Theorem grpasscan1 13818

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 2234	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
4		grpasscan1.n	. . . . 5 |- N = ( invg ` G )
5	1, 2, 3, 4	grprinv 13806	. . . 4 |- ( ( G e. Grp /\ X e. B ) -> ( X .+ ( N ` X ) ) = ( 0g ` G ) )
6	5	3adant3 1044	. . 3 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .+ ( N ` X ) ) = ( 0g ` G ) )
7	6	oveq1d 6073	. 2 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( X .+ ( N ` X ) ) .+ Y ) = ( ( 0g ` G ) .+ Y ) )
8	1, 4	grpinvcl 13803	. . . 4 |- ( ( G e. Grp /\ X e. B ) -> ( N ` X ) e. B )
9	1, 2	grpass 13764	. . . . . 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 1252	. . . . 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 1227	. 2 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( X .+ ( N ` X ) ) .+ Y ) = ( X .+ ( ( N ` X ) .+ Y ) ) )
14	1, 2, 3	grplid 13786	. . 3 |- ( ( G e. Grp /\ Y e. B ) -> ( ( 0g ` G ) .+ Y ) = Y )
15	14	3adant2 1043	. 2 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( 0g ` G ) .+ Y ) = Y )
16	7, 13, 15	3eqtr3d 2275	1 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .+ ( ( N ` X ) .+ Y ) ) = Y )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2205   ` cfv 5357  (class class class)co 6058   Basecbs 13296   +g cplusg 13374   0gc0g 13553   Grpcgrp 13755   invgcminusg 13756

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-inn 9255  df-2 9313  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-grp 13758  df-minusg 13759

This theorem is referenced by:  mulgaddcomlem  13898