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Theorem grporcan 23720
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 2443 . . . . . . . 8  |-  (GId `  G )  =  (GId
`  G )
31, 2grpoidinv2 23717 . . . . . . 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 461 . . . . . . . . 9  |-  ( ( ( y G C )  =  (GId `  G )  /\  ( C G y )  =  (GId `  G )
)  ->  ( C G y )  =  (GId `  G )
)
54reximi 2835 . . . . . . . 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 466 . . . . . . 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 16 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  C  e.  X )  ->  E. y  e.  X  ( C G y )  =  (GId `  G )
)
87ad2ant2rl 748 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  ( B  e.  X  /\  C  e.  X ) )  ->  E. y  e.  X  ( C G y )  =  (GId `  G
) )
9 oveq1 6110 . . . . . . . . . . . 12  |-  ( ( A G C )  =  ( B G C )  ->  (
( A G C ) G y )  =  ( ( B G C ) G y ) )
109ad2antll 728 . . . . . . . . . . 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 23702 . . . . . . . . . . . . . 14  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  C  e.  X  /\  y  e.  X )
)  ->  ( ( A G C ) G y )  =  ( A G ( C G y ) ) )
12113anassrs 1209 . . . . . . . . . . . . 13  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  C  e.  X )  /\  y  e.  X )  ->  (
( A G C ) G y )  =  ( A G ( C G y ) ) )
1312adantlrl 719 . . . . . . . . . . . 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 ) ) )
1413adantrr 716 . . . . . . . . . . 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 ) ) )
151grpoass 23702 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X  /\  y  e.  X )
)  ->  ( ( B G C ) G y )  =  ( B G ( C G y ) ) )
16153exp2 1205 . . . . . . . . . . . . . 14  |-  ( G  e.  GrpOp  ->  ( B  e.  X  ->  ( C  e.  X  ->  (
y  e.  X  -> 
( ( B G C ) G y )  =  ( B G ( C G y ) ) ) ) ) )
1716imp42 594 . . . . . . . . . . . . 13  |-  ( ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X
) )  /\  y  e.  X )  ->  (
( B G C ) G y )  =  ( B G ( C G y ) ) )
1817adantllr 718 . . . . . . . . . . . 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 ) ) )
1918adantrr 716 . . . . . . . . . . 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 ) ) )
2010, 14, 193eqtr3d 2483 . . . . . . . . . 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 ) ) )
2120adantrrl 723 . . . . . . . . 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 ) ) )
22 oveq2 6111 . . . . . . . . . . 11  |-  ( ( C G y )  =  (GId `  G
)  ->  ( A G ( C G y ) )  =  ( A G (GId
`  G ) ) )
2322ad2antrl 727 . . . . . . . . . 10  |-  ( ( y  e.  X  /\  ( ( C G y )  =  (GId
`  G )  /\  ( A G C )  =  ( B G C ) ) )  ->  ( A G ( C G y ) )  =  ( A G (GId `  G ) ) )
2423adantl 466 . . . . . . . . 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 ) ) )
25 oveq2 6111 . . . . . . . . . . 11  |-  ( ( C G y )  =  (GId `  G
)  ->  ( B G ( C G y ) )  =  ( B G (GId
`  G ) ) )
2625ad2antrl 727 . . . . . . . . . 10  |-  ( ( y  e.  X  /\  ( ( C G y )  =  (GId
`  G )  /\  ( A G C )  =  ( B G C ) ) )  ->  ( B G ( C G y ) )  =  ( B G (GId `  G ) ) )
2726adantl 466 . . . . . . . . 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 ) ) )
2821, 24, 273eqtr3d 2483 . . . . . . . 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
) ) )
291, 2grporid 23719 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G (GId `  G
) )  =  A )
3029ad2antrr 725 . . . . . . . 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 )
311, 2grporid 23719 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  ( B G (GId `  G
) )  =  B )
3231ad2ant2r 746 . . . . . . . . 9  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  ( B  e.  X  /\  C  e.  X ) )  -> 
( B G (GId
`  G ) )  =  B )
3332adantr 465 . . . . . . . 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 )
3428, 30, 333eqtr3d 2483 . . . . . . 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 )
3534exp45 614 . . . . . 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 ) ) ) )
3635rexlimdv 2852 . . . . 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 ) ) )
378, 36mpd 15 . . . 4  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  ( B  e.  X  /\  C  e.  X ) )  -> 
( ( A G C )  =  ( B G C )  ->  A  =  B ) )
38 oveq1 6110 . . . 4  |-  ( A  =  B  ->  ( A G C )  =  ( B G C ) )
3937, 38impbid1 203 . . 3  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  ( B  e.  X  /\  C  e.  X ) )  -> 
( ( A G C )  =  ( B G C )  <-> 
A  =  B ) )
4039exp43 612 . 2  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  ( B  e.  X  ->  ( C  e.  X  ->  ( ( A G C )  =  ( B G C )  <->  A  =  B ) ) ) ) )
41403imp2 1202 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   E.wrex 2728   ran crn 4853   ` cfv 5430  (class class class)co 6103   GrpOpcgr 23685  GIdcgi 23686
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-sep 4425  ax-nul 4433  ax-pr 4543  ax-un 6384
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-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-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-fo 5436  df-fv 5438  df-riota 6064  df-ov 6106  df-grpo 23690  df-gid 23691
This theorem is referenced by:  grpoinveu  23721  grpoid  23722  grpodiveq  23755  rngorcan  23895  rngorz  23901  vcrcan  23954  nvrcan  24015  ghomdiv  28761
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