Theorem ablpnpcan 14101

Description: Cancellation law for mixed addition and subtraction. (pnpcan 8555 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 14094	. . 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 1287	. 2 |- ( ph -> ( ( X .+ Y ) .- ( X .+ Z ) ) = ( ( X .- X ) .+ ( Y .- Z ) ) )
10		ablgrp 14069	. . . . 5 |- ( G e. Abel -> G e. Grp )
11	1, 10	syl 14	. . . 4 |- ( ph -> G e. Grp )
12		eqid 2238	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
13	5, 12, 7	grpsubid 13866	. . . 4 |- ( ( G e. Grp /\ X e. B ) -> ( X .- X ) = ( 0g ` G ) )
14	11, 2, 13	syl2anc 415	. . 3 |- ( ph -> ( X .- X ) = ( 0g ` G ) )
15	14	oveq1d 6090	. 2 |- ( ph -> ( ( X .- X ) .+ ( Y .- Z ) ) = ( ( 0g ` G ) .+ ( Y .- Z ) ) )
16	5, 7	grpsubcl 13862	. . . 4 |- ( ( G e. Grp /\ Y e. B /\ Z e. B ) -> ( Y .- Z ) e. B )
17	11, 3, 4, 16	syl3anc 1278	. . 3 |- ( ph -> ( Y .- Z ) e. B )
18	5, 6, 12	grplid 13813	. . 3 |- ( ( G e. Grp /\ ( Y .- Z ) e. B ) -> ( ( 0g ` G ) .+ ( Y .- Z ) ) = ( Y .- Z ) )
19	11, 17, 18	syl2anc 415	. 2 |- ( ph -> ( ( 0g ` G ) .+ ( Y .- Z ) ) = ( Y .- Z ) )
20	9, 15, 19	3eqtrd 2275	1 |- ( ph -> ( ( X .+ Y ) .- ( X .+ Z ) ) = ( Y .- Z ) )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   ` cfv 5372  (class class class)co 6075   Basecbs 13330   +g cplusg 13408   0gc0g 13587   Grpcgrp 13782   -gcsg 13784   Abelcabl 14065

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-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-inn 9284  df-2 9342  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-minusg 13786  df-sbg 13787  df-cmn 14066  df-abl 14067

This theorem is referenced by: (None)