Theorem isngp 23658

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 3899	. . 3 ⊢ (𝐺 ∈ (Grp ∩ MetSp) ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp))
2	1	anbi1i 623	. 2 ⊢ ((𝐺 ∈ (Grp ∩ MetSp) ∧ (𝑁 ∘ − ) ⊆ 𝐷) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp) ∧ (𝑁 ∘ − ) ⊆ 𝐷))
3		fveq2 6756	. . . . . 6 ⊢ (𝑔 = 𝐺 → (norm‘𝑔) = (norm‘𝐺))
4		isngp.n	. . . . . 6 ⊢ 𝑁 = (norm‘𝐺)
5	3, 4	eqtr4di 2797	. . . . 5 ⊢ (𝑔 = 𝐺 → (norm‘𝑔) = 𝑁)
6		fveq2 6756	. . . . . 6 ⊢ (𝑔 = 𝐺 → (-g‘𝑔) = (-g‘𝐺))
7		isngp.z	. . . . . 6 ⊢ − = (-g‘𝐺)
8	6, 7	eqtr4di 2797	. . . . 5 ⊢ (𝑔 = 𝐺 → (-g‘𝑔) = − )
9	5, 8	coeq12d 5762	. . . 4 ⊢ (𝑔 = 𝐺 → ((norm‘𝑔) ∘ (-g‘𝑔)) = (𝑁 ∘ − ))
10		fveq2 6756	. . . . 5 ⊢ (𝑔 = 𝐺 → (dist‘𝑔) = (dist‘𝐺))
11		isngp.d	. . . . 5 ⊢ 𝐷 = (dist‘𝐺)
12	10, 11	eqtr4di 2797	. . . 4 ⊢ (𝑔 = 𝐺 → (dist‘𝑔) = 𝐷)
13	9, 12	sseq12d 3950	. . 3 ⊢ (𝑔 = 𝐺 → (((norm‘𝑔) ∘ (-g‘𝑔)) ⊆ (dist‘𝑔) ↔ (𝑁 ∘ − ) ⊆ 𝐷))
14		df-ngp 23645	. . 3 ⊢ NrmGrp = {𝑔 ∈ (Grp ∩ MetSp) ∣ ((norm‘𝑔) ∘ (-g‘𝑔)) ⊆ (dist‘𝑔)}
15	13, 14	elrab2 3620	. 2 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ (Grp ∩ MetSp) ∧ (𝑁 ∘ − ) ⊆ 𝐷))
16		df-3an 1087	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ∘ − ) ⊆ 𝐷) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp) ∧ (𝑁 ∘ − ) ⊆ 𝐷))
17	2, 15, 16	3bitr4i 302	1 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ∘ − ) ⊆ 𝐷))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ∩ cin 3882   ⊆ wss 3883   ∘ ccom 5584  ‘cfv 6418  distcds 16897  Grpcgrp 18492  -_gcsg 18494  MetSpcms 23379  normcnm 23638  NrmGrpcngp 23639

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-co 5589  df-iota 6376  df-fv 6426  df-ngp 23645

This theorem is referenced by:  isngp2  23659  ngpgrp  23661  ngpms  23662  tngngp2  23722  cnngp  23849  zhmnrg  31817