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Theorem isngp 22310
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 3774 . . 3 (𝐺 ∈ (Grp ∩ MetSp) ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp))
21anbi1i 730 . 2 ((𝐺 ∈ (Grp ∩ MetSp) ∧ (𝑁 ) ⊆ 𝐷) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp) ∧ (𝑁 ) ⊆ 𝐷))
3 fveq2 6148 . . . . . 6 (𝑔 = 𝐺 → (norm‘𝑔) = (norm‘𝐺))
4 isngp.n . . . . . 6 𝑁 = (norm‘𝐺)
53, 4syl6eqr 2673 . . . . 5 (𝑔 = 𝐺 → (norm‘𝑔) = 𝑁)
6 fveq2 6148 . . . . . 6 (𝑔 = 𝐺 → (-g𝑔) = (-g𝐺))
7 isngp.z . . . . . 6 = (-g𝐺)
86, 7syl6eqr 2673 . . . . 5 (𝑔 = 𝐺 → (-g𝑔) = )
95, 8coeq12d 5246 . . . 4 (𝑔 = 𝐺 → ((norm‘𝑔) ∘ (-g𝑔)) = (𝑁 ))
10 fveq2 6148 . . . . 5 (𝑔 = 𝐺 → (dist‘𝑔) = (dist‘𝐺))
11 isngp.d . . . . 5 𝐷 = (dist‘𝐺)
1210, 11syl6eqr 2673 . . . 4 (𝑔 = 𝐺 → (dist‘𝑔) = 𝐷)
139, 12sseq12d 3613 . . 3 (𝑔 = 𝐺 → (((norm‘𝑔) ∘ (-g𝑔)) ⊆ (dist‘𝑔) ↔ (𝑁 ) ⊆ 𝐷))
14 df-ngp 22298 . . 3 NrmGrp = {𝑔 ∈ (Grp ∩ MetSp) ∣ ((norm‘𝑔) ∘ (-g𝑔)) ⊆ (dist‘𝑔)}
1513, 14elrab2 3348 . 2 (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ (Grp ∩ MetSp) ∧ (𝑁 ) ⊆ 𝐷))
16 df-3an 1038 . 2 ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ) ⊆ 𝐷) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp) ∧ (𝑁 ) ⊆ 𝐷))
172, 15, 163bitr4i 292 1 (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ) ⊆ 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  cin 3554  wss 3555  ccom 5078  cfv 5847  distcds 15871  Grpcgrp 17343  -gcsg 17345  MetSpcmt 22033  normcnm 22291  NrmGrpcngp 22292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-co 5083  df-iota 5810  df-fv 5855  df-ngp 22298
This theorem is referenced by:  isngp2  22311  ngpgrp  22313  ngpms  22314  tngngp2  22366  cnngp  22493  zhmnrg  29790
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