Theorem grppnpcan2 17509

Description: Cancellation law for mixed addition and subtraction. (pnpcan2 10321 analog.) (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.)

Hypotheses

Ref	Expression
grpsubadd.b	|- B = ( Base ` G )
grpsubadd.p	|- .+ = ( +g ` G )
grpsubadd.m	|- .- = ( -g ` G )

Assertion

Ref	Expression
grppnpcan2	|- ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Z ) .- ( Y .+ Z ) ) = ( X .- Y ) )

Proof of Theorem grppnpcan2

Step	Hyp	Ref	Expression
1		simpl 473	. . 3 |- ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> G e. Grp )
2		grpsubadd.b	. . . . 5 |- B = ( Base ` G )
3		grpsubadd.p	. . . . 5 |- .+ = ( +g ` G )
4	2, 3	grpcl 17430	. . . 4 |- ( ( G e. Grp /\ X e. B /\ Z e. B ) -> ( X .+ Z ) e. B )
5	4	3adant3r2 1275	. . 3 |- ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .+ Z ) e. B )
6		simpr3 1069	. . 3 |- ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Z e. B )
7		simpr2 1068	. . 3 |- ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Y e. B )
8		grpsubadd.m	. . . 4 |- .- = ( -g ` G )
9	2, 3, 8	grpsubsub4 17508	. . 3 |- ( ( G e. Grp /\ ( ( X .+ Z ) e. B /\ Z e. B /\ Y e. B ) ) -> ( ( ( X .+ Z ) .- Z ) .- Y ) = ( ( X .+ Z ) .- ( Y .+ Z ) ) )
10	1, 5, 6, 7, 9	syl13anc 1328	. 2 |- ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( ( X .+ Z ) .- Z ) .- Y ) = ( ( X .+ Z ) .- ( Y .+ Z ) ) )
11	2, 3, 8	grppncan 17506	. . . 4 |- ( ( G e. Grp /\ X e. B /\ Z e. B ) -> ( ( X .+ Z ) .- Z ) = X )
12	11	3adant3r2 1275	. . 3 |- ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Z ) .- Z ) = X )
13	12	oveq1d 6665	. 2 |- ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( ( X .+ Z ) .- Z ) .- Y ) = ( X .- Y ) )
14	10, 13	eqtr3d 2658	1 |- ( ( G e. Grp /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Z ) .- ( Y .+ Z ) ) = ( X .- Y ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   Grpcgrp 17422   -gcsg 17424

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427

This theorem is referenced by:  ngprcan  22414