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Theorem tendopl2 40290
Description: Value of result 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
tendopl2 ((𝑈𝐸𝑉𝐸𝐹𝑇) → ((𝑈𝑃𝑉)‘𝐹) = ((𝑈𝐹) ∘ (𝑉𝐹)))
Distinct variable groups:   𝑡,𝑠,𝐸   𝑓,𝑠,𝑡,𝑇   𝑓,𝑊,𝑠,𝑡
Allowed substitution hints:   𝑃(𝑡,𝑓,𝑠)   𝑈(𝑡,𝑓,𝑠)   𝐸(𝑓)   𝐹(𝑡,𝑓,𝑠)   𝐾(𝑡,𝑓,𝑠)   𝑉(𝑡,𝑓,𝑠)

Proof of Theorem tendopl2
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 tendoplcbv.p . . . 4 𝑃 = (𝑠𝐸, 𝑡𝐸 ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))
2 tendopl2.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
31, 2tendopl 40289 . . 3 ((𝑈𝐸𝑉𝐸) → (𝑈𝑃𝑉) = (𝑔𝑇 ↦ ((𝑈𝑔) ∘ (𝑉𝑔))))
433adant3 1129 . 2 ((𝑈𝐸𝑉𝐸𝐹𝑇) → (𝑈𝑃𝑉) = (𝑔𝑇 ↦ ((𝑈𝑔) ∘ (𝑉𝑔))))
5 fveq2 6902 . . . 4 (𝑔 = 𝐹 → (𝑈𝑔) = (𝑈𝐹))
6 fveq2 6902 . . . 4 (𝑔 = 𝐹 → (𝑉𝑔) = (𝑉𝐹))
75, 6coeq12d 5871 . . 3 (𝑔 = 𝐹 → ((𝑈𝑔) ∘ (𝑉𝑔)) = ((𝑈𝐹) ∘ (𝑉𝐹)))
87adantl 480 . 2 (((𝑈𝐸𝑉𝐸𝐹𝑇) ∧ 𝑔 = 𝐹) → ((𝑈𝑔) ∘ (𝑉𝑔)) = ((𝑈𝐹) ∘ (𝑉𝐹)))
9 simp3 1135 . 2 ((𝑈𝐸𝑉𝐸𝐹𝑇) → 𝐹𝑇)
10 fvex 6915 . . . 4 (𝑈𝐹) ∈ V
11 fvex 6915 . . . 4 (𝑉𝐹) ∈ V
1210, 11coex 7946 . . 3 ((𝑈𝐹) ∘ (𝑉𝐹)) ∈ V
1312a1i 11 . 2 ((𝑈𝐸𝑉𝐸𝐹𝑇) → ((𝑈𝐹) ∘ (𝑉𝐹)) ∈ V)
144, 8, 9, 13fvmptd 7017 1 ((𝑈𝐸𝑉𝐸𝐹𝑇) → ((𝑈𝑃𝑉)‘𝐹) = ((𝑈𝐹) ∘ (𝑉𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1533  wcel 2098  Vcvv 3473  cmpt 5235  ccom 5686  cfv 6553  (class class class)co 7426  cmpo 7428  LTrncltrn 39614
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-pow 5369  ax-pr 5433  ax-un 7748
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-pw 4608  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:  tendoplcl2  40291  tendoplco2  40292  tendopltp  40293  tendoplcom  40295  tendoplass  40296  tendodi1  40297  tendodi2  40298  tendo0pl  40304  tendoipl  40310  tendospdi2  40535
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