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Theorem tendopl 40159
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 6883 . . . 4 (𝑢 = 𝑈 → (𝑢𝑔) = (𝑈𝑔))
21coeq1d 5854 . . 3 (𝑢 = 𝑈 → ((𝑢𝑔) ∘ (𝑣𝑔)) = ((𝑈𝑔) ∘ (𝑣𝑔)))
32mpteq2dv 5243 . 2 (𝑢 = 𝑈 → (𝑔𝑇 ↦ ((𝑢𝑔) ∘ (𝑣𝑔))) = (𝑔𝑇 ↦ ((𝑈𝑔) ∘ (𝑣𝑔))))
4 fveq1 6883 . . . 4 (𝑣 = 𝑉 → (𝑣𝑔) = (𝑉𝑔))
54coeq2d 5855 . . 3 (𝑣 = 𝑉 → ((𝑈𝑔) ∘ (𝑣𝑔)) = ((𝑈𝑔) ∘ (𝑉𝑔)))
65mpteq2dv 5243 . 2 (𝑣 = 𝑉 → (𝑔𝑇 ↦ ((𝑈𝑔) ∘ (𝑣𝑔))) = (𝑔𝑇 ↦ ((𝑈𝑔) ∘ (𝑉𝑔))))
7 tendoplcbv.p . . 3 𝑃 = (𝑠𝐸, 𝑡𝐸 ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))
87tendoplcbv 40158 . 2 𝑃 = (𝑢𝐸, 𝑣𝐸 ↦ (𝑔𝑇 ↦ ((𝑢𝑔) ∘ (𝑣𝑔))))
9 tendopl2.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
109fvexi 6898 . . 3 𝑇 ∈ V
1110mptex 7219 . 2 (𝑔𝑇 ↦ ((𝑈𝑔) ∘ (𝑉𝑔))) ∈ V
123, 6, 8, 11ovmpo 7563 1 ((𝑈𝐸𝑉𝐸) → (𝑈𝑃𝑉) = (𝑔𝑇 ↦ ((𝑈𝑔) ∘ (𝑉𝑔))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  cmpt 5224  ccom 5673  cfv 6536  (class class class)co 7404  cmpo 7406  LTrncltrn 39484
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409
This theorem is referenced by:  tendopl2  40160  tendoplcl  40164  erngplus  40186  erngplus-rN  40194  dvaplusg  40392
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