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Theorem tendopl 38065
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 6648 . . . 4 (𝑢 = 𝑈 → (𝑢𝑔) = (𝑈𝑔))
21coeq1d 5700 . . 3 (𝑢 = 𝑈 → ((𝑢𝑔) ∘ (𝑣𝑔)) = ((𝑈𝑔) ∘ (𝑣𝑔)))
32mpteq2dv 5129 . 2 (𝑢 = 𝑈 → (𝑔𝑇 ↦ ((𝑢𝑔) ∘ (𝑣𝑔))) = (𝑔𝑇 ↦ ((𝑈𝑔) ∘ (𝑣𝑔))))
4 fveq1 6648 . . . 4 (𝑣 = 𝑉 → (𝑣𝑔) = (𝑉𝑔))
54coeq2d 5701 . . 3 (𝑣 = 𝑉 → ((𝑈𝑔) ∘ (𝑣𝑔)) = ((𝑈𝑔) ∘ (𝑉𝑔)))
65mpteq2dv 5129 . 2 (𝑣 = 𝑉 → (𝑔𝑇 ↦ ((𝑈𝑔) ∘ (𝑣𝑔))) = (𝑔𝑇 ↦ ((𝑈𝑔) ∘ (𝑉𝑔))))
7 tendoplcbv.p . . 3 𝑃 = (𝑠𝐸, 𝑡𝐸 ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))
87tendoplcbv 38064 . 2 𝑃 = (𝑢𝐸, 𝑣𝐸 ↦ (𝑔𝑇 ↦ ((𝑢𝑔) ∘ (𝑣𝑔))))
9 tendopl2.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
109fvexi 6663 . . 3 𝑇 ∈ V
1110mptex 6967 . 2 (𝑔𝑇 ↦ ((𝑈𝑔) ∘ (𝑉𝑔))) ∈ V
123, 6, 8, 11ovmpo 7293 1 ((𝑈𝐸𝑉𝐸) → (𝑈𝑃𝑉) = (𝑔𝑇 ↦ ((𝑈𝑔) ∘ (𝑉𝑔))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  cmpt 5113  ccom 5527  cfv 6328  (class class class)co 7139  cmpo 7141  LTrncltrn 37390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144
This theorem is referenced by:  tendopl2  38066  tendoplcl  38070  erngplus  38092  erngplus-rN  38100  dvaplusg  38298
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