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Theorem grpinv11 15607
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 5703 . . . . 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 15605 . . . . . 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 15605 . . . . . 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 2483 . . 3  |-  ( (
ph  /\  ( N `  X )  =  ( N `  Y ) )  ->  X  =  Y )
1514ex 434 . 2  |-  ( ph  ->  ( ( N `  X )  =  ( N `  Y )  ->  X  =  Y ) )
16 fveq2 5703 . 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 1369    e. wcel 1756   ` cfv 5430   Basecbs 14186   Grpcgrp 15422   invgcminusg 15423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-0g 14392  df-mnd 15427  df-grp 15557  df-minusg 15558
This theorem is referenced by:  gexdvds  16095  dchrisum0re  22774  mapdpglem30  35359
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