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Theorem isgrpinv 14532
Description: Properties showing that a function  M is the inverse function of a group. (Contributed by NM, 7-Aug-2013.) (Revised by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
grpinv.b  |-  B  =  ( Base `  G
)
grpinv.p  |-  .+  =  ( +g  `  G )
grpinv.u  |-  .0.  =  ( 0g `  G )
grpinv.n  |-  N  =  ( inv g `  G )
Assertion
Ref Expression
isgrpinv  |-  ( G  e.  Grp  ->  (
( M : B --> B  /\  A. x  e.  B  ( ( M `
 x )  .+  x )  =  .0.  )  <->  N  =  M
) )
Distinct variable groups:    x, B    x, G    x,  .0.    x,  .+    x, M    x, N

Proof of Theorem isgrpinv
Dummy variable  e is distinct from all other variables.
StepHypRef Expression
1 grpinv.b . . . . . . . . . 10  |-  B  =  ( Base `  G
)
2 grpinv.p . . . . . . . . . 10  |-  .+  =  ( +g  `  G )
3 grpinv.u . . . . . . . . . 10  |-  .0.  =  ( 0g `  G )
4 grpinv.n . . . . . . . . . 10  |-  N  =  ( inv g `  G )
51, 2, 3, 4grpinvval 14521 . . . . . . . . 9  |-  ( x  e.  B  ->  ( N `  x )  =  ( iota_ e  e.  B ( e  .+  x )  =  .0.  ) )
65ad2antlr 707 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  M : B
--> B )  /\  x  e.  B )  /\  (
( M `  x
)  .+  x )  =  .0.  )  ->  ( N `  x )  =  ( iota_ e  e.  B ( e  .+  x )  =  .0.  ) )
7 simpr 447 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Grp  /\  M : B
--> B )  /\  x  e.  B )  /\  (
( M `  x
)  .+  x )  =  .0.  )  ->  (
( M `  x
)  .+  x )  =  .0.  )
8 simpllr 735 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
Grp  /\  M : B
--> B )  /\  x  e.  B )  /\  (
( M `  x
)  .+  x )  =  .0.  )  ->  M : B --> B )
9 simplr 731 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
Grp  /\  M : B
--> B )  /\  x  e.  B )  /\  (
( M `  x
)  .+  x )  =  .0.  )  ->  x  e.  B )
10 ffvelrn 5663 . . . . . . . . . . 11  |-  ( ( M : B --> B  /\  x  e.  B )  ->  ( M `  x
)  e.  B )
118, 9, 10syl2anc 642 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  M : B
--> B )  /\  x  e.  B )  /\  (
( M `  x
)  .+  x )  =  .0.  )  ->  ( M `  x )  e.  B )
12 simplll 734 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
Grp  /\  M : B
--> B )  /\  x  e.  B )  /\  (
( M `  x
)  .+  x )  =  .0.  )  ->  G  e.  Grp )
131, 2, 3grpinveu 14516 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  x  e.  B )  ->  E! e  e.  B  ( e  .+  x
)  =  .0.  )
1412, 9, 13syl2anc 642 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  M : B
--> B )  /\  x  e.  B )  /\  (
( M `  x
)  .+  x )  =  .0.  )  ->  E! e  e.  B  (
e  .+  x )  =  .0.  )
15 oveq1 5865 . . . . . . . . . . . 12  |-  ( e  =  ( M `  x )  ->  (
e  .+  x )  =  ( ( M `
 x )  .+  x ) )
1615eqeq1d 2291 . . . . . . . . . . 11  |-  ( e  =  ( M `  x )  ->  (
( e  .+  x
)  =  .0.  <->  ( ( M `  x )  .+  x )  =  .0.  ) )
1716riota2 6327 . . . . . . . . . 10  |-  ( ( ( M `  x
)  e.  B  /\  E! e  e.  B  ( e  .+  x
)  =  .0.  )  ->  ( ( ( M `
 x )  .+  x )  =  .0.  <->  (
iota_ e  e.  B
( e  .+  x
)  =  .0.  )  =  ( M `  x ) ) )
1811, 14, 17syl2anc 642 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Grp  /\  M : B
--> B )  /\  x  e.  B )  /\  (
( M `  x
)  .+  x )  =  .0.  )  ->  (
( ( M `  x )  .+  x
)  =  .0.  <->  ( iota_ e  e.  B ( e 
.+  x )  =  .0.  )  =  ( M `  x ) ) )
197, 18mpbid 201 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  M : B
--> B )  /\  x  e.  B )  /\  (
( M `  x
)  .+  x )  =  .0.  )  ->  ( iota_ e  e.  B ( e  .+  x )  =  .0.  )  =  ( M `  x
) )
206, 19eqtrd 2315 . . . . . . 7  |-  ( ( ( ( G  e. 
Grp  /\  M : B
--> B )  /\  x  e.  B )  /\  (
( M `  x
)  .+  x )  =  .0.  )  ->  ( N `  x )  =  ( M `  x ) )
2120ex 423 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  M : B --> B )  /\  x  e.  B
)  ->  ( (
( M `  x
)  .+  x )  =  .0.  ->  ( N `  x )  =  ( M `  x ) ) )
2221ralimdva 2621 . . . . 5  |-  ( ( G  e.  Grp  /\  M : B --> B )  ->  ( A. x  e.  B  ( ( M `  x )  .+  x )  =  .0. 
->  A. x  e.  B  ( N `  x )  =  ( M `  x ) ) )
2322impr 602 . . . 4  |-  ( ( G  e.  Grp  /\  ( M : B --> B  /\  A. x  e.  B  ( ( M `  x
)  .+  x )  =  .0.  ) )  ->  A. x  e.  B  ( N `  x )  =  ( M `  x ) )
241, 4grpinvfn 14522 . . . . 5  |-  N  Fn  B
25 ffn 5389 . . . . . 6  |-  ( M : B --> B  ->  M  Fn  B )
2625ad2antrl 708 . . . . 5  |-  ( ( G  e.  Grp  /\  ( M : B --> B  /\  A. x  e.  B  ( ( M `  x
)  .+  x )  =  .0.  ) )  ->  M  Fn  B )
27 eqfnfv 5622 . . . . 5  |-  ( ( N  Fn  B  /\  M  Fn  B )  ->  ( N  =  M  <->  A. x  e.  B  ( N `  x )  =  ( M `  x ) ) )
2824, 26, 27sylancr 644 . . . 4  |-  ( ( G  e.  Grp  /\  ( M : B --> B  /\  A. x  e.  B  ( ( M `  x
)  .+  x )  =  .0.  ) )  -> 
( N  =  M  <->  A. x  e.  B  ( N `  x )  =  ( M `  x ) ) )
2923, 28mpbird 223 . . 3  |-  ( ( G  e.  Grp  /\  ( M : B --> B  /\  A. x  e.  B  ( ( M `  x
)  .+  x )  =  .0.  ) )  ->  N  =  M )
3029ex 423 . 2  |-  ( G  e.  Grp  ->  (
( M : B --> B  /\  A. x  e.  B  ( ( M `
 x )  .+  x )  =  .0.  )  ->  N  =  M ) )
311, 4grpinvf 14526 . . . 4  |-  ( G  e.  Grp  ->  N : B --> B )
321, 2, 3, 4grplinv 14528 . . . . 5  |-  ( ( G  e.  Grp  /\  x  e.  B )  ->  ( ( N `  x )  .+  x
)  =  .0.  )
3332ralrimiva 2626 . . . 4  |-  ( G  e.  Grp  ->  A. x  e.  B  ( ( N `  x )  .+  x )  =  .0.  )
3431, 33jca 518 . . 3  |-  ( G  e.  Grp  ->  ( N : B --> B  /\  A. x  e.  B  ( ( N `  x
)  .+  x )  =  .0.  ) )
35 feq1 5375 . . . 4  |-  ( N  =  M  ->  ( N : B --> B  <->  M : B
--> B ) )
36 fveq1 5524 . . . . . . 7  |-  ( N  =  M  ->  ( N `  x )  =  ( M `  x ) )
3736oveq1d 5873 . . . . . 6  |-  ( N  =  M  ->  (
( N `  x
)  .+  x )  =  ( ( M `
 x )  .+  x ) )
3837eqeq1d 2291 . . . . 5  |-  ( N  =  M  ->  (
( ( N `  x )  .+  x
)  =  .0.  <->  ( ( M `  x )  .+  x )  =  .0.  ) )
3938ralbidv 2563 . . . 4  |-  ( N  =  M  ->  ( A. x  e.  B  ( ( N `  x )  .+  x
)  =  .0.  <->  A. x  e.  B  ( ( M `  x )  .+  x )  =  .0.  ) )
4035, 39anbi12d 691 . . 3  |-  ( N  =  M  ->  (
( N : B --> B  /\  A. x  e.  B  ( ( N `
 x )  .+  x )  =  .0.  )  <->  ( M : B
--> B  /\  A. x  e.  B  ( ( M `  x )  .+  x )  =  .0.  ) ) )
4134, 40syl5ibcom 211 . 2  |-  ( G  e.  Grp  ->  ( N  =  M  ->  ( M : B --> B  /\  A. x  e.  B  ( ( M `  x
)  .+  x )  =  .0.  ) ) )
4230, 41impbid 183 1  |-  ( G  e.  Grp  ->  (
( M : B --> B  /\  A. x  e.  B  ( ( M `
 x )  .+  x )  =  .0.  )  <->  N  =  M
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E!wreu 2545    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   iota_crio 6297   Basecbs 13148   +g cplusg 13208   0gc0g 13400   Grpcgrp 14362   inv gcminusg 14363
This theorem is referenced by:  oppginv  14832
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-pow 4188  ax-pr 4214
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-rmo 2551  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-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490
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