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Theorem tendopl2 40161
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 40160 . . 3 ((𝑈𝐸𝑉𝐸) → (𝑈𝑃𝑉) = (𝑔𝑇 ↦ ((𝑈𝑔) ∘ (𝑉𝑔))))
433adant3 1129 . 2 ((𝑈𝐸𝑉𝐸𝐹𝑇) → (𝑈𝑃𝑉) = (𝑔𝑇 ↦ ((𝑈𝑔) ∘ (𝑉𝑔))))
5 fveq2 6885 . . . 4 (𝑔 = 𝐹 → (𝑈𝑔) = (𝑈𝐹))
6 fveq2 6885 . . . 4 (𝑔 = 𝐹 → (𝑉𝑔) = (𝑉𝐹))
75, 6coeq12d 5858 . . 3 (𝑔 = 𝐹 → ((𝑈𝑔) ∘ (𝑉𝑔)) = ((𝑈𝐹) ∘ (𝑉𝐹)))
87adantl 481 . 2 (((𝑈𝐸𝑉𝐸𝐹𝑇) ∧ 𝑔 = 𝐹) → ((𝑈𝑔) ∘ (𝑉𝑔)) = ((𝑈𝐹) ∘ (𝑉𝐹)))
9 simp3 1135 . 2 ((𝑈𝐸𝑉𝐸𝐹𝑇) → 𝐹𝑇)
10 fvex 6898 . . . 4 (𝑈𝐹) ∈ V
11 fvex 6898 . . . 4 (𝑉𝐹) ∈ V
1210, 11coex 7920 . . 3 ((𝑈𝐹) ∘ (𝑉𝐹)) ∈ V
1312a1i 11 . 2 ((𝑈𝐸𝑉𝐸𝐹𝑇) → ((𝑈𝐹) ∘ (𝑉𝐹)) ∈ V)
144, 8, 9, 13fvmptd 6999 1 ((𝑈𝐸𝑉𝐸𝐹𝑇) → ((𝑈𝑃𝑉)‘𝐹) = ((𝑈𝐹) ∘ (𝑉𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1533  wcel 2098  Vcvv 3468  cmpt 5224  ccom 5673  cfv 6537  (class class class)co 7405  cmpo 7407  LTrncltrn 39485
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-pow 5356  ax-pr 5420  ax-un 7722
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-pw 4599  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 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410
This theorem is referenced by:  tendoplcl2  40162  tendoplco2  40163  tendopltp  40164  tendoplcom  40166  tendoplass  40167  tendodi1  40168  tendodi2  40169  tendo0pl  40175  tendoipl  40181  tendospdi2  40406
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