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Theorem grporcan 23627
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 2441 . . . . . . . 8  |-  (GId `  G )  =  (GId
`  G )
31, 2grpoidinv2 23624 . . . . . . 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 458 . . . . . . . . 9  |-  ( ( ( y G C )  =  (GId `  G )  /\  ( C G y )  =  (GId `  G )
)  ->  ( C G y )  =  (GId `  G )
)
54reximi 2821 . . . . . . . 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 463 . . . . . . 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 743 . . . . 5  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  ( B  e.  X  /\  C  e.  X ) )  ->  E. y  e.  X  ( C G y )  =  (GId `  G
) )
9 oveq1 6097 . . . . . . . . . . . 12  |-  ( ( A G C )  =  ( B G C )  ->  (
( A G C ) G y )  =  ( ( B G C ) G y ) )
109ad2antll 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 )  =  ( ( B G C ) G y ) )
111grpoass 23609 . . . . . . . . . . . . . 14  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  C  e.  X  /\  y  e.  X )
)  ->  ( ( A G C ) G y )  =  ( A G ( C G y ) ) )
12113anassrs 1204 . . . . . . . . . . . . 13  |-  ( ( ( ( G  e. 
GrpOp  /\  A  e.  X
)  /\  C  e.  X )  /\  y  e.  X )  ->  (
( A G C ) G y )  =  ( A G ( C G y ) ) )
1312adantlrl 714 . . . . . . . . . . . 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 711 . . . . . . . . . . 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 23609 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X  /\  y  e.  X )
)  ->  ( ( B G C ) G y )  =  ( B G ( C G y ) ) )
16153exp2 1200 . . . . . . . . . . . . . 14  |-  ( G  e.  GrpOp  ->  ( B  e.  X  ->  ( C  e.  X  ->  (
y  e.  X  -> 
( ( B G C ) G y )  =  ( B G ( C G y ) ) ) ) ) )
1716imp42 591 . . . . . . . . . . . . 13  |-  ( ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X
) )  /\  y  e.  X )  ->  (
( B G C ) G y )  =  ( B G ( C G y ) ) )
1817adantllr 713 . . . . . . . . . . . 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 711 . . . . . . . . . . 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 2481 . . . . . . . . . 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 718 . . . . . . . . 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 6098 . . . . . . . . . . 11  |-  ( ( C G y )  =  (GId `  G
)  ->  ( A G ( C G y ) )  =  ( A G (GId
`  G ) ) )
2322ad2antrl 722 . . . . . . . . . 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 463 . . . . . . . . 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 6098 . . . . . . . . . . 11  |-  ( ( C G y )  =  (GId `  G
)  ->  ( B G ( C G y ) )  =  ( B G (GId
`  G ) ) )
2625ad2antrl 722 . . . . . . . . . 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 463 . . . . . . . . 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 2481 . . . . . . . 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 23626 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G (GId `  G
) )  =  A )
3029ad2antrr 720 . . . . . . . 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 23626 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  ( B G (GId `  G
) )  =  B )
3231ad2ant2r 741 . . . . . . . . 9  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X )  /\  ( B  e.  X  /\  C  e.  X ) )  -> 
( B G (GId
`  G ) )  =  B )
3332adantr 462 . . . . . . . 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 2481 . . . . . . 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 611 . . . . . 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 2838 . . . . 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 6097 . . . 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 609 . 2  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  ( B  e.  X  ->  ( C  e.  X  ->  ( ( A G C )  =  ( B G C )  <->  A  =  B ) ) ) ) )
41403imp2 1197 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 960    = wceq 1364    e. wcel 1761   E.wrex 2714   ran crn 4837   ` cfv 5415  (class class class)co 6090   GrpOpcgr 23592  GIdcgi 23593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-fo 5421  df-fv 5423  df-riota 6049  df-ov 6093  df-grpo 23597  df-gid 23598
This theorem is referenced by:  grpoinveu  23628  grpoid  23629  grpodiveq  23662  rngorcan  23802  rngorz  23808  vcrcan  23861  nvrcan  23922  ghomdiv  28658
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