Theorem nmval2 20302

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 20300	. . 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 6205	. . 3 |- ( A E .0. ) = ( A ( D |` ( X X. X ) ) .0. )
9		id 22	. . . 4 |- ( A e. X -> A e. X )
10	2, 3	grpidcl 15670	. . . 4 |- ( W e. Grp -> .0. e. X )
11		ovres 6332	. . . 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 2505	. 2 |- ( ( W e. Grp /\ A e. X ) -> ( A D .0. ) = ( A E .0. ) )
14	6, 13	eqtrd 2492	1 |- ( ( W e. Grp /\ A e. X ) -> ( N ` A ) = ( A E .0. ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1370   e. wcel 1758   X. cxp 4938   |` cres 4942   ` cfv 5518  (class class class)co 6192   Basecbs 14278   distcds 14351   0gc0g 14482   Grpcgrp 15514   normcnm 20287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-fv 5526  df-riota 6153  df-ov 6195  df-0g 14484  df-mnd 15519  df-grp 15649  df-nm 20293

This theorem is referenced by:  nmhmcn  20793