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Theorem tendoplcbv 39241
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 6842 . . . . 5 (𝑠 = 𝑢 → (𝑠𝑓) = (𝑢𝑓))
32coeq1d 5818 . . . 4 (𝑠 = 𝑢 → ((𝑠𝑓) ∘ (𝑡𝑓)) = ((𝑢𝑓) ∘ (𝑡𝑓)))
43mpteq2dv 5208 . . 3 (𝑠 = 𝑢 → (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))) = (𝑓𝑇 ↦ ((𝑢𝑓) ∘ (𝑡𝑓))))
5 fveq1 6842 . . . . . 6 (𝑡 = 𝑣 → (𝑡𝑓) = (𝑣𝑓))
65coeq2d 5819 . . . . 5 (𝑡 = 𝑣 → ((𝑢𝑓) ∘ (𝑡𝑓)) = ((𝑢𝑓) ∘ (𝑣𝑓)))
76mpteq2dv 5208 . . . 4 (𝑡 = 𝑣 → (𝑓𝑇 ↦ ((𝑢𝑓) ∘ (𝑡𝑓))) = (𝑓𝑇 ↦ ((𝑢𝑓) ∘ (𝑣𝑓))))
8 fveq2 6843 . . . . . 6 (𝑓 = 𝑔 → (𝑢𝑓) = (𝑢𝑔))
9 fveq2 6843 . . . . . 6 (𝑓 = 𝑔 → (𝑣𝑓) = (𝑣𝑔))
108, 9coeq12d 5821 . . . . 5 (𝑓 = 𝑔 → ((𝑢𝑓) ∘ (𝑣𝑓)) = ((𝑢𝑔) ∘ (𝑣𝑔)))
1110cbvmptv 5219 . . . 4 (𝑓𝑇 ↦ ((𝑢𝑓) ∘ (𝑣𝑓))) = (𝑔𝑇 ↦ ((𝑢𝑔) ∘ (𝑣𝑔)))
127, 11eqtrdi 2793 . . 3 (𝑡 = 𝑣 → (𝑓𝑇 ↦ ((𝑢𝑓) ∘ (𝑡𝑓))) = (𝑔𝑇 ↦ ((𝑢𝑔) ∘ (𝑣𝑔))))
134, 12cbvmpov 7453 . 2 (𝑠𝐸, 𝑡𝐸 ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓)))) = (𝑢𝐸, 𝑣𝐸 ↦ (𝑔𝑇 ↦ ((𝑢𝑔) ∘ (𝑣𝑔))))
141, 13eqtri 2765 1 𝑃 = (𝑢𝐸, 𝑣𝐸 ↦ (𝑔𝑇 ↦ ((𝑢𝑔) ∘ (𝑣𝑔))))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  cmpt 5189  ccom 5638  cfv 6497  cmpo 7360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-co 5643  df-iota 6449  df-fv 6505  df-oprab 7362  df-mpo 7363
This theorem is referenced by:  tendopl  39242
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