Theorem grpsubeq0 12807

Description: If the difference between two group elements is zero, they are equal. (subeq0 8149 analog.) (Contributed by NM, 31-Mar-2014.)

Hypotheses

Ref	Expression
grpsubid.b	|- B = ( Base ` G )
grpsubid.o	|- .0. = ( 0g ` G )
grpsubid.m	|- .- = ( -g ` G )

Assertion

Ref	Expression
grpsubeq0	|- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( X .- Y ) = .0. <-> X = Y ) )

Proof of Theorem grpsubeq0

Step	Hyp	Ref	Expression
1		grpsubid.b	. . . . 5 |- B = ( Base ` G )
2		eqid 2171	. . . . 5 |- ( +g ` G ) = ( +g ` G )
3		eqid 2171	. . . . 5 |- ( invg ` G ) = ( invg ` G )
4		grpsubid.m	. . . . 5 |- .- = ( -g ` G )
5	1, 2, 3, 4	grpsubval 12771	. . . 4 |- ( ( X e. B /\ Y e. B ) -> ( X .- Y ) = ( X ( +g ` G ) ( ( invg ` G ) ` Y ) ) )
6	5	3adant1 1011	. . 3 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .- Y ) = ( X ( +g ` G ) ( ( invg ` G ) ` Y ) ) )
7	6	eqeq1d 2180	. 2 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( X .- Y ) = .0. <-> ( X ( +g ` G ) ( ( invg ` G ) ` Y ) ) = .0. ) )
8		simp1 993	. . 3 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> G e. Grp )
9	1, 3	grpinvcl 12773	. . . 4 |- ( ( G e. Grp /\ Y e. B ) -> ( ( invg ` G ) ` Y ) e. B )
10	9	3adant2 1012	. . 3 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( invg ` G ) ` Y ) e. B )
11		simp2 994	. . 3 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> X e. B )
12		grpsubid.o	. . . 4 |- .0. = ( 0g ` G )
13	1, 2, 12, 3	grpinvid2 12777	. . 3 |- ( ( G e. Grp /\ ( ( invg ` G ) ` Y ) e. B /\ X e. B ) -> ( ( ( invg ` G ) ` ( ( invg ` G ) ` Y ) ) = X <-> ( X ( +g ` G ) ( ( invg ` G ) ` Y ) ) = .0. ) )
14	8, 10, 11, 13	syl3anc 1234	. 2 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( ( invg ` G ) ` ( ( invg ` G ) ` Y ) ) = X <-> ( X ( +g ` G ) ( ( invg ` G ) ` Y ) ) = .0. ) )
15	1, 3	grpinvinv 12788	. . . . 5 |- ( ( G e. Grp /\ Y e. B ) -> ( ( invg ` G ) ` ( ( invg ` G ) ` Y ) ) = Y )
16	15	3adant2 1012	. . . 4 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( invg ` G ) ` ( ( invg ` G ) ` Y ) ) = Y )
17	16	eqeq1d 2180	. . 3 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( ( invg ` G ) ` ( ( invg ` G ) ` Y ) ) = X <-> Y = X ) )
18		eqcom 2173	. . 3 |- ( Y = X <-> X = Y )
19	17, 18	bitrdi 195	. 2 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( ( invg ` G ) ` ( ( invg ` G ) ` Y ) ) = X <-> X = Y ) )
20	7, 14, 19	3bitr2d 215	1 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( X .- Y ) = .0. <-> X = Y ) )

Syntax hints:   -> wi 4   <-> wb 104   /\ w3a 974   = wceq 1349   e. wcel 2142   ` cfv 5200  (class class class)co 5857   Basecbs 12420   +g cplusg 12484   0gc0g 12618   Grpcgrp 12730   invgcminusg 12731   -gcsg 12732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 610  ax-in2 611  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-13 2144  ax-14 2145  ax-ext 2153  ax-coll 4105  ax-sep 4108  ax-pow 4161  ax-pr 4195  ax-un 4419  ax-setind 4522  ax-cnex 7869  ax-resscn 7870  ax-1re 7872  ax-addrcl 7875

This theorem depends on definitions:  df-bi 116  df-3an 976  df-tru 1352  df-fal 1355  df-nf 1455  df-sb 1757  df-eu 2023  df-mo 2024  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ne 2342  df-ral 2454  df-rex 2455  df-reu 2456  df-rmo 2457  df-rab 2458  df-v 2733  df-sbc 2957  df-csb 3051  df-dif 3124  df-un 3126  df-in 3128  df-ss 3135  df-pw 3569  df-sn 3590  df-pr 3591  df-op 3593  df-uni 3798  df-int 3833  df-iun 3876  df-br 3991  df-opab 4052  df-mpt 4053  df-id 4279  df-xp 4618  df-rel 4619  df-cnv 4620  df-co 4621  df-dm 4622  df-rn 4623  df-res 4624  df-ima 4625  df-iota 5162  df-fun 5202  df-fn 5203  df-f 5204  df-f1 5205  df-fo 5206  df-f1o 5207  df-fv 5208  df-riota 5813  df-ov 5860  df-oprab 5861  df-mpo 5862  df-1st 6123  df-2nd 6124  df-inn 8883  df-2 8941  df-ndx 12423  df-slot 12424  df-base 12426  df-plusg 12497  df-0g 12620  df-mgm 12632  df-sgrp 12665  df-mnd 12675  df-grp 12733  df-minusg 12734  df-sbg 12735

This theorem is referenced by: (None)