Theorem grpasscan2 12763

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 992	. . 3 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> G e. Grp )
2		simp2 993	. . 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 12751	. . . 4 |- ( ( G e. Grp /\ Y e. B ) -> ( N ` Y ) e. B )
6	5	3adant2 1011	. . 3 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( N ` Y ) e. B )
7		simp3 994	. . 3 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> Y e. B )
8		grplcan.p	. . . 4 |- .+ = ( +g ` G )
9	3, 8	grpass 12717	. . 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 1235	. 2 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( X .+ ( N ` Y ) ) .+ Y ) = ( X .+ ( ( N ` Y ) .+ Y ) ) )
11		eqid 2170	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
12	3, 8, 11, 4	grplinv 12752	. . . 4 |- ( ( G e. Grp /\ Y e. B ) -> ( ( N ` Y ) .+ Y ) = ( 0g ` G ) )
13	12	3adant2 1011	. . 3 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( N ` Y ) .+ Y ) = ( 0g ` G ) )
14	13	oveq2d 5869	. 2 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .+ ( ( N ` Y ) .+ Y ) ) = ( X .+ ( 0g ` G ) ) )
15	3, 8, 11	grprid 12737	. . 3 |- ( ( G e. Grp /\ X e. B ) -> ( X .+ ( 0g ` G ) ) = X )
16	15	3adant3 1012	. 2 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .+ ( 0g ` G ) ) = X )
17	10, 14, 16	3eqtrd 2207	1 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( X .+ ( N ` Y ) ) .+ Y ) = X )

Syntax hints:   -> wi 4   /\ w3a 973   = wceq 1348   e. wcel 2141   ` cfv 5198  (class class class)co 5853   Basecbs 12416   +g cplusg 12480   0gc0g 12596   Grpcgrp 12708   invgcminusg 12709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-cnex 7865  ax-resscn 7866  ax-1re 7868  ax-addrcl 7871

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-inn 8879  df-2 8937  df-ndx 12419  df-slot 12420  df-base 12422  df-plusg 12493  df-0g 12598  df-mgm 12610  df-sgrp 12643  df-mnd 12653  df-grp 12711  df-minusg 12712

This theorem is referenced by: (None)