Theorem grpasscan2 12960

Description: An associative cancellation law for groups. (Contributed by Paul Chapman, 17-Apr-2009.) (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
grpasscan2	|- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( X .+ ( N ` Y ) ) .+ Y ) = X )

Proof of Theorem grpasscan2

Step	Hyp	Ref	Expression
1		simp1 998	. . 3 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> G e. Grp )
2		simp2 999	. . 3 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> X e. B )
3		grplcan.b	. . . . 5 |- B = ( Base ` G )
4		grpasscan1.n	. . . . 5 |- N = ( invg ` G )
5	3, 4	grpinvcl 12944	. . . 4 |- ( ( G e. Grp /\ Y e. B ) -> ( N ` Y ) e. B )
6	5	3adant2 1017	. . 3 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( N ` Y ) e. B )
7		simp3 1000	. . 3 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> Y e. B )
8		grplcan.p	. . . 4 |- .+ = ( +g ` G )
9	3, 8	grpass 12907	. . 3 |- ( ( G e. Grp /\ ( X e. B /\ ( N ` Y ) e. B /\ Y e. B ) ) -> ( ( X .+ ( N ` Y ) ) .+ Y ) = ( X .+ ( ( N ` Y ) .+ Y ) ) )
10	1, 2, 6, 7, 9	syl13anc 1250	. 2 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( X .+ ( N ` Y ) ) .+ Y ) = ( X .+ ( ( N ` Y ) .+ Y ) ) )
11		eqid 2187	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
12	3, 8, 11, 4	grplinv 12946	. . . 4 |- ( ( G e. Grp /\ Y e. B ) -> ( ( N ` Y ) .+ Y ) = ( 0g ` G ) )
13	12	3adant2 1017	. . 3 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( N ` Y ) .+ Y ) = ( 0g ` G ) )
14	13	oveq2d 5904	. 2 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .+ ( ( N ` Y ) .+ Y ) ) = ( X .+ ( 0g ` G ) ) )
15	3, 8, 11	grprid 12928	. . 3 |- ( ( G e. Grp /\ X e. B ) -> ( X .+ ( 0g ` G ) ) = X )
16	15	3adant3 1018	. 2 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .+ ( 0g ` G ) ) = X )
17	10, 14, 16	3eqtrd 2224	1 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( X .+ ( N ` Y ) ) .+ Y ) = X )

Syntax hints:   -> wi 4   /\ w3a 979   = wceq 1363   e. wcel 2158   ` cfv 5228  (class class class)co 5888   Basecbs 12475   +g cplusg 12550   0gc0g 12722   Grpcgrp 12898   invgcminusg 12899

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-cnex 7915  ax-resscn 7916  ax-1re 7918  ax-addrcl 7921

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-inn 8933  df-2 8991  df-ndx 12478  df-slot 12479  df-base 12481  df-plusg 12563  df-0g 12724  df-mgm 12793  df-sgrp 12826  df-mnd 12839  df-grp 12901  df-minusg 12902

This theorem is referenced by:  mulgaddcomlem  13037