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Theorem galcan 17737
Description: The action of a particular group element is left-cancelable. (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 13-Jan-2015.)
Hypothesis
Ref Expression
galcan.1 |- X = ( Base ` G )
Assertion
Ref Expression
galcan |- ( ( .(+) e. ( G GrpAct Y ) /\ ( A e. X /\ B e. Y /\ C e. Y ) ) -> ( ( A .(+) B ) = ( A .(+) C ) <-> B = C ) )
Proof of Theorem galcan
StepHypRef Expression
1 oveq2 6658 . . 3 |- ( ( A .(+) B ) = ( A .(+) C ) -> ( ( ( invg ` G ) ` A ) .(+) ( A .(+) B ) ) = ( ( ( invg ` G ) ` A ) .(+) ( A .(+) C ) ) )
2 simpl 473 . . . . . . . 8 |- ( ( .(+) e. ( G GrpAct Y ) /\ ( A e. X /\ B e. Y /\ C e. Y ) ) -> .(+) e. ( G GrpAct Y ) )
3 gagrp 17725 . . . . . . . 8 |- ( .(+) e. ( G GrpAct Y ) -> G e. Grp )
42, 3syl 17 . . . . . . 7 |- ( ( .(+) e. ( G GrpAct Y ) /\ ( A e. X /\ B e. Y /\ C e. Y ) ) -> G e. Grp )
5 simpr1 1067 . . . . . . 7 |- ( ( .(+) e. ( G GrpAct Y ) /\ ( A e. X /\ B e. Y /\ C e. Y ) ) -> A e. X )
6 galcan.1 . . . . . . . 8 |- X = ( Base ` G )
7 eqid 2622 . . . . . . . 8 |- ( +g ` G ) = ( +g ` G )
8 eqid 2622 . . . . . . . 8 |- ( 0g ` G ) = ( 0g ` G )
9 eqid 2622 . . . . . . . 8 |- ( invg ` G ) = ( invg ` G )
106, 7, 8, 9grplinv 17468 . . . . . . 7 |- ( ( G e. Grp /\ A e. X ) -> ( ( ( invg ` G ) ` A ) ( +g ` G ) A ) = ( 0g ` G ) )
114, 5, 10syl2anc 693 . . . . . 6 |- ( ( .(+) e. ( G GrpAct Y ) /\ ( A e. X /\ B e. Y /\ C e. Y ) ) -> ( ( ( invg ` G ) ` A ) ( +g ` G ) A ) = ( 0g ` G ) )
1211oveq1d 6665 . . . . 5 |- ( ( .(+) e. ( G GrpAct Y ) /\ ( A e. X /\ B e. Y /\ C e. Y ) ) -> ( ( ( ( invg ` G ) ` A ) ( +g ` G ) A ) .(+) B ) = ( ( 0g ` G ) .(+) B ) )
136, 9grpinvcl 17467 . . . . . . 7 |- ( ( G e. Grp /\ A e. X ) -> ( ( invg ` G ) ` A ) e. X )
144, 5, 13syl2anc 693 . . . . . 6 |- ( ( .(+) e. ( G GrpAct Y ) /\ ( A e. X /\ B e. Y /\ C e. Y ) ) -> ( ( invg ` G ) ` A ) e. X )
15 simpr2 1068 . . . . . 6 |- ( ( .(+) e. ( G GrpAct Y ) /\ ( A e. X /\ B e. Y /\ C e. Y ) ) -> B e. Y )
166, 7gaass 17730 . . . . . 6 |- ( ( .(+) e. ( G GrpAct Y ) /\ ( ( ( invg ` G ) ` A ) e. X /\ A e. X /\ B e. Y ) ) -> ( ( ( ( invg ` G ) ` A ) ( +g ` G ) A ) .(+) B ) = ( ( ( invg ` G ) ` A ) .(+) ( A .(+) B ) ) )
172, 14, 5, 15, 16syl13anc 1328 . . . . 5 |- ( ( .(+) e. ( G GrpAct Y ) /\ ( A e. X /\ B e. Y /\ C e. Y ) ) -> ( ( ( ( invg ` G ) ` A ) ( +g ` G ) A ) .(+) B ) = ( ( ( invg ` G ) ` A ) .(+) ( A .(+) B ) ) )
188gagrpid 17727 . . . . . 6 |- ( ( .(+) e. ( G GrpAct Y ) /\ B e. Y ) -> ( ( 0g ` G ) .(+) B ) = B )
192, 15, 18syl2anc 693 . . . . 5 |- ( ( .(+) e. ( G GrpAct Y ) /\ ( A e. X /\ B e. Y /\ C e. Y ) ) -> ( ( 0g ` G ) .(+) B ) = B )
2012, 17, 193eqtr3d 2664 . . . 4 |- ( ( .(+) e. ( G GrpAct Y ) /\ ( A e. X /\ B e. Y /\ C e. Y ) ) -> ( ( ( invg ` G ) ` A ) .(+) ( A .(+) B ) ) = B )
2111oveq1d 6665 . . . . 5 |- ( ( .(+) e. ( G GrpAct Y ) /\ ( A e. X /\ B e. Y /\ C e. Y ) ) -> ( ( ( ( invg ` G ) ` A ) ( +g ` G ) A ) .(+) C ) = ( ( 0g ` G ) .(+) C ) )
22 simpr3 1069 . . . . . 6 |- ( ( .(+) e. ( G GrpAct Y ) /\ ( A e. X /\ B e. Y /\ C e. Y ) ) -> C e. Y )
236, 7gaass 17730 . . . . . 6 |- ( ( .(+) e. ( G GrpAct Y ) /\ ( ( ( invg ` G ) ` A ) e. X /\ A e. X /\ C e. Y ) ) -> ( ( ( ( invg ` G ) ` A ) ( +g ` G ) A ) .(+) C ) = ( ( ( invg ` G ) ` A ) .(+) ( A .(+) C ) ) )
242, 14, 5, 22, 23syl13anc 1328 . . . . 5 |- ( ( .(+) e. ( G GrpAct Y ) /\ ( A e. X /\ B e. Y /\ C e. Y ) ) -> ( ( ( ( invg ` G ) ` A ) ( +g ` G ) A ) .(+) C ) = ( ( ( invg ` G ) ` A ) .(+) ( A .(+) C ) ) )
258gagrpid 17727 . . . . . 6 |- ( ( .(+) e. ( G GrpAct Y ) /\ C e. Y ) -> ( ( 0g ` G ) .(+) C ) = C )
262, 22, 25syl2anc 693 . . . . 5 |- ( ( .(+) e. ( G GrpAct Y ) /\ ( A e. X /\ B e. Y /\ C e. Y ) ) -> ( ( 0g ` G ) .(+) C ) = C )
2721, 24, 263eqtr3d 2664 . . . 4 |- ( ( .(+) e. ( G GrpAct Y ) /\ ( A e. X /\ B e. Y /\ C e. Y ) ) -> ( ( ( invg ` G ) ` A ) .(+) ( A .(+) C ) ) = C )
2820, 27eqeq12d 2637 . . 3 |- ( ( .(+) e. ( G GrpAct Y ) /\ ( A e. X /\ B e. Y /\ C e. Y ) ) -> ( ( ( ( invg ` G ) ` A ) .(+) ( A .(+) B ) ) = ( ( ( invg ` G ) ` A ) .(+) ( A .(+) C ) ) <-> B = C ) )
291, 28syl5ib 234 . 2 |- ( ( .(+) e. ( G GrpAct Y ) /\ ( A e. X /\ B e. Y /\ C e. Y ) ) -> ( ( A .(+) B ) = ( A .(+) C ) -> B = C ) )
30 oveq2 6658 . 2 |- ( B = C -> ( A .(+) B ) = ( A .(+) C ) )
3129, 30impbid1 215 1 |- ( ( .(+) e. ( G GrpAct Y ) /\ ( A e. X /\ B e. Y /\ C e. Y ) ) -> ( ( A .(+) B ) = ( A .(+) C ) <-> B = C ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   0gc0g 16100   Grpcgrp 17422   invgcminusg 17423   GrpAct cga 17722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-ga 17723
This theorem is referenced by:  gacan  17738
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