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Theorem grpinvid1 12918
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
grpinvid1  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( N `  X )  =  Y  <-> 
( X  .+  Y
)  =  .0.  )
)

Proof of Theorem grpinvid1
StepHypRef Expression
1 oveq2 5882 . . . 4  |-  ( ( N `  X )  =  Y  ->  ( X  .+  ( N `  X ) )  =  ( X  .+  Y
) )
21adantl 277 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( N `  X
)  =  Y )  ->  ( X  .+  ( N `  X ) )  =  ( X 
.+  Y ) )
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, 6grprinv 12917 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X  .+  ( N `  X )
)  =  .0.  )
873adant3 1017 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  ( N `  X )
)  =  .0.  )
98adantr 276 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( N `  X
)  =  Y )  ->  ( X  .+  ( N `  X ) )  =  .0.  )
102, 9eqtr3d 2212 . 2  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( N `  X
)  =  Y )  ->  ( X  .+  Y )  =  .0.  )
11 oveq2 5882 . . . 4  |-  ( ( X  .+  Y )  =  .0.  ->  (
( N `  X
)  .+  ( X  .+  Y ) )  =  ( ( N `  X )  .+  .0.  ) )
1211adantl 277 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( X  .+  Y
)  =  .0.  )  ->  ( ( N `  X )  .+  ( X  .+  Y ) )  =  ( ( N `
 X )  .+  .0.  ) )
133, 4, 5, 6grplinv 12916 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( N `  X )  .+  X
)  =  .0.  )
1413oveq1d 5889 . . . . . . 7  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( ( N `
 X )  .+  X )  .+  Y
)  =  (  .0.  .+  Y ) )
15143adant3 1017 . . . . . 6  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( N `
 X )  .+  X )  .+  Y
)  =  (  .0.  .+  Y ) )
163, 6grpinvcl 12915 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  X
)  e.  B )
1716adantrr 479 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  ( N `  X )  e.  B )
18 simprl 529 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  X  e.  B )
19 simprr 531 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  Y  e.  B )
2017, 18, 193jca 1177 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  (
( N `  X
)  e.  B  /\  X  e.  B  /\  Y  e.  B )
)
213, 4grpass 12880 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  ( ( N `  X )  e.  B  /\  X  e.  B  /\  Y  e.  B
) )  ->  (
( ( N `  X )  .+  X
)  .+  Y )  =  ( ( N `
 X )  .+  ( X  .+  Y ) ) )
2220, 21syldan 282 . . . . . . 7  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B
) )  ->  (
( ( N `  X )  .+  X
)  .+  Y )  =  ( ( N `
 X )  .+  ( X  .+  Y ) ) )
23223impb 1199 . . . . . 6  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( N `
 X )  .+  X )  .+  Y
)  =  ( ( N `  X ) 
.+  ( X  .+  Y ) ) )
2415, 23eqtr3d 2212 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  (  .0.  .+  Y
)  =  ( ( N `  X ) 
.+  ( X  .+  Y ) ) )
253, 4, 5grplid 12900 . . . . . 6  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  (  .0.  .+  Y
)  =  Y )
26253adant2 1016 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  (  .0.  .+  Y
)  =  Y )
2724, 26eqtr3d 2212 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( N `  X )  .+  ( X  .+  Y ) )  =  Y )
2827adantr 276 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( X  .+  Y
)  =  .0.  )  ->  ( ( N `  X )  .+  ( X  .+  Y ) )  =  Y )
293, 4, 5grprid 12901 . . . . . 6  |-  ( ( G  e.  Grp  /\  ( N `  X )  e.  B )  -> 
( ( N `  X )  .+  .0.  )  =  ( N `  X ) )
3016, 29syldan 282 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( N `  X )  .+  .0.  )  =  ( N `  X ) )
31303adant3 1017 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( N `  X )  .+  .0.  )  =  ( N `  X ) )
3231adantr 276 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( X  .+  Y
)  =  .0.  )  ->  ( ( N `  X )  .+  .0.  )  =  ( N `  X ) )
3312, 28, 323eqtr3rd 2219 . 2  |-  ( ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  /\  ( X  .+  Y
)  =  .0.  )  ->  ( N `  X
)  =  Y )
3410, 33impbida 596 1  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( N `  X )  =  Y  <-> 
( X  .+  Y
)  =  .0.  )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   ` cfv 5216  (class class class)co 5874   Basecbs 12456   +g cplusg 12530   0gc0g 12699   Grpcgrp 12871   invgcminusg 12872
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 4118  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-cnex 7901  ax-resscn 7902  ax-1re 7904  ax-addrcl 7907
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 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-inn 8918  df-2 8976  df-ndx 12459  df-slot 12460  df-base 12462  df-plusg 12543  df-0g 12701  df-mgm 12769  df-sgrp 12802  df-mnd 12812  df-grp 12874  df-minusg 12875
This theorem is referenced by:  grpinvid  12924  grpinvcnv  12932  grpinvadd  12942  subginv  13034  ringnegl  13221  cnfldneg  13398  zringinvg  13425
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