Theorem ablpnpcan 18225

Description: Cancellation law for mixed addition and subtraction. (pnpcan 10320 analog.) (Contributed by NM, 29-May-2015.)

Hypotheses

Ref	Expression
ablsubadd.b	|- B = ( Base ` G )
ablsubadd.p	|- .+ = ( +g ` G )
ablsubadd.m	|- .- = ( -g ` G )
ablsubsub.g	|- ( ph -> G e. Abel )
ablsubsub.x	|- ( ph -> X e. B )
ablsubsub.y	|- ( ph -> Y e. B )
ablsubsub.z	|- ( ph -> Z e. B )
ablpnpcan.g	|- ( ph -> G e. Abel )
ablpnpcan.x	|- ( ph -> X e. B )
ablpnpcan.y	|- ( ph -> Y e. B )
ablpnpcan.z	|- ( ph -> Z e. B )

Assertion

Ref	Expression
ablpnpcan	|- ( ph -> ( ( X .+ Y ) .- ( X .+ Z ) ) = ( Y .- Z ) )

Proof of Theorem ablpnpcan

Step	Hyp	Ref	Expression
1		ablsubsub.g	. . 3 |- ( ph -> G e. Abel )
2		ablsubsub.x	. . 3 |- ( ph -> X e. B )
3		ablsubsub.y	. . 3 |- ( ph -> Y e. B )
4		ablsubsub.z	. . 3 |- ( ph -> Z e. B )
5		ablsubadd.b	. . . 4 |- B = ( Base ` G )
6		ablsubadd.p	. . . 4 |- .+ = ( +g ` G )
7		ablsubadd.m	. . . 4 |- .- = ( -g ` G )
8	5, 6, 7	ablsub4 18218	. . 3 |- ( ( G e. Abel /\ ( X e. B /\ Y e. B ) /\ ( X e. B /\ Z e. B ) ) -> ( ( X .+ Y ) .- ( X .+ Z ) ) = ( ( X .- X ) .+ ( Y .- Z ) ) )
9	1, 2, 3, 2, 4, 8	syl122anc 1335	. 2 |- ( ph -> ( ( X .+ Y ) .- ( X .+ Z ) ) = ( ( X .- X ) .+ ( Y .- Z ) ) )
10		ablgrp 18198	. . . . 5 |- ( G e. Abel -> G e. Grp )
11	1, 10	syl 17	. . . 4 |- ( ph -> G e. Grp )
12		eqid 2622	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
13	5, 12, 7	grpsubid 17499	. . . 4 |- ( ( G e. Grp /\ X e. B ) -> ( X .- X ) = ( 0g ` G ) )
14	11, 2, 13	syl2anc 693	. . 3 |- ( ph -> ( X .- X ) = ( 0g ` G ) )
15	14	oveq1d 6665	. 2 |- ( ph -> ( ( X .- X ) .+ ( Y .- Z ) ) = ( ( 0g ` G ) .+ ( Y .- Z ) ) )
16	5, 7	grpsubcl 17495	. . . 4 |- ( ( G e. Grp /\ Y e. B /\ Z e. B ) -> ( Y .- Z ) e. B )
17	11, 3, 4, 16	syl3anc 1326	. . 3 |- ( ph -> ( Y .- Z ) e. B )
18	5, 6, 12	grplid 17452	. . 3 |- ( ( G e. Grp /\ ( Y .- Z ) e. B ) -> ( ( 0g ` G ) .+ ( Y .- Z ) ) = ( Y .- Z ) )
19	11, 17, 18	syl2anc 693	. 2 |- ( ph -> ( ( 0g ` G ) .+ ( Y .- Z ) ) = ( Y .- Z ) )
20	9, 15, 19	3eqtrd 2660	1 |- ( ph -> ( ( X .+ Y ) .- ( X .+ Z ) ) = ( Y .- Z ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   0gc0g 16100   Grpcgrp 17422   -gcsg 17424   Abelcabl 18194

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  df-cmn 18195  df-abl 18196

This theorem is referenced by:  hdmaprnlem7N  37147