Theorem isngp 24549

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 3960	. . 3 ⊢ (𝐺 ∈ (Grp ∩ MetSp) ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp))
2	1	anbi1i 622	. 2 ⊢ ((𝐺 ∈ (Grp ∩ MetSp) ∧ (𝑁 ∘ − ) ⊆ 𝐷) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp) ∧ (𝑁 ∘ − ) ⊆ 𝐷))
3		fveq2 6896	. . . . . 6 ⊢ (𝑔 = 𝐺 → (norm‘𝑔) = (norm‘𝐺))
4		isngp.n	. . . . . 6 ⊢ 𝑁 = (norm‘𝐺)
5	3, 4	eqtr4di 2783	. . . . 5 ⊢ (𝑔 = 𝐺 → (norm‘𝑔) = 𝑁)
6		fveq2 6896	. . . . . 6 ⊢ (𝑔 = 𝐺 → (-g‘𝑔) = (-g‘𝐺))
7		isngp.z	. . . . . 6 ⊢ − = (-g‘𝐺)
8	6, 7	eqtr4di 2783	. . . . 5 ⊢ (𝑔 = 𝐺 → (-g‘𝑔) = − )
9	5, 8	coeq12d 5867	. . . 4 ⊢ (𝑔 = 𝐺 → ((norm‘𝑔) ∘ (-g‘𝑔)) = (𝑁 ∘ − ))
10		fveq2 6896	. . . . 5 ⊢ (𝑔 = 𝐺 → (dist‘𝑔) = (dist‘𝐺))
11		isngp.d	. . . . 5 ⊢ 𝐷 = (dist‘𝐺)
12	10, 11	eqtr4di 2783	. . . 4 ⊢ (𝑔 = 𝐺 → (dist‘𝑔) = 𝐷)
13	9, 12	sseq12d 4010	. . 3 ⊢ (𝑔 = 𝐺 → (((norm‘𝑔) ∘ (-g‘𝑔)) ⊆ (dist‘𝑔) ↔ (𝑁 ∘ − ) ⊆ 𝐷))
14		df-ngp 24536	. . 3 ⊢ NrmGrp = {𝑔 ∈ (Grp ∩ MetSp) ∣ ((norm‘𝑔) ∘ (-g‘𝑔)) ⊆ (dist‘𝑔)}
15	13, 14	elrab2 3682	. 2 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ (Grp ∩ MetSp) ∧ (𝑁 ∘ − ) ⊆ 𝐷))
16		df-3an 1086	. 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ∩ cin 3943   ⊆ wss 3944   ∘ ccom 5682  ‘cfv 6549  distcds 17245  Grpcgrp 18898  -_gcsg 18900  MetSpcms 24268  normcnm 24529  NrmGrpcngp 24530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-co 5687  df-iota 6501  df-fv 6557  df-ngp 24536

This theorem is referenced by:  isngp2  24550  ngpgrp  24552  ngpms  24553  tngngp2  24613  cnngp  24740  zhmnrg  33696