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Theorem isgrpinv 12803
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 12793 . . . . . . . . 9  |-  ( x  e.  B  ->  ( N `  x )  =  ( iota_ e  e.  B  ( e  .+  x )  =  .0.  ) )
65ad2antlr 489 . . . . . . . 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 110 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Grp  /\  M : B
--> B )  /\  x  e.  B )  /\  (
( M `  x
)  .+  x )  =  .0.  )  ->  (
( M `  x
)  .+  x )  =  .0.  )
8 simpllr 534 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
Grp  /\  M : B
--> B )  /\  x  e.  B )  /\  (
( M `  x
)  .+  x )  =  .0.  )  ->  M : B --> B )
9 simplr 528 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
Grp  /\  M : B
--> B )  /\  x  e.  B )  /\  (
( M `  x
)  .+  x )  =  .0.  )  ->  x  e.  B )
108, 9ffvelcdmd 5647 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  M : B
--> B )  /\  x  e.  B )  /\  (
( M `  x
)  .+  x )  =  .0.  )  ->  ( M `  x )  e.  B )
111, 2, 3grpinveu 12788 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  x  e.  B )  ->  E! e  e.  B  ( e  .+  x
)  =  .0.  )
1211ad4ant13 513 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  M : B
--> B )  /\  x  e.  B )  /\  (
( M `  x
)  .+  x )  =  .0.  )  ->  E! e  e.  B  (
e  .+  x )  =  .0.  )
13 oveq1 5875 . . . . . . . . . . . 12  |-  ( e  =  ( M `  x )  ->  (
e  .+  x )  =  ( ( M `
 x )  .+  x ) )
1413eqeq1d 2186 . . . . . . . . . . 11  |-  ( e  =  ( M `  x )  ->  (
( e  .+  x
)  =  .0.  <->  ( ( M `  x )  .+  x )  =  .0.  ) )
1514riota2 5846 . . . . . . . . . 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 ) ) )
1610, 12, 15syl2anc 411 . . . . . . . . 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 ) ) )
177, 16mpbid 147 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  M : B
--> B )  /\  x  e.  B )  /\  (
( M `  x
)  .+  x )  =  .0.  )  ->  ( iota_ e  e.  B  ( e  .+  x )  =  .0.  )  =  ( M `  x
) )
186, 17eqtrd 2210 . . . . . . 7  |-  ( ( ( ( G  e. 
Grp  /\  M : B
--> B )  /\  x  e.  B )  /\  (
( M `  x
)  .+  x )  =  .0.  )  ->  ( N `  x )  =  ( M `  x ) )
1918ex 115 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  M : B --> B )  /\  x  e.  B
)  ->  ( (
( M `  x
)  .+  x )  =  .0.  ->  ( N `  x )  =  ( M `  x ) ) )
2019ralimdva 2544 . . . . 5  |-  ( ( G  e.  Grp  /\  M : B --> B )  ->  ( A. x  e.  B  ( ( M `  x )  .+  x )  =  .0. 
->  A. x  e.  B  ( N `  x )  =  ( M `  x ) ) )
2120impr 379 . . . 4  |-  ( ( G  e.  Grp  /\  ( M : B --> B  /\  A. x  e.  B  ( ( M `  x
)  .+  x )  =  .0.  ) )  ->  A. x  e.  B  ( N `  x )  =  ( M `  x ) )
221, 4grpinvfng 12794 . . . . 5  |-  ( G  e.  Grp  ->  N  Fn  B )
23 ffn 5360 . . . . . 6  |-  ( M : B --> B  ->  M  Fn  B )
2423ad2antrl 490 . . . . 5  |-  ( ( G  e.  Grp  /\  ( M : B --> B  /\  A. x  e.  B  ( ( M `  x
)  .+  x )  =  .0.  ) )  ->  M  Fn  B )
25 eqfnfv 5608 . . . . 5  |-  ( ( N  Fn  B  /\  M  Fn  B )  ->  ( N  =  M  <->  A. x  e.  B  ( N `  x )  =  ( M `  x ) ) )
2622, 24, 25syl2an2r 595 . . . 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 ) ) )
2721, 26mpbird 167 . . 3  |-  ( ( G  e.  Grp  /\  ( M : B --> B  /\  A. x  e.  B  ( ( M `  x
)  .+  x )  =  .0.  ) )  ->  N  =  M )
2827ex 115 . 2  |-  ( G  e.  Grp  ->  (
( M : B --> B  /\  A. x  e.  B  ( ( M `
 x )  .+  x )  =  .0.  )  ->  N  =  M ) )
291, 4grpinvf 12797 . . . 4  |-  ( G  e.  Grp  ->  N : B --> B )
301, 2, 3, 4grplinv 12799 . . . . 5  |-  ( ( G  e.  Grp  /\  x  e.  B )  ->  ( ( N `  x )  .+  x
)  =  .0.  )
3130ralrimiva 2550 . . . 4  |-  ( G  e.  Grp  ->  A. x  e.  B  ( ( N `  x )  .+  x )  =  .0.  )
3229, 31jca 306 . . 3  |-  ( G  e.  Grp  ->  ( N : B --> B  /\  A. x  e.  B  ( ( N `  x
)  .+  x )  =  .0.  ) )
33 feq1 5343 . . . 4  |-  ( N  =  M  ->  ( N : B --> B  <->  M : B
--> B ) )
34 fveq1 5509 . . . . . . 7  |-  ( N  =  M  ->  ( N `  x )  =  ( M `  x ) )
3534oveq1d 5883 . . . . . 6  |-  ( N  =  M  ->  (
( N `  x
)  .+  x )  =  ( ( M `
 x )  .+  x ) )
3635eqeq1d 2186 . . . . 5  |-  ( N  =  M  ->  (
( ( N `  x )  .+  x
)  =  .0.  <->  ( ( M `  x )  .+  x )  =  .0.  ) )
3736ralbidv 2477 . . . 4  |-  ( N  =  M  ->  ( A. x  e.  B  ( ( N `  x )  .+  x
)  =  .0.  <->  A. x  e.  B  ( ( M `  x )  .+  x )  =  .0.  ) )
3833, 37anbi12d 473 . . 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.  ) ) )
3932, 38syl5ibcom 155 . 2  |-  ( G  e.  Grp  ->  ( N  =  M  ->  ( M : B --> B  /\  A. x  e.  B  ( ( M `  x
)  .+  x )  =  .0.  ) ) )
4028, 39impbid 129 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    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   A.wral 2455   E!wreu 2457    Fn wfn 5206   -->wf 5207   ` cfv 5211   iota_crio 5823  (class class class)co 5868   Basecbs 12432   +g cplusg 12505   0gc0g 12640   Grpcgrp 12754   invgcminusg 12755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-cnex 7880  ax-resscn 7881  ax-1re 7883  ax-addrcl 7886
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-riota 5824  df-ov 5871  df-inn 8896  df-2 8954  df-ndx 12435  df-slot 12436  df-base 12438  df-plusg 12518  df-0g 12642  df-mgm 12654  df-sgrp 12687  df-mnd 12697  df-grp 12757  df-minusg 12758
This theorem is referenced by: (None)
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