Theorem nmval2 20840

Description: The value of the norm function using a restricted metric. (Contributed by Mario Carneiro, 2-Oct-2015.)

Hypotheses

Ref	Expression
nmfval.n	|- N = ( norm ` W )
nmfval.x	|- X = ( Base ` W )
nmfval.z	|- .0. = ( 0g ` W )
nmfval.d	|- D = ( dist ` W )
nmfval.e	|- E = ( D |` ( X X. X ) )

Assertion

Ref	Expression
nmval2	|- ( ( W e. Grp /\ A e. X ) -> ( N ` A ) = ( A E .0. ) )

Proof of Theorem nmval2

Step	Hyp	Ref	Expression
1		nmfval.n	. . . 4 |- N = ( norm ` W )
2		nmfval.x	. . . 4 |- X = ( Base ` W )
3		nmfval.z	. . . 4 |- .0. = ( 0g ` W )
4		nmfval.d	. . . 4 |- D = ( dist ` W )
5	1, 2, 3, 4	nmval 20838	. . 3 |- ( A e. X -> ( N ` A ) = ( A D .0. ) )
6	5	adantl 466	. 2 |- ( ( W e. Grp /\ A e. X ) -> ( N ` A ) = ( A D .0. ) )
7		nmfval.e	. . . 4 |- E = ( D |` ( X X. X ) )
8	7	oveqi 6288	. . 3 |- ( A E .0. ) = ( A ( D |` ( X X. X ) ) .0. )
9		id 22	. . . 4 |- ( A e. X -> A e. X )
10	2, 3	grpidcl 15872	. . . 4 |- ( W e. Grp -> .0. e. X )
11		ovres 6417	. . . 4 |- ( ( A e. X /\ .0. e. X ) -> ( A ( D |` ( X X. X ) ) .0. ) = ( A D .0. ) )
12	9, 10, 11	syl2anr 478	. . 3 |- ( ( W e. Grp /\ A e. X ) -> ( A ( D |` ( X X. X ) ) .0. ) = ( A D .0. ) )
13	8, 12	syl5req 2514	. 2 |- ( ( W e. Grp /\ A e. X ) -> ( A D .0. ) = ( A E .0. ) )
14	6, 13	eqtrd 2501	1 |- ( ( W e. Grp /\ A e. X ) -> ( N ` A ) = ( A E .0. ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1374   e. wcel 1762   X. cxp 4990   |` cres 4994   ` cfv 5579  (class class class)co 6275   Basecbs 14479   distcds 14553   0gc0g 14684   Grpcgrp 15716   normcnm 20825

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fv 5587  df-riota 6236  df-ov 6278  df-0g 14686  df-mnd 15721  df-grp 15851  df-nm 20831

This theorem is referenced by:  nmhmcn  21331