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Theorem isgrpinv 16728
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 16717 . . . . . . . . 9  |-  ( x  e.  B  ->  ( N `  x )  =  ( iota_ e  e.  B  ( e  .+  x )  =  .0.  ) )
65ad2antlr 734 . . . . . . . 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 463 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Grp  /\  M : B
--> B )  /\  x  e.  B )  /\  (
( M `  x
)  .+  x )  =  .0.  )  ->  (
( M `  x
)  .+  x )  =  .0.  )
8 simpllr 770 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
Grp  /\  M : B
--> B )  /\  x  e.  B )  /\  (
( M `  x
)  .+  x )  =  .0.  )  ->  M : B --> B )
9 simplr 763 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
Grp  /\  M : B
--> B )  /\  x  e.  B )  /\  (
( M `  x
)  .+  x )  =  .0.  )  ->  x  e.  B )
108, 9ffvelrnd 6028 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  M : B
--> B )  /\  x  e.  B )  /\  (
( M `  x
)  .+  x )  =  .0.  )  ->  ( M `  x )  e.  B )
11 simplll 769 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
Grp  /\  M : B
--> B )  /\  x  e.  B )  /\  (
( M `  x
)  .+  x )  =  .0.  )  ->  G  e.  Grp )
121, 2, 3grpinveu 16712 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  x  e.  B )  ->  E! e  e.  B  ( e  .+  x
)  =  .0.  )
1311, 9, 12syl2anc 667 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  M : B
--> B )  /\  x  e.  B )  /\  (
( M `  x
)  .+  x )  =  .0.  )  ->  E! e  e.  B  (
e  .+  x )  =  .0.  )
14 oveq1 6302 . . . . . . . . . . . 12  |-  ( e  =  ( M `  x )  ->  (
e  .+  x )  =  ( ( M `
 x )  .+  x ) )
1514eqeq1d 2455 . . . . . . . . . . 11  |-  ( e  =  ( M `  x )  ->  (
( e  .+  x
)  =  .0.  <->  ( ( M `  x )  .+  x )  =  .0.  ) )
1615riota2 6279 . . . . . . . . . 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 667 . . . . . . . . 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 214 . . . . . . . 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 2487 . . . . . . 7  |-  ( ( ( ( G  e. 
Grp  /\  M : B
--> B )  /\  x  e.  B )  /\  (
( M `  x
)  .+  x )  =  .0.  )  ->  ( N `  x )  =  ( M `  x ) )
2019ex 436 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  M : B --> B )  /\  x  e.  B
)  ->  ( (
( M `  x
)  .+  x )  =  .0.  ->  ( N `  x )  =  ( M `  x ) ) )
2120ralimdva 2798 . . . . 5  |-  ( ( G  e.  Grp  /\  M : B --> B )  ->  ( A. x  e.  B  ( ( M `  x )  .+  x )  =  .0. 
->  A. x  e.  B  ( N `  x )  =  ( M `  x ) ) )
2221impr 625 . . . 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 16718 . . . . 5  |-  N  Fn  B
24 ffn 5733 . . . . . 6  |-  ( M : B --> B  ->  M  Fn  B )
2524ad2antrl 735 . . . . 5  |-  ( ( G  e.  Grp  /\  ( M : B --> B  /\  A. x  e.  B  ( ( M `  x
)  .+  x )  =  .0.  ) )  ->  M  Fn  B )
26 eqfnfv 5981 . . . . 5  |-  ( ( N  Fn  B  /\  M  Fn  B )  ->  ( N  =  M  <->  A. x  e.  B  ( N `  x )  =  ( M `  x ) ) )
2723, 25, 26sylancr 670 . . . 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 236 . . 3  |-  ( ( G  e.  Grp  /\  ( M : B --> B  /\  A. x  e.  B  ( ( M `  x
)  .+  x )  =  .0.  ) )  ->  N  =  M )
2928ex 436 . 2  |-  ( G  e.  Grp  ->  (
( M : B --> B  /\  A. x  e.  B  ( ( M `
 x )  .+  x )  =  .0.  )  ->  N  =  M ) )
301, 4grpinvf 16722 . . . 4  |-  ( G  e.  Grp  ->  N : B --> B )
311, 2, 3, 4grplinv 16724 . . . . 5  |-  ( ( G  e.  Grp  /\  x  e.  B )  ->  ( ( N `  x )  .+  x
)  =  .0.  )
3231ralrimiva 2804 . . . 4  |-  ( G  e.  Grp  ->  A. x  e.  B  ( ( N `  x )  .+  x )  =  .0.  )
3330, 32jca 535 . . 3  |-  ( G  e.  Grp  ->  ( N : B --> B  /\  A. x  e.  B  ( ( N `  x
)  .+  x )  =  .0.  ) )
34 feq1 5715 . . . 4  |-  ( N  =  M  ->  ( N : B --> B  <->  M : B
--> B ) )
35 fveq1 5869 . . . . . . 7  |-  ( N  =  M  ->  ( N `  x )  =  ( M `  x ) )
3635oveq1d 6310 . . . . . 6  |-  ( N  =  M  ->  (
( N `  x
)  .+  x )  =  ( ( M `
 x )  .+  x ) )
3736eqeq1d 2455 . . . . 5  |-  ( N  =  M  ->  (
( ( N `  x )  .+  x
)  =  .0.  <->  ( ( M `  x )  .+  x )  =  .0.  ) )
3837ralbidv 2829 . . . 4  |-  ( N  =  M  ->  ( A. x  e.  B  ( ( N `  x )  .+  x
)  =  .0.  <->  A. x  e.  B  ( ( M `  x )  .+  x )  =  .0.  ) )
3934, 38anbi12d 718 . . 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 224 . 2  |-  ( G  e.  Grp  ->  ( N  =  M  ->  ( M : B --> B  /\  A. x  e.  B  ( ( M `  x
)  .+  x )  =  .0.  ) ) )
4129, 40impbid 194 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 188    /\ wa 371    = wceq 1446    e. wcel 1889   A.wral 2739   E!wreu 2741    Fn wfn 5580   -->wf 5581   ` cfv 5585   iota_crio 6256  (class class class)co 6295   Basecbs 15133   +g cplusg 15202   0gc0g 15350   Grpcgrp 16681   invgcminusg 16682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-0g 15352  df-mgm 16500  df-sgrp 16539  df-mnd 16549  df-grp 16685  df-minusg 16686
This theorem is referenced by:  oppginv  17022
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