝑠𝐸, 𝑡𝐸 ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓)))) = (𝑢𝐸, 𝑣𝐸 ↦ (𝑔𝑇 ↦ ((𝑢𝑔) ∘ (𝑣𝑔))))
141, 13eqtri 2760 1 𝑃 = (𝑢𝐸, 𝑣𝐸 ↦ (𝑔𝑇 ↦ ((𝑢𝑔) ∘ (𝑣𝑔))))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  cmpt 5231  ccom 5680  cfv 6543  cmpo 7413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-co 5685  df-iota 6495  df-fv 6551  df-oprab 7415  df-mpo 7416
This theorem is referenced by:  tendopl  39739
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