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Theorem grpinv11 16234
Description: The group inverse is one-to-one. (Contributed by NM, 22-Mar-2015.)
Hypotheses
Ref Expression
grpinvinv.b  |-  B  =  ( Base `  G
)
grpinvinv.n  |-  N  =  ( invg `  G )
grpinv11.g  |-  ( ph  ->  G  e.  Grp )
grpinv11.x  |-  ( ph  ->  X  e.  B )
grpinv11.y  |-  ( ph  ->  Y  e.  B )
Assertion
Ref Expression
grpinv11  |-  ( ph  ->  ( ( N `  X )  =  ( N `  Y )  <-> 
X  =  Y ) )

Proof of Theorem grpinv11
StepHypRef Expression
1 fveq2 5872 . . . . 5  |-  ( ( N `  X )  =  ( N `  Y )  ->  ( N `  ( N `  X ) )  =  ( N `  ( N `  Y )
) )
21adantl 466 . . . 4  |-  ( (
ph  /\  ( N `  X )  =  ( N `  Y ) )  ->  ( N `  ( N `  X
) )  =  ( N `  ( N `
 Y ) ) )
3 grpinv11.g . . . . . 6  |-  ( ph  ->  G  e.  Grp )
4 grpinv11.x . . . . . 6  |-  ( ph  ->  X  e.  B )
5 grpinvinv.b . . . . . . 7  |-  B  =  ( Base `  G
)
6 grpinvinv.n . . . . . . 7  |-  N  =  ( invg `  G )
75, 6grpinvinv 16232 . . . . . 6  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  ( N `  X )
)  =  X )
83, 4, 7syl2anc 661 . . . . 5  |-  ( ph  ->  ( N `  ( N `  X )
)  =  X )
98adantr 465 . . . 4  |-  ( (
ph  /\  ( N `  X )  =  ( N `  Y ) )  ->  ( N `  ( N `  X
) )  =  X )
10 grpinv11.y . . . . . 6  |-  ( ph  ->  Y  e.  B )
115, 6grpinvinv 16232 . . . . . 6  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( N `  ( N `  Y )
)  =  Y )
123, 10, 11syl2anc 661 . . . . 5  |-  ( ph  ->  ( N `  ( N `  Y )
)  =  Y )
1312adantr 465 . . . 4  |-  ( (
ph  /\  ( N `  X )  =  ( N `  Y ) )  ->  ( N `  ( N `  Y
) )  =  Y )
142, 9, 133eqtr3d 2506 . . 3  |-  ( (
ph  /\  ( N `  X )  =  ( N `  Y ) )  ->  X  =  Y )
1514ex 434 . 2  |-  ( ph  ->  ( ( N `  X )  =  ( N `  Y )  ->  X  =  Y ) )
16 fveq2 5872 . 2  |-  ( X  =  Y  ->  ( N `  X )  =  ( N `  Y ) )
1715, 16impbid1 203 1  |-  ( ph  ->  ( ( N `  X )  =  ( N `  Y )  <-> 
X  =  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   ` cfv 5594   Basecbs 14644   Grpcgrp 16180   invgcminusg 16181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-0g 14859  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-grp 16184  df-minusg 16185
This theorem is referenced by:  gexdvds  16731  dchrisum0re  23824  mapdpglem30  37572
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