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Theorem isngp 23132
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 = (-g𝐺)
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.
StepHypRef Expression
1 elin 4166 . . 3 (𝐺 ∈ (Grp ∩ MetSp) ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp))
21anbi1i 623 . 2 ((𝐺 ∈ (Grp ∩ MetSp) ∧ (𝑁 ) ⊆ 𝐷) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp) ∧ (𝑁 ) ⊆ 𝐷))
3 fveq2 6663 . . . . . 6 (𝑔 = 𝐺 → (norm‘𝑔) = (norm‘𝐺))
4 isngp.n . . . . . 6 𝑁 = (norm‘𝐺)
53, 4syl6eqr 2871 . . . . 5 (𝑔 = 𝐺 → (norm‘𝑔) = 𝑁)
6 fveq2 6663 . . . . . 6 (𝑔 = 𝐺 → (-g𝑔) = (-g𝐺))
7 isngp.z . . . . . 6 = (-g𝐺)
86, 7syl6eqr 2871 . . . . 5 (𝑔 = 𝐺 → (-g𝑔) = )
95, 8coeq12d 5728 . . . 4 (𝑔 = 𝐺 → ((norm‘𝑔) ∘ (-g𝑔)) = (𝑁 ))
10 fveq2 6663 . . . . 5 (𝑔 = 𝐺 → (dist‘𝑔) = (dist‘𝐺))
11 isngp.d . . . . 5 𝐷 = (dist‘𝐺)
1210, 11syl6eqr 2871 . . . 4 (𝑔 = 𝐺 → (dist‘𝑔) = 𝐷)
139, 12sseq12d 3997 . . 3 (𝑔 = 𝐺 → (((norm‘𝑔) ∘ (-g𝑔)) ⊆ (dist‘𝑔) ↔ (𝑁 ) ⊆ 𝐷))
14 df-ngp 23120 . . 3 NrmGrp = {𝑔 ∈ (Grp ∩ MetSp) ∣ ((norm‘𝑔) ∘ (-g𝑔)) ⊆ (dist‘𝑔)}
1513, 14elrab2 3680 . 2 (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ (Grp ∩ MetSp) ∧ (𝑁 ) ⊆ 𝐷))
16 df-3an 1081 . 2 ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ) ⊆ 𝐷) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp) ∧ (𝑁 ) ⊆ 𝐷))
172, 15, 163bitr4i 304 1 (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ) ⊆ 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  cin 3932  wss 3933  ccom 5552  cfv 6348  distcds 16562  Grpcgrp 18041  -gcsg 18043  MetSpcms 22855  normcnm 23113  NrmGrpcngp 23114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-co 5557  df-iota 6307  df-fv 6356  df-ngp 23120
This theorem is referenced by:  isngp2  23133  ngpgrp  23135  ngpms  23136  tngngp2  23188  cnngp  23315  zhmnrg  31107
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