Theorem isngp 24764

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 3920	. . 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 2815	. . . . 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 2815	. . . . 5 ⊢ (𝑔 = 𝐺 → (-g‘𝑔) = − )
9	5, 8	coeq12d 5849	. . . 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 2815	. . . 4 ⊢ (𝑔 = 𝐺 → (dist‘𝑔) = 𝐷)
13	9, 12	sseq12d 3969	. . 3 ⊢ (𝑔 = 𝐺 → (((norm‘𝑔) ∘ (-g‘𝑔)) ⊆ (dist‘𝑔) ↔ (𝑁 ∘ − ) ⊆ 𝐷))
14		df-ngp 24751	. . 3 ⊢ NrmGrp = {𝑔 ∈ (Grp ∩ MetSp) ∣ ((norm‘𝑔) ∘ (-g‘𝑔)) ⊆ (dist‘𝑔)}
15	13, 14	elrab2 3653	. 2 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ (Grp ∩ MetSp) ∧ (𝑁 ∘ − ) ⊆ 𝐷))
16		df-3an 1104	. 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))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142   ∩ cin 3903   ⊆ wss 3904   ∘ ccom 5664  ‘cfv 6536  distcds 17325  Grpcgrp 19006  -_gcsg 19008  MetSpcms 24486  normcnm 24744  NrmGrpcngp 24745

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-co 5669  df-iota 6492  df-fv 6544  df-ngp 24751

This theorem is used by:  isngp2  24765  ngpgrp  24767  ngpms  24768  tngngp2  24820  cnngp  24947  zhmnrg  34364