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

Theorem grpinvfval 14835
Description: The inverse function of a group. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.)
Hypotheses
Ref Expression
grpinvval.b  |-  B  =  ( Base `  G
)
grpinvval.p  |-  .+  =  ( +g  `  G )
grpinvval.o  |-  .0.  =  ( 0g `  G )
grpinvval.n  |-  N  =  ( inv g `  G )
Assertion
Ref Expression
grpinvfval  |-  N  =  ( x  e.  B  |->  ( iota_ y  e.  B
( y  .+  x
)  =  .0.  )
)
Distinct variable groups:    x, y, B    x, G, y    x,  .0.    x,  .+
Allowed substitution hints:    .+ ( y)    N( x, y)    .0. ( y)

Proof of Theorem grpinvfval
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 grpinvval.n . 2  |-  N  =  ( inv g `  G )
2 fveq2 5720 . . . . . 6  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
3 grpinvval.b . . . . . 6  |-  B  =  ( Base `  G
)
42, 3syl6eqr 2485 . . . . 5  |-  ( g  =  G  ->  ( Base `  g )  =  B )
5 fveq2 5720 . . . . . . . . 9  |-  ( g  =  G  ->  ( +g  `  g )  =  ( +g  `  G
) )
6 grpinvval.p . . . . . . . . 9  |-  .+  =  ( +g  `  G )
75, 6syl6eqr 2485 . . . . . . . 8  |-  ( g  =  G  ->  ( +g  `  g )  = 
.+  )
87oveqd 6090 . . . . . . 7  |-  ( g  =  G  ->  (
y ( +g  `  g
) x )  =  ( y  .+  x
) )
9 fveq2 5720 . . . . . . . 8  |-  ( g  =  G  ->  ( 0g `  g )  =  ( 0g `  G
) )
10 grpinvval.o . . . . . . . 8  |-  .0.  =  ( 0g `  G )
119, 10syl6eqr 2485 . . . . . . 7  |-  ( g  =  G  ->  ( 0g `  g )  =  .0.  )
128, 11eqeq12d 2449 . . . . . 6  |-  ( g  =  G  ->  (
( y ( +g  `  g ) x )  =  ( 0g `  g )  <->  ( y  .+  x )  =  .0.  ) )
134, 12riotaeqbidv 6544 . . . . 5  |-  ( g  =  G  ->  ( iota_ y  e.  ( Base `  g ) ( y ( +g  `  g
) x )  =  ( 0g `  g
) )  =  (
iota_ y  e.  B
( y  .+  x
)  =  .0.  )
)
144, 13mpteq12dv 4279 . . . 4  |-  ( g  =  G  ->  (
x  e.  ( Base `  g )  |->  ( iota_ y  e.  ( Base `  g
) ( y ( +g  `  g ) x )  =  ( 0g `  g ) ) )  =  ( x  e.  B  |->  (
iota_ y  e.  B
( y  .+  x
)  =  .0.  )
) )
15 df-minusg 14805 . . . 4  |-  inv g  =  ( g  e. 
_V  |->  ( x  e.  ( Base `  g
)  |->  ( iota_ y  e.  ( Base `  g
) ( y ( +g  `  g ) x )  =  ( 0g `  g ) ) ) )
16 fvex 5734 . . . . . 6  |-  ( Base `  G )  e.  _V
173, 16eqeltri 2505 . . . . 5  |-  B  e. 
_V
1817mptex 5958 . . . 4  |-  ( x  e.  B  |->  ( iota_ y  e.  B ( y 
.+  x )  =  .0.  ) )  e. 
_V
1914, 15, 18fvmpt 5798 . . 3  |-  ( G  e.  _V  ->  ( inv g `  G )  =  ( x  e.  B  |->  ( iota_ y  e.  B ( y  .+  x )  =  .0.  ) ) )
20 fvprc 5714 . . . . 5  |-  ( -.  G  e.  _V  ->  ( inv g `  G
)  =  (/) )
21 mpt0 5564 . . . . 5  |-  ( x  e.  (/)  |->  ( iota_ y  e.  B ( y  .+  x )  =  .0.  ) )  =  (/)
2220, 21syl6eqr 2485 . . . 4  |-  ( -.  G  e.  _V  ->  ( inv g `  G
)  =  ( x  e.  (/)  |->  ( iota_ y  e.  B ( y  .+  x )  =  .0.  ) ) )
23 fvprc 5714 . . . . . 6  |-  ( -.  G  e.  _V  ->  (
Base `  G )  =  (/) )
243, 23syl5eq 2479 . . . . 5  |-  ( -.  G  e.  _V  ->  B  =  (/) )
2524mpteq1d 4282 . . . 4  |-  ( -.  G  e.  _V  ->  ( x  e.  B  |->  (
iota_ y  e.  B
( y  .+  x
)  =  .0.  )
)  =  ( x  e.  (/)  |->  ( iota_ y  e.  B ( y  .+  x )  =  .0.  ) ) )
2622, 25eqtr4d 2470 . . 3  |-  ( -.  G  e.  _V  ->  ( inv g `  G
)  =  ( x  e.  B  |->  ( iota_ y  e.  B ( y 
.+  x )  =  .0.  ) ) )
2719, 26pm2.61i 158 . 2  |-  ( inv g `  G )  =  ( x  e.  B  |->  ( iota_ y  e.  B ( y  .+  x )  =  .0.  ) )
281, 27eqtri 2455 1  |-  N  =  ( x  e.  B  |->  ( iota_ y  e.  B
( y  .+  x
)  =  .0.  )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1652    e. wcel 1725   _Vcvv 2948   (/)c0 3620    e. cmpt 4258   ` cfv 5446  (class class class)co 6073   iota_crio 6534   Basecbs 13461   +g cplusg 13521   0gc0g 13715   inv gcminusg 14678
This theorem is referenced by:  grpinvval  14836  grpinvfn  14837  grpinvf  14841  grpinvpropd  14858  opprneg  15732
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-riota 6541  df-minusg 14805
  Copyright terms: Public domain W3C validator