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

Theorem grpoinvfval 24930
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 6716 . . . . 5  |-  ( G  e.  GrpOp  ->  ran  G  e. 
_V )
42, 3syl5eqel 2559 . . . 4  |-  ( G  e.  GrpOp  ->  X  e.  _V )
5 mptexg 6130 . . . 4  |-  ( X  e.  _V  ->  (
x  e.  X  |->  (
iota_ y  e.  X  ( y G x )  =  U ) )  e.  _V )
64, 5syl 16 . . 3  |-  ( G  e.  GrpOp  ->  ( x  e.  X  |->  ( iota_ y  e.  X  ( y G x )  =  U ) )  e. 
_V )
7 rneq 5228 . . . . . 6  |-  ( g  =  G  ->  ran  g  =  ran  G )
87, 2syl6eqr 2526 . . . . 5  |-  ( g  =  G  ->  ran  g  =  X )
9 oveq 6290 . . . . . . 7  |-  ( g  =  G  ->  (
y g x )  =  ( y G x ) )
10 fveq2 5866 . . . . . . . 8  |-  ( g  =  G  ->  (GId `  g )  =  (GId
`  G ) )
11 grpinvfval.2 . . . . . . . 8  |-  U  =  (GId `  G )
1210, 11syl6eqr 2526 . . . . . . 7  |-  ( g  =  G  ->  (GId `  g )  =  U )
139, 12eqeq12d 2489 . . . . . 6  |-  ( g  =  G  ->  (
( y g x )  =  (GId `  g )  <->  ( y G x )  =  U ) )
148, 13riotaeqbidv 6248 . . . . 5  |-  ( g  =  G  ->  ( iota_ y  e.  ran  g
( y g x )  =  (GId `  g ) )  =  ( iota_ y  e.  X  ( y G x )  =  U ) )
158, 14mpteq12dv 4525 . . . 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 24899 . . . 4  |-  inv  =  ( g  e.  GrpOp  |->  ( x  e.  ran  g  |->  ( iota_ y  e. 
ran  g ( y g x )  =  (GId `  g )
) ) )
1715, 16fvmptg 5948 . . 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 668 . 2  |-  ( G  e.  GrpOp  ->  ( inv `  G )  =  ( x  e.  X  |->  (
iota_ y  e.  X  ( y G x )  =  U ) ) )
191, 18syl5eq 2520 1  |-  ( G  e.  GrpOp  ->  N  =  ( x  e.  X  |->  ( iota_ y  e.  X  ( y G x )  =  U ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   _Vcvv 3113    |-> cmpt 4505   ran crn 5000   ` cfv 5588   iota_crio 6244  (class class class)co 6284   GrpOpcgr 24892  GIdcgi 24893   invcgn 24894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-ginv 24899
This theorem is referenced by:  grpoinvval  24931  grpoinvf  24946
  Copyright terms: Public domain W3C validator