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Theorem isngp 24625
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 3979 . . 3 (𝐺 ∈ (Grp ∩ MetSp) ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp))
21anbi1i 624 . 2 ((𝐺 ∈ (Grp ∩ MetSp) ∧ (𝑁 ) ⊆ 𝐷) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp) ∧ (𝑁 ) ⊆ 𝐷))
3 fveq2 6907 . . . . . 6 (𝑔 = 𝐺 → (norm‘𝑔) = (norm‘𝐺))
4 isngp.n . . . . . 6 𝑁 = (norm‘𝐺)
53, 4eqtr4di 2793 . . . . 5 (𝑔 = 𝐺 → (norm‘𝑔) = 𝑁)
6 fveq2 6907 . . . . . 6 (𝑔 = 𝐺 → (-g𝑔) = (-g𝐺))
7 isngp.z . . . . . 6 = (-g𝐺)
86, 7eqtr4di 2793 . . . . 5 (𝑔 = 𝐺 → (-g𝑔) = )
95, 8coeq12d 5878 . . . 4 (𝑔 = 𝐺 → ((norm‘𝑔) ∘ (-g𝑔)) = (𝑁 ))
10 fveq2 6907 . . . . 5 (𝑔 = 𝐺 → (dist‘𝑔) = (dist‘𝐺))
11 isngp.d . . . . 5 𝐷 = (dist‘𝐺)
1210, 11eqtr4di 2793 . . . 4 (𝑔 = 𝐺 → (dist‘𝑔) = 𝐷)
139, 12sseq12d 4029 . . 3 (𝑔 = 𝐺 → (((norm‘𝑔) ∘ (-g𝑔)) ⊆ (dist‘𝑔) ↔ (𝑁 ) ⊆ 𝐷))
14 df-ngp 24612 . . 3 NrmGrp = {𝑔 ∈ (Grp ∩ MetSp) ∣ ((norm‘𝑔) ∘ (-g𝑔)) ⊆ (dist‘𝑔)}
1513, 14elrab2 3698 . 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))
172, 15, 163bitr4i 303 1 (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ) ⊆ 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  cin 3962  wss 3963  ccom 5693  cfv 6563  distcds 17307  Grpcgrp 18964  -gcsg 18966  MetSpcms 24344  normcnm 24605  NrmGrpcngp 24606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-co 5698  df-iota 6516  df-fv 6571  df-ngp 24612
This theorem is referenced by:  isngp2  24626  ngpgrp  24628  ngpms  24629  tngngp2  24689  cnngp  24816  zhmnrg  33928
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