Theorem isngp 24491

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 3933	. . 3 ⊢ (𝐺 ∈ (Grp ∩ MetSp) ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp))
2	1	anbi1i 624	. 2 ⊢ ((𝐺 ∈ (Grp ∩ MetSp) ∧ (𝑁 ∘ − ) ⊆ 𝐷) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp) ∧ (𝑁 ∘ − ) ⊆ 𝐷))
3		fveq2 6861	. . . . . 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 6861	. . . . . 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 5831	. . . 4 ⊢ (𝑔 = 𝐺 → ((norm‘𝑔) ∘ (-g‘𝑔)) = (𝑁 ∘ − ))
10		fveq2 6861	. . . . 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 3983	. . 3 ⊢ (𝑔 = 𝐺 → (((norm‘𝑔) ∘ (-g‘𝑔)) ⊆ (dist‘𝑔) ↔ (𝑁 ∘ − ) ⊆ 𝐷))
14		df-ngp 24478	. . 3 ⊢ NrmGrp = {𝑔 ∈ (Grp ∩ MetSp) ∣ ((norm‘𝑔) ∘ (-g‘𝑔)) ⊆ (dist‘𝑔)}
15	13, 14	elrab2 3665	. 2 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ (Grp ∩ MetSp) ∧ (𝑁 ∘ − ) ⊆ 𝐷))
16		df-3an 1088	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ∘ − ) ⊆ 𝐷) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp) ∧ (𝑁 ∘ − ) ⊆ 𝐷))
17	2, 15, 16	3bitr4i 303	1 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ∘ − ) ⊆ 𝐷))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ∩ cin 3916   ⊆ wss 3917   ∘ ccom 5645  ‘cfv 6514  distcds 17236  Grpcgrp 18872  -_gcsg 18874  MetSpcms 24213  normcnm 24471  NrmGrpcngp 24472

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-co 5650  df-iota 6467  df-fv 6522  df-ngp 24478

This theorem is referenced by:  isngp2  24492  ngpgrp  24494  ngpms  24495  tngngp2  24547  cnngp  24674  zhmnrg  33962