Theorem nmval2 22396

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 22394	. . 3 |- ( A e. X -> ( N ` A ) = ( A D .0. ) )
6	5	adantl 482	. 2 |- ( ( W e. Grp /\ A e. X ) -> ( N ` A ) = ( A D .0. ) )
7		nmfval.e	. . . 4 |- E = ( D |` ( X X. X ) )
8	7	oveqi 6663	. . 3 |- ( A E .0. ) = ( A ( D |` ( X X. X ) ) .0. )
9		id 22	. . . 4 |- ( A e. X -> A e. X )
10	2, 3	grpidcl 17450	. . . 4 |- ( W e. Grp -> .0. e. X )
11		ovres 6800	. . . 4 |- ( ( A e. X /\ .0. e. X ) -> ( A ( D |` ( X X. X ) ) .0. ) = ( A D .0. ) )
12	9, 10, 11	syl2anr 495	. . 3 |- ( ( W e. Grp /\ A e. X ) -> ( A ( D |` ( X X. X ) ) .0. ) = ( A D .0. ) )
13	8, 12	syl5req 2669	. 2 |- ( ( W e. Grp /\ A e. X ) -> ( A D .0. ) = ( A E .0. ) )
14	6, 13	eqtrd 2656	1 |- ( ( W e. Grp /\ A e. X ) -> ( N ` A ) = ( A E .0. ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   X. cxp 5112   |` cres 5116   ` cfv 5888  (class class class)co 6650   Basecbs 15857   distcds 15950   0gc0g 16100   Grpcgrp 17422   normcnm 22381

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-riota 6611  df-ov 6653  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-nm 22387

This theorem is referenced by:  nmhmcn  22920  nglmle  23100