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Theorem tendopl 40281
Description: Value of endomorphism sum operation. (Contributed by NM, 10-Jun-2013.)
Hypotheses
Ref Expression
tendoplcbv.p 𝑃 = (𝑠𝐸, 𝑡𝐸 ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))
tendopl2.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
Assertion
Ref Expression
tendopl ((𝑈𝐸𝑉𝐸) → (𝑈𝑃𝑉) = (𝑔𝑇 ↦ ((𝑈𝑔) ∘ (𝑉𝑔))))
Distinct variable groups:   𝑡,𝑠,𝐸   𝑓,𝑔,𝑠,𝑡,𝑇   𝑓,𝑊,𝑔,𝑠,𝑡   𝑈,𝑔   𝑔,𝑉
Allowed substitution hints:   𝑃(𝑡,𝑓,𝑔,𝑠)   𝑈(𝑡,𝑓,𝑠)   𝐸(𝑓,𝑔)   𝐾(𝑡,𝑓,𝑔,𝑠)   𝑉(𝑡,𝑓,𝑠)

Proof of Theorem tendopl
Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq1 6901 . . . 4 (𝑢 = 𝑈 → (𝑢𝑔) = (𝑈𝑔))
21coeq1d 5868 . . 3 (𝑢 = 𝑈 → ((𝑢𝑔) ∘ (𝑣𝑔)) = ((𝑈𝑔) ∘ (𝑣𝑔)))
32mpteq2dv 5254 . 2 (𝑢 = 𝑈 → (𝑔𝑇 ↦ ((𝑢𝑔) ∘ (𝑣𝑔))) = (𝑔𝑇 ↦ ((𝑈𝑔) ∘ (𝑣𝑔))))
4 fveq1 6901 . . . 4 (𝑣 = 𝑉 → (𝑣𝑔) = (𝑉𝑔))
54coeq2d 5869 . . 3 (𝑣 = 𝑉 → ((𝑈𝑔) ∘ (𝑣𝑔)) = ((𝑈𝑔) ∘ (𝑉𝑔)))
65mpteq2dv 5254 . 2 (𝑣 = 𝑉 → (𝑔𝑇 ↦ ((𝑈𝑔) ∘ (𝑣𝑔))) = (𝑔𝑇 ↦ ((𝑈𝑔) ∘ (𝑉𝑔))))
7 tendoplcbv.p . . 3 𝑃 = (𝑠𝐸, 𝑡𝐸 ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))
87tendoplcbv 40280 . 2 𝑃 = (𝑢𝐸, 𝑣𝐸 ↦ (𝑔𝑇 ↦ ((𝑢𝑔) ∘ (𝑣𝑔))))
9 tendopl2.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
109fvexi 6916 . . 3 𝑇 ∈ V
1110mptex 7241 . 2 (𝑔𝑇 ↦ ((𝑈𝑔) ∘ (𝑉𝑔))) ∈ V
123, 6, 8, 11ovmpo 7587 1 ((𝑈𝐸𝑉𝐸) → (𝑈𝑃𝑉) = (𝑔𝑇 ↦ ((𝑈𝑔) ∘ (𝑉𝑔))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  cmpt 5235  ccom 5686  cfv 6553  (class class class)co 7426  cmpo 7428  LTrncltrn 39606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431
This theorem is referenced by:  tendopl2  40282  tendoplcl  40286  erngplus  40308  erngplus-rN  40316  dvaplusg  40514
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