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Theorem tendopl2 40760
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 40759 . . 3 ((𝑈𝐸𝑉𝐸) → (𝑈𝑃𝑉) = (𝑔𝑇 ↦ ((𝑈𝑔) ∘ (𝑉𝑔))))
433adant3 1131 . 2 ((𝑈𝐸𝑉𝐸𝐹𝑇) → (𝑈𝑃𝑉) = (𝑔𝑇 ↦ ((𝑈𝑔) ∘ (𝑉𝑔))))
5 fveq2 6907 . . . 4 (𝑔 = 𝐹 → (𝑈𝑔) = (𝑈𝐹))
6 fveq2 6907 . . . 4 (𝑔 = 𝐹 → (𝑉𝑔) = (𝑉𝐹))
75, 6coeq12d 5878 . . 3 (𝑔 = 𝐹 → ((𝑈𝑔) ∘ (𝑉𝑔)) = ((𝑈𝐹) ∘ (𝑉𝐹)))
87adantl 481 . 2 (((𝑈𝐸𝑉𝐸𝐹𝑇) ∧ 𝑔 = 𝐹) → ((𝑈𝑔) ∘ (𝑉𝑔)) = ((𝑈𝐹) ∘ (𝑉𝐹)))
9 simp3 1137 . 2 ((𝑈𝐸𝑉𝐸𝐹𝑇) → 𝐹𝑇)
10 fvex 6920 . . . 4 (𝑈𝐹) ∈ V
11 fvex 6920 . . . 4 (𝑉𝐹) ∈ V
1210, 11coex 7953 . . 3 ((𝑈𝐹) ∘ (𝑉𝐹)) ∈ V
1312a1i 11 . 2 ((𝑈𝐸𝑉𝐸𝐹𝑇) → ((𝑈𝐹) ∘ (𝑉𝐹)) ∈ V)
144, 8, 9, 13fvmptd 7023 1 ((𝑈𝐸𝑉𝐸𝐹𝑇) → ((𝑈𝑃𝑉)‘𝐹) = ((𝑈𝐹) ∘ (𝑉𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1537  wcel 2106  Vcvv 3478  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-pow 5371  ax-pr 5438  ax-un 7754
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-pw 4607  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:  tendoplcl2  40761  tendoplco2  40762  tendopltp  40763  tendoplcom  40765  tendoplass  40766  tendodi1  40767  tendodi2  40768  tendo0pl  40774  tendoipl  40780  tendospdi2  41005
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