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Theorem grporcan 20888
Description: Right cancellation law for groups. (Contributed by NM, 26-Oct-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
grprcan.1  |-  X  =  ran  G
Assertion
Ref Expression
grporcan  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G C )  =  ( B G C )  <->  A  =  B
) )

Proof of Theorem grporcan
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 grprcan.1 . . . . . . . 8  |-  X  =  ran  G
2 eqid 2283 . . . . . . . 8  |-  (GId `  G )  =  (GId
`  G )
31, 2grpoidinv2 20885 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  C  e.  X )  ->  (
( ( (GId `  G ) G C )  =  C  /\  ( C G (GId `  G ) )  =  C )  /\  E. y  e.  X  (
( y G C )  =  (GId `  G )  /\  ( C G y )  =  (GId `  G )
) ) )
4 simpr 447 . . . . . . . . 9  |-  ( ( ( y G C )  =  (GId `  G )  /\  ( C G y )  =  (GId `  G )
)  ->  ( C G y )  =  (GId `  G )
)
54reximi 2650 . . . . . . . 8  |-  ( E. y  e.  X  ( ( y G C )  =  (GId `  G )  /\  ( C G y )  =  (GId `  G )
)  ->  E. y  e.  X  ( C G y )  =  (GId `  G )
)
65adantl 452 . . . . . . 7  |-  ( ( ( ( (GId `  G ) G C )  =  C  /\  ( C G (GId `  G ) )  =  C )  /\  E. y  e.  X  (
( y G C )  =  (GId `  G )  /\  ( C G y )  =  (GId `  G )
) )  ->  E. y  e.  X  ( C G y )  =  (GId `  G )
)
73, 6syl 15 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  C  e.  X )  ->  E. y  e.  X  ( C G y )  =  (GId `  G )
)
87ad2ant2rl 729 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  ( B  e.  X  /\  C  e.  X ) )  ->  E. y  e.  X  ( C G y )  =  (GId `  G
) )
9 oveq1 5865 . . . . . . . . . . . 12  |-  ( ( A G C )  =  ( B G C )  ->  (
( A G C ) G y )  =  ( ( B G C ) G y ) )
109ad2antll 709 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  ( B  e.  X  /\  C  e.  X ) )  /\  ( y  e.  X  /\  ( A G C )  =  ( B G C ) ) )  ->  ( ( A G C ) G y )  =  ( ( B G C ) G y ) )
111grpoass 20870 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  C  e.  X  /\  y  e.  X )
)  ->  ( ( A G C ) G y )  =  ( A G ( C G y ) ) )
12113exp2 1169 . . . . . . . . . . . . . 14  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  ( C  e.  X  ->  (
y  e.  X  -> 
( ( A G C ) G y )  =  ( A G ( C G y ) ) ) ) ) )
1312imp41 576 . . . . . . . . . . . . 13  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  C  e.  X )  /\  y  e.  X )  ->  (
( A G C ) G y )  =  ( A G ( C G y ) ) )
1413adantlrl 700 . . . . . . . . . . . 12  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  ( B  e.  X  /\  C  e.  X ) )  /\  y  e.  X )  ->  ( ( A G C ) G y )  =  ( A G ( C G y ) ) )
1514adantrr 697 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  ( B  e.  X  /\  C  e.  X ) )  /\  ( y  e.  X  /\  ( A G C )  =  ( B G C ) ) )  ->  ( ( A G C ) G y )  =  ( A G ( C G y ) ) )
161grpoass 20870 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X  /\  y  e.  X )
)  ->  ( ( B G C ) G y )  =  ( B G ( C G y ) ) )
17163exp2 1169 . . . . . . . . . . . . . 14  |-  ( G  e.  GrpOp  ->  ( B  e.  X  ->  ( C  e.  X  ->  (
y  e.  X  -> 
( ( B G C ) G y )  =  ( B G ( C G y ) ) ) ) ) )
1817imp42 577 . . . . . . . . . . . . 13  |-  ( ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X
) )  /\  y  e.  X )  ->  (
( B G C ) G y )  =  ( B G ( C G y ) ) )
1918adantllr 699 . . . . . . . . . . . 12  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  ( B  e.  X  /\  C  e.  X ) )  /\  y  e.  X )  ->  ( ( B G C ) G y )  =  ( B G ( C G y ) ) )
2019adantrr 697 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  ( B  e.  X  /\  C  e.  X ) )  /\  ( y  e.  X  /\  ( A G C )  =  ( B G C ) ) )  ->  ( ( B G C ) G y )  =  ( B G ( C G y ) ) )
2110, 15, 203eqtr3d 2323 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  ( B  e.  X  /\  C  e.  X ) )  /\  ( y  e.  X  /\  ( A G C )  =  ( B G C ) ) )  ->  ( A G ( C G y ) )  =  ( B G ( C G y ) ) )
2221adantrrl 704 . . . . . . . . 9  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  ( B  e.  X  /\  C  e.  X ) )  /\  ( y  e.  X  /\  ( ( C G y )  =  (GId
`  G )  /\  ( A G C )  =  ( B G C ) ) ) )  ->  ( A G ( C G y ) )  =  ( B G ( C G y ) ) )
23 oveq2 5866 . . . . . . . . . . 11  |-  ( ( C G y )  =  (GId `  G
)  ->  ( A G ( C G y ) )  =  ( A G (GId
`  G ) ) )
2423ad2antrl 708 . . . . . . . . . 10  |-  ( ( y  e.  X  /\  ( ( C G y )  =  (GId
`  G )  /\  ( A G C )  =  ( B G C ) ) )  ->  ( A G ( C G y ) )  =  ( A G (GId `  G ) ) )
2524adantl 452 . . . . . . . . 9  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  ( B  e.  X  /\  C  e.  X ) )  /\  ( y  e.  X  /\  ( ( C G y )  =  (GId
`  G )  /\  ( A G C )  =  ( B G C ) ) ) )  ->  ( A G ( C G y ) )  =  ( A G (GId
`  G ) ) )
26 oveq2 5866 . . . . . . . . . . 11  |-  ( ( C G y )  =  (GId `  G
)  ->  ( B G ( C G y ) )  =  ( B G (GId
`  G ) ) )
2726ad2antrl 708 . . . . . . . . . 10  |-  ( ( y  e.  X  /\  ( ( C G y )  =  (GId
`  G )  /\  ( A G C )  =  ( B G C ) ) )  ->  ( B G ( C G y ) )  =  ( B G (GId `  G ) ) )
2827adantl 452 . . . . . . . . 9  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  ( B  e.  X  /\  C  e.  X ) )  /\  ( y  e.  X  /\  ( ( C G y )  =  (GId
`  G )  /\  ( A G C )  =  ( B G C ) ) ) )  ->  ( B G ( C G y ) )  =  ( B G (GId
`  G ) ) )
2922, 25, 283eqtr3d 2323 . . . . . . . 8  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  ( B  e.  X  /\  C  e.  X ) )  /\  ( y  e.  X  /\  ( ( C G y )  =  (GId
`  G )  /\  ( A G C )  =  ( B G C ) ) ) )  ->  ( A G (GId `  G )
)  =  ( B G (GId `  G
) ) )
301, 2grporid 20887 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G (GId `  G
) )  =  A )
3130ad2antrr 706 . . . . . . . 8  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  ( B  e.  X  /\  C  e.  X ) )  /\  ( y  e.  X  /\  ( ( C G y )  =  (GId
`  G )  /\  ( A G C )  =  ( B G C ) ) ) )  ->  ( A G (GId `  G )
)  =  A )
321, 2grporid 20887 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  ( B G (GId `  G
) )  =  B )
3332ad2ant2r 727 . . . . . . . . 9  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  ( B  e.  X  /\  C  e.  X ) )  -> 
( B G (GId
`  G ) )  =  B )
3433adantr 451 . . . . . . . 8  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  ( B  e.  X  /\  C  e.  X ) )  /\  ( y  e.  X  /\  ( ( C G y )  =  (GId
`  G )  /\  ( A G C )  =  ( B G C ) ) ) )  ->  ( B G (GId `  G )
)  =  B )
3529, 31, 343eqtr3d 2323 . . . . . . 7  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  ( B  e.  X  /\  C  e.  X ) )  /\  ( y  e.  X  /\  ( ( C G y )  =  (GId
`  G )  /\  ( A G C )  =  ( B G C ) ) ) )  ->  A  =  B )
3635exp45 597 . . . . . 6  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  ( B  e.  X  /\  C  e.  X ) )  -> 
( y  e.  X  ->  ( ( C G y )  =  (GId
`  G )  -> 
( ( A G C )  =  ( B G C )  ->  A  =  B ) ) ) )
3736rexlimdv 2666 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  ( B  e.  X  /\  C  e.  X ) )  -> 
( E. y  e.  X  ( C G y )  =  (GId
`  G )  -> 
( ( A G C )  =  ( B G C )  ->  A  =  B ) ) )
388, 37mpd 14 . . . 4  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  ( B  e.  X  /\  C  e.  X ) )  -> 
( ( A G C )  =  ( B G C )  ->  A  =  B ) )
39 oveq1 5865 . . . 4  |-  ( A  =  B  ->  ( A G C )  =  ( B G C ) )
4038, 39impbid1 194 . . 3  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  ( B  e.  X  /\  C  e.  X ) )  -> 
( ( A G C )  =  ( B G C )  <-> 
A  =  B ) )
4140exp43 595 . 2  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  ( B  e.  X  ->  ( C  e.  X  ->  ( ( A G C )  =  ( B G C )  <->  A  =  B ) ) ) ) )
42413imp2 1166 1  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G C )  =  ( B G C )  <->  A  =  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544   ran crn 4690   ` cfv 5255  (class class class)co 5858   GrpOpcgr 20853  GIdcgi 20854
This theorem is referenced by:  grpoinveu  20889  grpoid  20890  grpodiveq  20923  rngorcan  21063  rngorz  21069  vcrcan  21120  nvrcan  21181  grpodrcan  25375  trooo  25394  rltrooo  25415  vecrcan  25475  ghomdiv  26574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-riota 6304  df-grpo 20858  df-gid 20859
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