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Theorem grpinvid2 12812
Description: The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 24-Aug-2011.)
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
grpinvid2  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( N `  X )  =  Y  <-> 
( Y  .+  X
)  =  .0.  )
)

Proof of Theorem grpinvid2
StepHypRef Expression
1 oveq1 5876 . . . 4  |-  ( ( N `  X )  =  Y  ->  (
( N `  X
)  .+  X )  =  ( Y  .+  X ) )
21adantl 277 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( N `  X
)  =  Y )  ->  ( ( N `
 X )  .+  X )  =  ( Y  .+  X ) )
3 grpinv.b . . . . . 6  |-  B  =  ( Base `  G
)
4 grpinv.p . . . . . 6  |-  .+  =  ( +g  `  G )
5 grpinv.u . . . . . 6  |-  .0.  =  ( 0g `  G )
6 grpinv.n . . . . . 6  |-  N  =  ( invg `  G )
73, 4, 5, 6grplinv 12809 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( N `  X )  .+  X
)  =  .0.  )
873adant3 1017 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( N `  X )  .+  X
)  =  .0.  )
98adantr 276 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( N `  X
)  =  Y )  ->  ( ( N `
 X )  .+  X )  =  .0.  )
102, 9eqtr3d 2212 . 2  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( N `  X
)  =  Y )  ->  ( Y  .+  X )  =  .0.  )
113, 6grpinvcl 12808 . . . . . . 7  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  X
)  e.  B )
123, 4, 5grplid 12793 . . . . . . 7  |-  ( ( G  e.  Grp  /\  ( N `  X )  e.  B )  -> 
(  .0.  .+  ( N `  X )
)  =  ( N `
 X ) )
1311, 12syldan 282 . . . . . 6  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  (  .0.  .+  ( N `  X )
)  =  ( N `
 X ) )
14133adant3 1017 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  (  .0.  .+  ( N `  X )
)  =  ( N `
 X ) )
1514eqcomd 2183 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  X
)  =  (  .0.  .+  ( N `  X
) ) )
1615adantr 276 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( Y  .+  X
)  =  .0.  )  ->  ( N `  X
)  =  (  .0.  .+  ( N `  X
) ) )
17 oveq1 5876 . . . 4  |-  ( ( Y  .+  X )  =  .0.  ->  (
( Y  .+  X
)  .+  ( N `  X ) )  =  (  .0.  .+  ( N `  X )
) )
1817adantl 277 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( Y  .+  X
)  =  .0.  )  ->  ( ( Y  .+  X )  .+  ( N `  X )
)  =  (  .0.  .+  ( N `  X
) ) )
19 simprr 531 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  Y  e.  B )
20 simprl 529 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  X  e.  B )
2111adantrr 479 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  ( N `  X )  e.  B )
2219, 20, 213jca 1177 . . . . . . 7  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  ( Y  e.  B  /\  X  e.  B  /\  ( N `  X )  e.  B ) )
233, 4grpass 12773 . . . . . . 7  |-  ( ( G  e.  Grp  /\  ( Y  e.  B  /\  X  e.  B  /\  ( N `  X
)  e.  B ) )  ->  ( ( Y  .+  X )  .+  ( N `  X ) )  =  ( Y 
.+  ( X  .+  ( N `  X ) ) ) )
2422, 23syldan 282 . . . . . 6  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  (
( Y  .+  X
)  .+  ( N `  X ) )  =  ( Y  .+  ( X  .+  ( N `  X ) ) ) )
25243impb 1199 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( Y  .+  X )  .+  ( N `  X )
)  =  ( Y 
.+  ( X  .+  ( N `  X ) ) ) )
263, 4, 5, 6grprinv 12810 . . . . . . 7  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X  .+  ( N `  X )
)  =  .0.  )
2726oveq2d 5885 . . . . . 6  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( Y  .+  ( X  .+  ( N `  X ) ) )  =  ( Y  .+  .0.  ) )
28273adant3 1017 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  .+  ( X  .+  ( N `  X ) ) )  =  ( Y  .+  .0.  ) )
293, 4, 5grprid 12794 . . . . . 6  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( Y  .+  .0.  )  =  Y )
30293adant2 1016 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  .+  .0.  )  =  Y )
3125, 28, 303eqtrd 2214 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( Y  .+  X )  .+  ( N `  X )
)  =  Y )
3231adantr 276 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( Y  .+  X
)  =  .0.  )  ->  ( ( Y  .+  X )  .+  ( N `  X )
)  =  Y )
3316, 18, 323eqtr2d 2216 . 2  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( Y  .+  X
)  =  .0.  )  ->  ( N `  X
)  =  Y )
3410, 33impbida 596 1  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( N `  X )  =  Y  <-> 
( Y  .+  X
)  =  .0.  )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   ` cfv 5212  (class class class)co 5869   Basecbs 12442   +g cplusg 12515   0gc0g 12650   Grpcgrp 12764   invgcminusg 12765
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 4206  ax-un 4430  ax-cnex 7890  ax-resscn 7891  ax-1re 7893  ax-addrcl 7896
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 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-inn 8906  df-2 8964  df-ndx 12445  df-slot 12446  df-base 12448  df-plusg 12528  df-0g 12652  df-mgm 12664  df-sgrp 12697  df-mnd 12707  df-grp 12767  df-minusg 12768
This theorem is referenced by:  grpinvcnv  12824  grpsubeq0  12842  rngnegr  13052
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