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Theorem isgrpinv 15579
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  =  ( invg `  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  =  ( invg `  G )
51, 2, 3, 4grpinvval 15568 . . . . . . . . 9  |-  ( x  e.  B  ->  ( N `  x )  =  ( iota_ e  e.  B  ( e  .+  x )  =  .0.  ) )
65ad2antlr 726 . . . . . . . 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 461 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Grp  /\  M : B
--> B )  /\  x  e.  B )  /\  (
( M `  x
)  .+  x )  =  .0.  )  ->  (
( M `  x
)  .+  x )  =  .0.  )
8 simpllr 758 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
Grp  /\  M : B
--> B )  /\  x  e.  B )  /\  (
( M `  x
)  .+  x )  =  .0.  )  ->  M : B --> B )
9 simplr 754 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
Grp  /\  M : B
--> B )  /\  x  e.  B )  /\  (
( M `  x
)  .+  x )  =  .0.  )  ->  x  e.  B )
108, 9ffvelrnd 5839 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  M : B
--> B )  /\  x  e.  B )  /\  (
( M `  x
)  .+  x )  =  .0.  )  ->  ( M `  x )  e.  B )
11 simplll 757 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
Grp  /\  M : B
--> B )  /\  x  e.  B )  /\  (
( M `  x
)  .+  x )  =  .0.  )  ->  G  e.  Grp )
121, 2, 3grpinveu 15563 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  x  e.  B )  ->  E! e  e.  B  ( e  .+  x
)  =  .0.  )
1311, 9, 12syl2anc 661 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  M : B
--> B )  /\  x  e.  B )  /\  (
( M `  x
)  .+  x )  =  .0.  )  ->  E! e  e.  B  (
e  .+  x )  =  .0.  )
14 oveq1 6093 . . . . . . . . . . . 12  |-  ( e  =  ( M `  x )  ->  (
e  .+  x )  =  ( ( M `
 x )  .+  x ) )
1514eqeq1d 2446 . . . . . . . . . . 11  |-  ( e  =  ( M `  x )  ->  (
( e  .+  x
)  =  .0.  <->  ( ( M `  x )  .+  x )  =  .0.  ) )
1615riota2 6070 . . . . . . . . . 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 ) ) )
1710, 13, 16syl2anc 661 . . . . . . . . 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 ) ) )
187, 17mpbid 210 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  M : B
--> B )  /\  x  e.  B )  /\  (
( M `  x
)  .+  x )  =  .0.  )  ->  ( iota_ e  e.  B  ( e  .+  x )  =  .0.  )  =  ( M `  x
) )
196, 18eqtrd 2470 . . . . . . 7  |-  ( ( ( ( G  e. 
Grp  /\  M : B
--> B )  /\  x  e.  B )  /\  (
( M `  x
)  .+  x )  =  .0.  )  ->  ( N `  x )  =  ( M `  x ) )
2019ex 434 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  M : B --> B )  /\  x  e.  B
)  ->  ( (
( M `  x
)  .+  x )  =  .0.  ->  ( N `  x )  =  ( M `  x ) ) )
2120ralimdva 2789 . . . . 5  |-  ( ( G  e.  Grp  /\  M : B --> B )  ->  ( A. x  e.  B  ( ( M `  x )  .+  x )  =  .0. 
->  A. x  e.  B  ( N `  x )  =  ( M `  x ) ) )
2221impr 619 . . . 4  |-  ( ( G  e.  Grp  /\  ( M : B --> B  /\  A. x  e.  B  ( ( M `  x
)  .+  x )  =  .0.  ) )  ->  A. x  e.  B  ( N `  x )  =  ( M `  x ) )
231, 4grpinvfn 15569 . . . . 5  |-  N  Fn  B
24 ffn 5554 . . . . . 6  |-  ( M : B --> B  ->  M  Fn  B )
2524ad2antrl 727 . . . . 5  |-  ( ( G  e.  Grp  /\  ( M : B --> B  /\  A. x  e.  B  ( ( M `  x
)  .+  x )  =  .0.  ) )  ->  M  Fn  B )
26 eqfnfv 5792 . . . . 5  |-  ( ( N  Fn  B  /\  M  Fn  B )  ->  ( N  =  M  <->  A. x  e.  B  ( N `  x )  =  ( M `  x ) ) )
2723, 25, 26sylancr 663 . . . 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 ) ) )
2822, 27mpbird 232 . . 3  |-  ( ( G  e.  Grp  /\  ( M : B --> B  /\  A. x  e.  B  ( ( M `  x
)  .+  x )  =  .0.  ) )  ->  N  =  M )
2928ex 434 . 2  |-  ( G  e.  Grp  ->  (
( M : B --> B  /\  A. x  e.  B  ( ( M `
 x )  .+  x )  =  .0.  )  ->  N  =  M ) )
301, 4grpinvf 15573 . . . 4  |-  ( G  e.  Grp  ->  N : B --> B )
311, 2, 3, 4grplinv 15575 . . . . 5  |-  ( ( G  e.  Grp  /\  x  e.  B )  ->  ( ( N `  x )  .+  x
)  =  .0.  )
3231ralrimiva 2794 . . . 4  |-  ( G  e.  Grp  ->  A. x  e.  B  ( ( N `  x )  .+  x )  =  .0.  )
3330, 32jca 532 . . 3  |-  ( G  e.  Grp  ->  ( N : B --> B  /\  A. x  e.  B  ( ( N `  x
)  .+  x )  =  .0.  ) )
34 feq1 5537 . . . 4  |-  ( N  =  M  ->  ( N : B --> B  <->  M : B
--> B ) )
35 fveq1 5685 . . . . . . 7  |-  ( N  =  M  ->  ( N `  x )  =  ( M `  x ) )
3635oveq1d 6101 . . . . . 6  |-  ( N  =  M  ->  (
( N `  x
)  .+  x )  =  ( ( M `
 x )  .+  x ) )
3736eqeq1d 2446 . . . . 5  |-  ( N  =  M  ->  (
( ( N `  x )  .+  x
)  =  .0.  <->  ( ( M `  x )  .+  x )  =  .0.  ) )
3837ralbidv 2730 . . . 4  |-  ( N  =  M  ->  ( A. x  e.  B  ( ( N `  x )  .+  x
)  =  .0.  <->  A. x  e.  B  ( ( M `  x )  .+  x )  =  .0.  ) )
3934, 38anbi12d 710 . . 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.  ) ) )
4033, 39syl5ibcom 220 . 2  |-  ( G  e.  Grp  ->  ( N  =  M  ->  ( M : B --> B  /\  A. x  e.  B  ( ( M `  x
)  .+  x )  =  .0.  ) ) )
4129, 40impbid 191 1  |-  ( G  e.  Grp  ->  (
( M : B --> B  /\  A. x  e.  B  ( ( M `
 x )  .+  x )  =  .0.  )  <->  N  =  M
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2710   E!wreu 2712    Fn wfn 5408   -->wf 5409   ` cfv 5413   iota_crio 6046  (class class class)co 6086   Basecbs 14166   +g cplusg 14230   0gc0g 14370   Grpcgrp 15402   invgcminusg 15403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-0g 14372  df-mnd 15407  df-grp 15536  df-minusg 15537
This theorem is referenced by:  oppginv  15865
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