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Theorem tendoi 40176
Description: Value of 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
tendoi (𝑆𝐸 → (𝐼𝑆) = (𝑔𝑇(𝑆𝑔)))
Distinct variable groups:   𝐸,𝑠   𝑓,𝑔,𝑠,𝑇   𝑓,𝑊,𝑔,𝑠   𝑆,𝑔
Allowed substitution hints:   𝑆(𝑓,𝑠)   𝐸(𝑓,𝑔)   𝐼(𝑓,𝑔,𝑠)   𝐾(𝑓,𝑔,𝑠)

Proof of Theorem tendoi
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 fveq1 6883 . . . 4 (𝑢 = 𝑆 → (𝑢𝑔) = (𝑆𝑔))
21cnveqd 5868 . . 3 (𝑢 = 𝑆(𝑢𝑔) = (𝑆𝑔))
32mpteq2dv 5243 . 2 (𝑢 = 𝑆 → (𝑔𝑇(𝑢𝑔)) = (𝑔𝑇(𝑆𝑔)))
4 tendoi.i . . 3 𝐼 = (𝑠𝐸 ↦ (𝑓𝑇(𝑠𝑓)))
54tendoicbv 40175 . 2 𝐼 = (𝑢𝐸 ↦ (𝑔𝑇(𝑢𝑔)))
6 tendoi.t . 2 𝑇 = ((LTrn‘𝐾)‘𝑊)
73, 5, 6mptfvmpt 7224 1 (𝑆𝐸 → (𝐼𝑆) = (𝑔𝑇(𝑆𝑔)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cmpt 5224  ccnv 5668  cfv 6536  LTrncltrn 39483
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544
This theorem is referenced by:  tendoi2  40177  tendoicl  40178
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