Theorem grpasscan1 12940

Description: An associative cancellation law for groups. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by AV, 30-Aug-2021.)

Hypotheses

Ref	Expression
grplcan.b	|- B = ( Base ` G )
grplcan.p	|- .+ = ( +g ` G )
grpasscan1.n	|- N = ( invg ` G )

Assertion

Ref	Expression
grpasscan1	|- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .+ ( ( N ` X ) .+ Y ) ) = Y )

Proof of Theorem grpasscan1

Step	Hyp	Ref	Expression
1		grplcan.b	. . . . 5 |- B = ( Base ` G )
2		grplcan.p	. . . . 5 |- .+ = ( +g ` G )
3		eqid 2177	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
4		grpasscan1.n	. . . . 5 |- N = ( invg ` G )
5	1, 2, 3, 4	grprinv 12930	. . . 4 |- ( ( G e. Grp /\ X e. B ) -> ( X .+ ( N ` X ) ) = ( 0g ` G ) )
6	5	3adant3 1017	. . 3 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .+ ( N ` X ) ) = ( 0g ` G ) )
7	6	oveq1d 5893	. 2 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( X .+ ( N ` X ) ) .+ Y ) = ( ( 0g ` G ) .+ Y ) )
8	1, 4	grpinvcl 12928	. . . 4 |- ( ( G e. Grp /\ X e. B ) -> ( N ` X ) e. B )
9	1, 2	grpass 12893	. . . . . 6 |- ( ( G e. Grp /\ ( X e. B /\ ( N ` X ) e. B /\ Y e. B ) ) -> ( ( X .+ ( N ` X ) ) .+ Y ) = ( X .+ ( ( N ` X ) .+ Y ) ) )
10	9	3exp2 1225	. . . . 5 |- ( G e. Grp -> ( X e. B -> ( ( N ` X ) e. B -> ( Y e. B -> ( ( X .+ ( N ` X ) ) .+ Y ) = ( X .+ ( ( N ` X ) .+ Y ) ) ) ) ) )
11	10	imp 124	. . . 4 |- ( ( G e. Grp /\ X e. B ) -> ( ( N ` X ) e. B -> ( Y e. B -> ( ( X .+ ( N ` X ) ) .+ Y ) = ( X .+ ( ( N ` X ) .+ Y ) ) ) ) )
12	8, 11	mpd 13	. . 3 |- ( ( G e. Grp /\ X e. B ) -> ( Y e. B -> ( ( X .+ ( N ` X ) ) .+ Y ) = ( X .+ ( ( N ` X ) .+ Y ) ) ) )
13	12	3impia 1200	. 2 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( X .+ ( N ` X ) ) .+ Y ) = ( X .+ ( ( N ` X ) .+ Y ) ) )
14	1, 2, 3	grplid 12913	. . 3 |- ( ( G e. Grp /\ Y e. B ) -> ( ( 0g ` G ) .+ Y ) = Y )
15	14	3adant2 1016	. 2 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( 0g ` G ) .+ Y ) = Y )
16	7, 13, 15	3eqtr3d 2218	1 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .+ ( ( N ` X ) .+ Y ) ) = Y )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2148   ` cfv 5218  (class class class)co 5878   Basecbs 12465   +g cplusg 12539   0gc0g 12711   Grpcgrp 12884   invgcminusg 12885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-cnex 7905  ax-resscn 7906  ax-1re 7908  ax-addrcl 7911

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5834  df-ov 5881  df-inn 8923  df-2 8981  df-ndx 12468  df-slot 12469  df-base 12471  df-plusg 12552  df-0g 12713  df-mgm 12782  df-sgrp 12815  df-mnd 12825  df-grp 12887  df-minusg 12888

This theorem is referenced by:  mulgaddcomlem  13016