MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpoinvfval Unicode version

Theorem grpoinvfval 20891
Description: The inverse function of a group. (Contributed by NM, 26-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpinvfval.1  |-  X  =  ran  G
grpinvfval.2  |-  U  =  (GId `  G )
grpinvfval.3  |-  N  =  ( inv `  G
)
Assertion
Ref Expression
grpoinvfval  |-  ( G  e.  GrpOp  ->  N  =  ( x  e.  X  |->  ( iota_ y  e.  X
( y G x )  =  U ) ) )
Distinct variable groups:    x, y, G    x, X, y    x, U
Allowed substitution hints:    U( y)    N( x, y)

Proof of Theorem grpoinvfval
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 grpinvfval.3 . 2  |-  N  =  ( inv `  G
)
2 grpinvfval.1 . . . . 5  |-  X  =  ran  G
3 rnexg 4940 . . . . 5  |-  ( G  e.  GrpOp  ->  ran  G  e. 
_V )
42, 3syl5eqel 2367 . . . 4  |-  ( G  e.  GrpOp  ->  X  e.  _V )
5 mptexg 5745 . . . 4  |-  ( X  e.  _V  ->  (
x  e.  X  |->  (
iota_ y  e.  X
( y G x )  =  U ) )  e.  _V )
64, 5syl 15 . . 3  |-  ( G  e.  GrpOp  ->  ( x  e.  X  |->  ( iota_ y  e.  X ( y G x )  =  U ) )  e. 
_V )
7 rneq 4904 . . . . . 6  |-  ( g  =  G  ->  ran  g  =  ran  G )
87, 2syl6eqr 2333 . . . . 5  |-  ( g  =  G  ->  ran  g  =  X )
9 oveq 5864 . . . . . . 7  |-  ( g  =  G  ->  (
y g x )  =  ( y G x ) )
10 fveq2 5525 . . . . . . . 8  |-  ( g  =  G  ->  (GId `  g )  =  (GId
`  G ) )
11 grpinvfval.2 . . . . . . . 8  |-  U  =  (GId `  G )
1210, 11syl6eqr 2333 . . . . . . 7  |-  ( g  =  G  ->  (GId `  g )  =  U )
139, 12eqeq12d 2297 . . . . . 6  |-  ( g  =  G  ->  (
( y g x )  =  (GId `  g )  <->  ( y G x )  =  U ) )
148, 13riotaeqbidv 6307 . . . . 5  |-  ( g  =  G  ->  ( iota_ y  e.  ran  g
( y g x )  =  (GId `  g ) )  =  ( iota_ y  e.  X
( y G x )  =  U ) )
158, 14mpteq12dv 4098 . . . 4  |-  ( g  =  G  ->  (
x  e.  ran  g  |->  ( iota_ y  e.  ran  g ( y g x )  =  (GId
`  g ) ) )  =  ( x  e.  X  |->  ( iota_ y  e.  X ( y G x )  =  U ) ) )
16 df-ginv 20860 . . . 4  |-  inv  =  ( g  e.  GrpOp  |->  ( x  e.  ran  g  |->  ( iota_ y  e. 
ran  g ( y g x )  =  (GId `  g )
) ) )
1715, 16fvmptg 5600 . . 3  |-  ( ( G  e.  GrpOp  /\  (
x  e.  X  |->  (
iota_ y  e.  X
( y G x )  =  U ) )  e.  _V )  ->  ( inv `  G
)  =  ( x  e.  X  |->  ( iota_ y  e.  X ( y G x )  =  U ) ) )
186, 17mpdan 649 . 2  |-  ( G  e.  GrpOp  ->  ( inv `  G )  =  ( x  e.  X  |->  (
iota_ y  e.  X
( y G x )  =  U ) ) )
191, 18syl5eq 2327 1  |-  ( G  e.  GrpOp  ->  N  =  ( x  e.  X  |->  ( iota_ y  e.  X
( y G x )  =  U ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788    e. cmpt 4077   ran crn 4690   ` cfv 5255  (class class class)co 5858   iota_crio 6297   GrpOpcgr 20853  GIdcgi 20854   invcgn 20855
This theorem is referenced by:  grpoinvval  20892  grpoinvf  20907
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-riota 6304  df-ginv 20860
  Copyright terms: Public domain W3C validator