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Theorem grpinv11 16735
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 5870 . . . . 5  |-  ( ( N `  X )  =  ( N `  Y )  ->  ( N `  ( N `  X ) )  =  ( N `  ( N `  Y )
) )
21adantl 468 . . . 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 16733 . . . . . 6  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  ( N `  X )
)  =  X )
83, 4, 7syl2anc 667 . . . . 5  |-  ( ph  ->  ( N `  ( N `  X )
)  =  X )
98adantr 467 . . . 4  |-  ( (
ph  /\  ( N `  X )  =  ( N `  Y ) )  ->  ( N `  ( N `  X
) )  =  X )
10 grpinv11.y . . . . . 6  |-  ( ph  ->  Y  e.  B )
115, 6grpinvinv 16733 . . . . . 6  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( N `  ( N `  Y )
)  =  Y )
123, 10, 11syl2anc 667 . . . . 5  |-  ( ph  ->  ( N `  ( N `  Y )
)  =  Y )
1312adantr 467 . . . 4  |-  ( (
ph  /\  ( N `  X )  =  ( N `  Y ) )  ->  ( N `  ( N `  Y
) )  =  Y )
142, 9, 133eqtr3d 2495 . . 3  |-  ( (
ph  /\  ( N `  X )  =  ( N `  Y ) )  ->  X  =  Y )
1514ex 436 . 2  |-  ( ph  ->  ( ( N `  X )  =  ( N `  Y )  ->  X  =  Y ) )
16 fveq2 5870 . 2  |-  ( X  =  Y  ->  ( N `  X )  =  ( N `  Y ) )
1715, 16impbid1 207 1  |-  ( ph  ->  ( ( N `  X )  =  ( N `  Y )  <-> 
X  =  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1446    e. wcel 1889   ` cfv 5585   Basecbs 15133   Grpcgrp 16681   invgcminusg 16682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-0g 15352  df-mgm 16500  df-sgrp 16539  df-mnd 16549  df-grp 16685  df-minusg 16686
This theorem is referenced by:  gexdvds  17247  dchrisum0re  24363  mapdpglem30  35282
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