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Theorem grporcan 25949
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 2451 . . . . . . . 8  |-  (GId `  G )  =  (GId
`  G )
31, 2grpoidinv2 25946 . . . . . . 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 463 . . . . . . . . 9  |-  ( ( ( y G C )  =  (GId `  G )  /\  ( C G y )  =  (GId `  G )
)  ->  ( C G y )  =  (GId `  G )
)
54reximi 2855 . . . . . . . 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 468 . . . . . . 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 17 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  C  e.  X )  ->  E. y  e.  X  ( C G y )  =  (GId `  G )
)
87ad2ant2rl 755 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  ( B  e.  X  /\  C  e.  X ) )  ->  E. y  e.  X  ( C G y )  =  (GId `  G
) )
9 oveq1 6297 . . . . . . . . . . . 12  |-  ( ( A G C )  =  ( B G C )  ->  (
( A G C ) G y )  =  ( ( B G C ) G y ) )
109ad2antll 735 . . . . . . . . . . 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 25931 . . . . . . . . . . . . . 14  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  C  e.  X  /\  y  e.  X )
)  ->  ( ( A G C ) G y )  =  ( A G ( C G y ) ) )
12113anassrs 1232 . . . . . . . . . . . . 13  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  C  e.  X )  /\  y  e.  X )  ->  (
( A G C ) G y )  =  ( A G ( C G y ) ) )
1312adantlrl 726 . . . . . . . . . . . 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 723 . . . . . . . . . . 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 25931 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X  /\  y  e.  X )
)  ->  ( ( B G C ) G y )  =  ( B G ( C G y ) ) )
16153exp2 1227 . . . . . . . . . . . . . 14  |-  ( G  e.  GrpOp  ->  ( B  e.  X  ->  ( C  e.  X  ->  (
y  e.  X  -> 
( ( B G C ) G y )  =  ( B G ( C G y ) ) ) ) ) )
1716imp42 599 . . . . . . . . . . . . 13  |-  ( ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X
) )  /\  y  e.  X )  ->  (
( B G C ) G y )  =  ( B G ( C G y ) ) )
1817adantllr 725 . . . . . . . . . . . 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 723 . . . . . . . . . . 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 2493 . . . . . . . . . 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 730 . . . . . . . . 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 6298 . . . . . . . . . . 11  |-  ( ( C G y )  =  (GId `  G
)  ->  ( A G ( C G y ) )  =  ( A G (GId
`  G ) ) )
2322ad2antrl 734 . . . . . . . . . 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 468 . . . . . . . . 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 6298 . . . . . . . . . . 11  |-  ( ( C G y )  =  (GId `  G
)  ->  ( B G ( C G y ) )  =  ( B G (GId
`  G ) ) )
2625ad2antrl 734 . . . . . . . . . 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 468 . . . . . . . . 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 2493 . . . . . . . 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 25948 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G (GId `  G
) )  =  A )
3029ad2antrr 732 . . . . . . . 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 25948 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  ( B G (GId `  G
) )  =  B )
3231ad2ant2r 753 . . . . . . . . 9  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  ( B  e.  X  /\  C  e.  X ) )  -> 
( B G (GId
`  G ) )  =  B )
3332adantr 467 . . . . . . . 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 2493 . . . . . . 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 619 . . . . . 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 2877 . . . . 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 6297 . . . 4  |-  ( A  =  B  ->  ( A G C )  =  ( B G C ) )
3937, 38impbid1 207 . . 3  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  ( B  e.  X  /\  C  e.  X ) )  -> 
( ( A G C )  =  ( B G C )  <-> 
A  =  B ) )
4039exp43 617 . 2  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  ( B  e.  X  ->  ( C  e.  X  ->  ( ( A G C )  =  ( B G C )  <->  A  =  B ) ) ) ) )
41403imp2 1224 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 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   E.wrex 2738   ran crn 4835   ` cfv 5582  (class class class)co 6290   GrpOpcgr 25914  GIdcgi 25915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-fo 5588  df-fv 5590  df-riota 6252  df-ov 6293  df-grpo 25919  df-gid 25920
This theorem is referenced by:  grpoinveu  25950  grpoid  25951  grpodiveq  25984  rngorcan  26124  rngorz  26130  vcrcan  26183  nvrcan  26244  ghomdiv  32182
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