Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  tendoi Structured version   Visualization version   GIF version

Theorem tendoi 40267
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 6896 . . . 4 (𝑢 = 𝑆 → (𝑢𝑔) = (𝑆𝑔))
21cnveqd 5878 . . 3 (𝑢 = 𝑆(𝑢𝑔) = (𝑆𝑔))
32mpteq2dv 5250 . 2 (𝑢 = 𝑆 → (𝑔𝑇(𝑢𝑔)) = (𝑔𝑇(𝑆𝑔)))
4 tendoi.i . . 3 𝐼 = (𝑠𝐸 ↦ (𝑓𝑇(𝑠𝑓)))
54tendoicbv 40266 . 2 𝐼 = (𝑢𝐸 ↦ (𝑔𝑇(𝑢𝑔)))
6 tendoi.t . 2 𝑇 = ((LTrn‘𝐾)‘𝑊)
73, 5, 6mptfvmpt 7240 1 (𝑆𝐸 → (𝐼𝑆) = (𝑔𝑇(𝑆𝑔)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  cmpt 5231  ccnv 5677  cfv 6548  LTrncltrn 39574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556
This theorem is referenced by:  tendoi2  40268  tendoicl  40269
  Copyright terms: Public domain W3C validator