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Theorem tendopl 36585
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 6331 . . . 4 (𝑢 = 𝑈 → (𝑢𝑔) = (𝑈𝑔))
21coeq1d 5422 . . 3 (𝑢 = 𝑈 → ((𝑢𝑔) ∘ (𝑣𝑔)) = ((𝑈𝑔) ∘ (𝑣𝑔)))
32mpteq2dv 4879 . 2 (𝑢 = 𝑈 → (𝑔𝑇 ↦ ((𝑢𝑔) ∘ (𝑣𝑔))) = (𝑔𝑇 ↦ ((𝑈𝑔) ∘ (𝑣𝑔))))
4 fveq1 6331 . . . 4 (𝑣 = 𝑉 → (𝑣𝑔) = (𝑉𝑔))
54coeq2d 5423 . . 3 (𝑣 = 𝑉 → ((𝑈𝑔) ∘ (𝑣𝑔)) = ((𝑈𝑔) ∘ (𝑉𝑔)))
65mpteq2dv 4879 . 2 (𝑣 = 𝑉 → (𝑔𝑇 ↦ ((𝑈𝑔) ∘ (𝑣𝑔))) = (𝑔𝑇 ↦ ((𝑈𝑔) ∘ (𝑉𝑔))))
7 tendoplcbv.p . . 3 𝑃 = (𝑠𝐸, 𝑡𝐸 ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))
87tendoplcbv 36584 . 2 𝑃 = (𝑢𝐸, 𝑣𝐸 ↦ (𝑔𝑇 ↦ ((𝑢𝑔) ∘ (𝑣𝑔))))
9 tendopl2.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
10 fvex 6342 . . . 4 ((LTrn‘𝐾)‘𝑊) ∈ V
119, 10eqeltri 2846 . . 3 𝑇 ∈ V
1211mptex 6630 . 2 (𝑔𝑇 ↦ ((𝑈𝑔) ∘ (𝑉𝑔))) ∈ V
133, 6, 8, 12ovmpt2 6943 1 ((𝑈𝐸𝑉𝐸) → (𝑈𝑃𝑉) = (𝑔𝑇 ↦ ((𝑈𝑔) ∘ (𝑉𝑔))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  cmpt 4863  ccom 5253  cfv 6031  (class class class)co 6793  cmpt2 6795  LTrncltrn 35909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798
This theorem is referenced by:  tendopl2  36586  tendoplcl  36590  erngplus  36612  erngplus-rN  36620  dvaplusg  36818
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