Theorem nmeq0

Description: The identity is the only element of the group with zero norm. First part of Problem 2 of Kreyszig p. 64. (Contributed by NM, 24-Nov-2006) (Revised by Mario Carneiro, 4-Oct-2015)

	Ref	Expression
Hypotheses	nmf.x	|- X = ( Base ` G )
	nmf.n	|- N = ( norm ` G )
	nmeq0.z	|- .0. = ( 0g ` G )
Assertion	nmeq0	|- ( ( G e. NrmGrp /\ A e. X ) -> ( ( N ` A ) = 0 <-> A = .0. ) )

Proof

Step	Hyp	Ref	Expression
1		nmf.x	|- X = ( Base ` G )
2		nmf.n	|- N = ( norm ` G )
3		nmeq0.z	|- .0. = ( 0g ` G )
4		eqid	|- ( dist ` G ) = ( dist ` G )
5	2 1 3 4	nmval	|- ( A e. X -> ( N ` A ) = ( A ( dist ` G ) .0. ) )
6	5	adantl	|- ( ( G e. NrmGrp /\ A e. X ) -> ( N ` A ) = ( A ( dist ` G ) .0. ) )
7	6	eqeq1d	|- ( ( G e. NrmGrp /\ A e. X ) -> ( ( N ` A ) = 0 <-> ( A ( dist ` G ) .0. ) = 0 ) )
8		ngpgrp	|- ( G e. NrmGrp -> G e. Grp )
9	8	adantr	|- ( ( G e. NrmGrp /\ A e. X ) -> G e. Grp )
10	1 3	grpidcl	|- ( G e. Grp -> .0. e. X )
11	9 10	syl	|- ( ( G e. NrmGrp /\ A e. X ) -> .0. e. X )
12		ngpxms	|- ( G e. NrmGrp -> G e. *MetSp )
13	1 4	xmseq0	|- ( ( G e. *MetSp /\ A e. X /\ .0. e. X ) -> ( ( A ( dist ` G ) .0. ) = 0 <-> A = .0. ) )
14	12 13	syl3an1	|- ( ( G e. NrmGrp /\ A e. X /\ .0. e. X ) -> ( ( A ( dist ` G ) .0. ) = 0 <-> A = .0. ) )
15	11 14	mpd3an3	|- ( ( G e. NrmGrp /\ A e. X ) -> ( ( A ( dist ` G ) .0. ) = 0 <-> A = .0. ) )
16	7 15	bitrd	|- ( ( G e. NrmGrp /\ A e. X ) -> ( ( N ` A ) = 0 <-> A = .0. ) )