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Theorem tendoplcbv 35081
 Description: Define sum operation for trace-perserving 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 6102 . . . . 5 (𝑠 = 𝑢 → (𝑠𝑓) = (𝑢𝑓))
32coeq1d 5205 . . . 4 (𝑠 = 𝑢 → ((𝑠𝑓) ∘ (𝑡𝑓)) = ((𝑢𝑓) ∘ (𝑡𝑓)))
43mpteq2dv 4673 . . 3 (𝑠 = 𝑢 → (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))) = (𝑓𝑇 ↦ ((𝑢𝑓) ∘ (𝑡𝑓))))
5 fveq1 6102 . . . . . 6 (𝑡 = 𝑣 → (𝑡𝑓) = (𝑣𝑓))
65coeq2d 5206 . . . . 5 (𝑡 = 𝑣 → ((𝑢𝑓) ∘ (𝑡𝑓)) = ((𝑢𝑓) ∘ (𝑣𝑓)))
76mpteq2dv 4673 . . . 4 (𝑡 = 𝑣 → (𝑓𝑇 ↦ ((𝑢𝑓) ∘ (𝑡𝑓))) = (𝑓𝑇 ↦ ((𝑢𝑓) ∘ (𝑣𝑓))))
8 fveq2 6103 . . . . . 6 (𝑓 = 𝑔 → (𝑢𝑓) = (𝑢𝑔))
9 fveq2 6103 . . . . . 6 (𝑓 = 𝑔 → (𝑣𝑓) = (𝑣𝑔))
108, 9coeq12d 5208 . . . . 5 (𝑓 = 𝑔 → ((𝑢𝑓) ∘ (𝑣𝑓)) = ((𝑢𝑔) ∘ (𝑣𝑔)))
1110cbvmptv 4678 . . . 4 (𝑓𝑇 ↦ ((𝑢𝑓) ∘ (𝑣𝑓))) = (𝑔𝑇 ↦ ((𝑢𝑔) ∘ (𝑣𝑔)))
127, 11syl6eq 2660 . . 3 (𝑡 = 𝑣 → (𝑓𝑇 ↦ ((𝑢𝑓) ∘ (𝑡𝑓))) = (𝑔𝑇 ↦ ((𝑢𝑔) ∘ (𝑣𝑔))))
134, 12cbvmpt2v 6633 . 2 (𝑠𝐸, 𝑡𝐸 ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓)))) = (𝑢𝐸, 𝑣𝐸 ↦ (𝑔𝑇 ↦ ((𝑢𝑔) ∘ (𝑣𝑔))))
141, 13eqtri 2632 1 𝑃 = (𝑢𝐸, 𝑣𝐸 ↦ (𝑔𝑇 ↦ ((𝑢𝑔) ∘ (𝑣𝑔))))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ↦ cmpt 4643   ∘ ccom 5042  ‘cfv 5804   ↦ cmpt2 6551 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-co 5047  df-iota 5768  df-fv 5812  df-oprab 6553  df-mpt2 6554 This theorem is referenced by:  tendopl  35082
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