Theorem nmval2 20978

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 20976	. . 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 6290	. . 3 |- ( A E .0. ) = ( A ( D |` ( X X. X ) ) .0. )
9		id 22	. . . 4 |- ( A e. X -> A e. X )
10	2, 3	grpidcl 15947	. . . 4 |- ( W e. Grp -> .0. e. X )
11		ovres 6423	. . . 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 2495	. 2 |- ( ( W e. Grp /\ A e. X ) -> ( A D .0. ) = ( A E .0. ) )
14	6, 13	eqtrd 2482	1 |- ( ( W e. Grp /\ A e. X ) -> ( N ` A ) = ( A E .0. ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1381   e. wcel 1802   X. cxp 4983   |` cres 4987   ` cfv 5574  (class class class)co 6277   Basecbs 14504   distcds 14578   0gc0g 14709   Grpcgrp 15922   normcnm 20963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-fv 5582  df-riota 6238  df-ov 6280  df-0g 14711  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-grp 15926  df-nm 20969

This theorem is referenced by:  nmhmcn  21469