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Theorem tendoi2 37409
Description: Value of additive inverse endomorphism. (Contributed by NM, 12-Jun-2013.)
Hypotheses
Ref Expression
tendoi.i 𝐼 = (𝑠𝐸 ↦ (𝑓𝑇(𝑠𝑓)))
tendoi.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
Assertion
Ref Expression
tendoi2 ((𝑆𝐸𝐹𝑇) → ((𝐼𝑆)‘𝐹) = (𝑆𝐹))
Distinct variable groups:   𝐸,𝑠   𝑓,𝑠,𝑇   𝑓,𝑊,𝑠
Allowed substitution hints:   𝑆(𝑓,𝑠)   𝐸(𝑓)   𝐹(𝑓,𝑠)   𝐼(𝑓,𝑠)   𝐾(𝑓,𝑠)

Proof of Theorem tendoi2
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 tendoi.i . . . 4 𝐼 = (𝑠𝐸 ↦ (𝑓𝑇(𝑠𝑓)))
2 tendoi.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
31, 2tendoi 37408 . . 3 (𝑆𝐸 → (𝐼𝑆) = (𝑔𝑇(𝑆𝑔)))
43adantr 473 . 2 ((𝑆𝐸𝐹𝑇) → (𝐼𝑆) = (𝑔𝑇(𝑆𝑔)))
5 fveq2 6497 . . . 4 (𝑔 = 𝐹 → (𝑆𝑔) = (𝑆𝐹))
65cnveqd 5593 . . 3 (𝑔 = 𝐹(𝑆𝑔) = (𝑆𝐹))
76adantl 474 . 2 (((𝑆𝐸𝐹𝑇) ∧ 𝑔 = 𝐹) → (𝑆𝑔) = (𝑆𝐹))
8 simpr 477 . 2 ((𝑆𝐸𝐹𝑇) → 𝐹𝑇)
9 fvex 6510 . . . 4 (𝑆𝐹) ∈ V
109cnvex 7444 . . 3 (𝑆𝐹) ∈ V
1110a1i 11 . 2 ((𝑆𝐸𝐹𝑇) → (𝑆𝐹) ∈ V)
124, 7, 8, 11fvmptd 6600 1 ((𝑆𝐸𝐹𝑇) → ((𝐼𝑆)‘𝐹) = (𝑆𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1508  wcel 2051  Vcvv 3410  cmpt 5005  ccnv 5403  cfv 6186  LTrncltrn 36715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-rep 5046  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-ral 3088  df-rex 3089  df-reu 3090  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-op 4443  df-uni 4710  df-iun 4791  df-br 4927  df-opab 4989  df-mpt 5006  df-id 5309  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194
This theorem is referenced by:  tendoicl  37410  tendoipl  37411  dihjatcclem4  38035
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