Theorem isngp 22400

Description: The property of being a normed group. (Contributed by Mario Carneiro, 2-Oct-2015.)

Hypotheses

Ref	Expression
isngp.n	|- N = ( norm ` G )
isngp.z	|- .- = ( -g ` G )
isngp.d	|- D = ( dist ` G )

Assertion

Ref	Expression
isngp	|- ( G e. NrmGrp <-> ( G e. Grp /\ G e. MetSp /\ ( N o. .- ) C_ D ) )

Proof of Theorem isngp

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3796	. . 3 |- ( G e. ( Grp i^i MetSp ) <-> ( G e. Grp /\ G e. MetSp ) )
2	1	anbi1i 731	. 2 |- ( ( G e. ( Grp i^i MetSp ) /\ ( N o. .- ) C_ D ) <-> ( ( G e. Grp /\ G e. MetSp ) /\ ( N o. .- ) C_ D ) )
3		fveq2 6191	. . . . . 6 |- ( g = G -> ( norm ` g ) = ( norm ` G ) )
4		isngp.n	. . . . . 6 |- N = ( norm ` G )
5	3, 4	syl6eqr 2674	. . . . 5 |- ( g = G -> ( norm ` g ) = N )
6		fveq2 6191	. . . . . 6 |- ( g = G -> ( -g ` g ) = ( -g ` G ) )
7		isngp.z	. . . . . 6 |- .- = ( -g ` G )
8	6, 7	syl6eqr 2674	. . . . 5 |- ( g = G -> ( -g ` g ) = .- )
9	5, 8	coeq12d 5286	. . . 4 |- ( g = G -> ( ( norm ` g ) o. ( -g ` g ) ) = ( N o. .- ) )
10		fveq2 6191	. . . . 5 |- ( g = G -> ( dist ` g ) = ( dist ` G ) )
11		isngp.d	. . . . 5 |- D = ( dist ` G )
12	10, 11	syl6eqr 2674	. . . 4 |- ( g = G -> ( dist ` g ) = D )
13	9, 12	sseq12d 3634	. . 3 |- ( g = G -> ( ( ( norm ` g ) o. ( -g ` g ) ) C_ ( dist ` g ) <-> ( N o. .- ) C_ D ) )
14		df-ngp 22388	. . 3 |- NrmGrp = { g e. ( Grp i^i MetSp ) | ( ( norm ` g ) o. ( -g ` g ) ) C_ ( dist ` g ) }
15	13, 14	elrab2 3366	. 2 |- ( G e. NrmGrp <-> ( G e. ( Grp i^i MetSp ) /\ ( N o. .- ) C_ D ) )
16		df-3an 1039	. 2 |- ( ( G e. Grp /\ G e. MetSp /\ ( N o. .- ) C_ D ) <-> ( ( G e. Grp /\ G e. MetSp ) /\ ( N o. .- ) C_ D ) )
17	2, 15, 16	3bitr4i 292	1 |- ( G e. NrmGrp <-> ( G e. Grp /\ G e. MetSp /\ ( N o. .- ) C_ D ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   o. ccom 5118   ` cfv 5888   distcds 15950   Grpcgrp 17422   -gcsg 17424   MetSpcmt 22123   normcnm 22381  NrmGrpcngp 22382

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-co 5123  df-iota 5851  df-fv 5896  df-ngp 22388

This theorem is referenced by:  isngp2  22401  ngpgrp  22403  ngpms  22404  tngngp2  22456  cnngp  22583  zhmnrg  30011