Theorem ablnncan 18226

Description: Cancellation law for group subtraction. (nncan 10310 analog.) (Contributed by NM, 7-Apr-2015.)

Hypotheses

Ref	Expression
ablnncan.b	|- B = ( Base ` G )
ablnncan.m	|- .- = ( -g ` G )
ablnncan.g	|- ( ph -> G e. Abel )
ablnncan.x	|- ( ph -> X e. B )
ablnncan.y	|- ( ph -> Y e. B )

Assertion

Ref	Expression
ablnncan	|- ( ph -> ( X .- ( X .- Y ) ) = Y )

Proof of Theorem ablnncan

Step	Hyp	Ref	Expression
1		ablnncan.b	. . 3 |- B = ( Base ` G )
2		eqid 2622	. . 3 |- ( +g ` G ) = ( +g ` G )
3		ablnncan.m	. . 3 |- .- = ( -g ` G )
4		ablnncan.g	. . 3 |- ( ph -> G e. Abel )
5		ablnncan.x	. . 3 |- ( ph -> X e. B )
6		ablnncan.y	. . 3 |- ( ph -> Y e. B )
7	1, 2, 3, 4, 5, 5, 6	ablsubsub 18223	. 2 |- ( ph -> ( X .- ( X .- Y ) ) = ( ( X .- X ) ( +g ` G ) Y ) )
8		ablgrp 18198	. . . . 5 |- ( G e. Abel -> G e. Grp )
9	4, 8	syl 17	. . . 4 |- ( ph -> G e. Grp )
10		eqid 2622	. . . . 5 |- ( 0g ` G ) = ( 0g ` G )
11	1, 10, 3	grpsubid 17499	. . . 4 |- ( ( G e. Grp /\ X e. B ) -> ( X .- X ) = ( 0g ` G ) )
12	9, 5, 11	syl2anc 693	. . 3 |- ( ph -> ( X .- X ) = ( 0g ` G ) )
13	12	oveq1d 6665	. 2 |- ( ph -> ( ( X .- X ) ( +g ` G ) Y ) = ( ( 0g ` G ) ( +g ` G ) Y ) )
14	1, 2, 10	grplid 17452	. . 3 |- ( ( G e. Grp /\ Y e. B ) -> ( ( 0g ` G ) ( +g ` G ) Y ) = Y )
15	9, 6, 14	syl2anc 693	. 2 |- ( ph -> ( ( 0g ` G ) ( +g ` G ) Y ) = Y )
16	7, 13, 15	3eqtrd 2660	1 |- ( ph -> ( X .- ( X .- Y ) ) = Y )

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:  ablnnncan1  18229  pgpfac1lem3  18476  tsmsxplem1  21956  baerlem5blem2  37001