Theorem grpasscan1 13395

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 2205	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
4		grpasscan1.n	. . . . 5 |- N = ( invg ` G )
5	1, 2, 3, 4	grprinv 13383	. . . 4 |- ( ( G e. Grp /\ X e. B ) -> ( X .+ ( N ` X ) ) = ( 0g ` G ) )
6	5	3adant3 1020	. . 3 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .+ ( N ` X ) ) = ( 0g ` G ) )
7	6	oveq1d 5959	. 2 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( X .+ ( N ` X ) ) .+ Y ) = ( ( 0g ` G ) .+ Y ) )
8	1, 4	grpinvcl 13380	. . . 4 |- ( ( G e. Grp /\ X e. B ) -> ( N ` X ) e. B )
9	1, 2	grpass 13341	. . . . . 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 1228	. . . . 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 1203	. 2 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( X .+ ( N ` X ) ) .+ Y ) = ( X .+ ( ( N ` X ) .+ Y ) ) )
14	1, 2, 3	grplid 13363	. . 3 |- ( ( G e. Grp /\ Y e. B ) -> ( ( 0g ` G ) .+ Y ) = Y )
15	14	3adant2 1019	. 2 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( 0g ` G ) .+ Y ) = Y )
16	7, 13, 15	3eqtr3d 2246	1 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .+ ( ( N ` X ) .+ Y ) ) = Y )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 981   = wceq 1373   e. wcel 2176   ` cfv 5271  (class class class)co 5944   Basecbs 12832   +g cplusg 12909   0gc0g 13088   Grpcgrp 13332   invgcminusg 13333

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-cnex 8016  ax-resscn 8017  ax-1re 8019  ax-addrcl 8022

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-inn 9037  df-2 9095  df-ndx 12835  df-slot 12836  df-base 12838  df-plusg 12922  df-0g 13090  df-mgm 13188  df-sgrp 13234  df-mnd 13249  df-grp 13335  df-minusg 13336

This theorem is referenced by:  mulgaddcomlem  13481