Theorem isngp 24753

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 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ∘ − ) ⊆ 𝐷))

Proof of Theorem isngp

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3921	. . 3 ⊢ (𝐺 ∈ (Grp ∩ MetSp) ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp))
2	1	anbi1i 635	. 2 ⊢ ((𝐺 ∈ (Grp ∩ MetSp) ∧ (𝑁 ∘ − ) ⊆ 𝐷) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp) ∧ (𝑁 ∘ − ) ⊆ 𝐷))
3		fveq2 6881	. . . . . 6 ⊢ (𝑔 = 𝐺 → (norm‘𝑔) = (norm‘𝐺))
4		isngp.n	. . . . . 6 ⊢ 𝑁 = (norm‘𝐺)
5	3, 4	eqtr4di 2816	. . . . 5 ⊢ (𝑔 = 𝐺 → (norm‘𝑔) = 𝑁)
6		fveq2 6881	. . . . . 6 ⊢ (𝑔 = 𝐺 → (-g‘𝑔) = (-g‘𝐺))
7		isngp.z	. . . . . 6 ⊢ − = (-g‘𝐺)
8	6, 7	eqtr4di 2816	. . . . 5 ⊢ (𝑔 = 𝐺 → (-g‘𝑔) = − )
9	5, 8	coeq12d 5850	. . . 4 ⊢ (𝑔 = 𝐺 → ((norm‘𝑔) ∘ (-g‘𝑔)) = (𝑁 ∘ − ))
10		fveq2 6881	. . . . 5 ⊢ (𝑔 = 𝐺 → (dist‘𝑔) = (dist‘𝐺))
11		isngp.d	. . . . 5 ⊢ 𝐷 = (dist‘𝐺)
12	10, 11	eqtr4di 2816	. . . 4 ⊢ (𝑔 = 𝐺 → (dist‘𝑔) = 𝐷)
13	9, 12	sseq12d 3970	. . 3 ⊢ (𝑔 = 𝐺 → (((norm‘𝑔) ∘ (-g‘𝑔)) ⊆ (dist‘𝑔) ↔ (𝑁 ∘ − ) ⊆ 𝐷))
14		df-ngp 24740	. . 3 ⊢ NrmGrp = {𝑔 ∈ (Grp ∩ MetSp) ∣ ((norm‘𝑔) ∘ (-g‘𝑔)) ⊆ (dist‘𝑔)}
15	13, 14	elrab2 3654	. 2 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ (Grp ∩ MetSp) ∧ (𝑁 ∘ − ) ⊆ 𝐷))
16		df-3an 1105	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ∘ − ) ⊆ 𝐷) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp) ∧ (𝑁 ∘ − ) ⊆ 𝐷))
17	2, 15, 16	3bitr4i 306	1 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ∘ − ) ⊆ 𝐷))

Syntax hints:   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ∩ cin 3904   ⊆ wss 3905   ∘ ccom 5665  ‘cfv 6536  distcds 17314  Grpcgrp 18995  -_gcsg 18997  MetSpcms 24475  normcnm 24733  NrmGrpcngp 24734

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-co 5670  df-iota 6492  df-fv 6544  df-ngp 24740

This theorem is referenced by:  isngp2  24754  ngpgrp  24756  ngpms  24757  tngngp2  24809  cnngp  24936  zhmnrg  34355