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Theorem tendopl 40759
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 6906 . . . 4 (𝑢 = 𝑈 → (𝑢𝑔) = (𝑈𝑔))
21coeq1d 5875 . . 3 (𝑢 = 𝑈 → ((𝑢𝑔) ∘ (𝑣𝑔)) = ((𝑈𝑔) ∘ (𝑣𝑔)))
32mpteq2dv 5250 . 2 (𝑢 = 𝑈 → (𝑔𝑇 ↦ ((𝑢𝑔) ∘ (𝑣𝑔))) = (𝑔𝑇 ↦ ((𝑈𝑔) ∘ (𝑣𝑔))))
4 fveq1 6906 . . . 4 (𝑣 = 𝑉 → (𝑣𝑔) = (𝑉𝑔))
54coeq2d 5876 . . 3 (𝑣 = 𝑉 → ((𝑈𝑔) ∘ (𝑣𝑔)) = ((𝑈𝑔) ∘ (𝑉𝑔)))
65mpteq2dv 5250 . 2 (𝑣 = 𝑉 → (𝑔𝑇 ↦ ((𝑈𝑔) ∘ (𝑣𝑔))) = (𝑔𝑇 ↦ ((𝑈𝑔) ∘ (𝑉𝑔))))
7 tendoplcbv.p . . 3 𝑃 = (𝑠𝐸, 𝑡𝐸 ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))
87tendoplcbv 40758 . 2 𝑃 = (𝑢𝐸, 𝑣𝐸 ↦ (𝑔𝑇 ↦ ((𝑢𝑔) ∘ (𝑣𝑔))))
9 tendopl2.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
109fvexi 6921 . . 3 𝑇 ∈ V
1110mptex 7243 . 2 (𝑔𝑇 ↦ ((𝑈𝑔) ∘ (𝑉𝑔))) ∈ V
123, 6, 8, 11ovmpo 7593 1 ((𝑈𝐸𝑉𝐸) → (𝑈𝑃𝑉) = (𝑔𝑇 ↦ ((𝑈𝑔) ∘ (𝑉𝑔))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  cmpt 5231  ccom 5693  cfv 6563  (class class class)co 7431  cmpo 7433  LTrncltrn 40084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436
This theorem is referenced by:  tendopl2  40760  tendoplcl  40764  erngplus  40786  erngplus-rN  40794  dvaplusg  40992
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