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Theorem tendoi2 40300
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 40299 . . 3 (𝑆𝐸 → (𝐼𝑆) = (𝑔𝑇(𝑆𝑔)))
43adantr 479 . 2 ((𝑆𝐸𝐹𝑇) → (𝐼𝑆) = (𝑔𝑇(𝑆𝑔)))
5 fveq2 6902 . . . 4 (𝑔 = 𝐹 → (𝑆𝑔) = (𝑆𝐹))
65cnveqd 5882 . . 3 (𝑔 = 𝐹(𝑆𝑔) = (𝑆𝐹))
76adantl 480 . 2 (((𝑆𝐸𝐹𝑇) ∧ 𝑔 = 𝐹) → (𝑆𝑔) = (𝑆𝐹))
8 simpr 483 . 2 ((𝑆𝐸𝐹𝑇) → 𝐹𝑇)
9 fvex 6915 . . . 4 (𝑆𝐹) ∈ V
109cnvex 7939 . . 3 (𝑆𝐹) ∈ V
1110a1i 11 . 2 ((𝑆𝐸𝐹𝑇) → (𝑆𝐹) ∈ V)
124, 7, 8, 11fvmptd 7017 1 ((𝑆𝐸𝐹𝑇) → ((𝐼𝑆)‘𝐹) = (𝑆𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  Vcvv 3473  cmpt 5235  ccnv 5681  cfv 6553  LTrncltrn 39606
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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561
This theorem is referenced by:  tendoicl  40301  tendoipl  40302  dihjatcclem4  40926
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