Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  isngp Structured version   Visualization version   GIF version

Theorem isngp 22210
 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 3758 . . 3 (𝐺 ∈ (Grp ∩ MetSp) ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp))
21anbi1i 727 . 2 ((𝐺 ∈ (Grp ∩ MetSp) ∧ (𝑁 ) ⊆ 𝐷) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp) ∧ (𝑁 ) ⊆ 𝐷))
3 fveq2 6103 . . . . . 6 (𝑔 = 𝐺 → (norm‘𝑔) = (norm‘𝐺))
4 isngp.n . . . . . 6 𝑁 = (norm‘𝐺)
53, 4syl6eqr 2662 . . . . 5 (𝑔 = 𝐺 → (norm‘𝑔) = 𝑁)
6 fveq2 6103 . . . . . 6 (𝑔 = 𝐺 → (-g𝑔) = (-g𝐺))
7 isngp.z . . . . . 6 = (-g𝐺)
86, 7syl6eqr 2662 . . . . 5 (𝑔 = 𝐺 → (-g𝑔) = )
95, 8coeq12d 5208 . . . 4 (𝑔 = 𝐺 → ((norm‘𝑔) ∘ (-g𝑔)) = (𝑁 ))
10 fveq2 6103 . . . . 5 (𝑔 = 𝐺 → (dist‘𝑔) = (dist‘𝐺))
11 isngp.d . . . . 5 𝐷 = (dist‘𝐺)
1210, 11syl6eqr 2662 . . . 4 (𝑔 = 𝐺 → (dist‘𝑔) = 𝐷)
139, 12sseq12d 3597 . . 3 (𝑔 = 𝐺 → (((norm‘𝑔) ∘ (-g𝑔)) ⊆ (dist‘𝑔) ↔ (𝑁 ) ⊆ 𝐷))
14 df-ngp 22198 . . 3 NrmGrp = {𝑔 ∈ (Grp ∩ MetSp) ∣ ((norm‘𝑔) ∘ (-g𝑔)) ⊆ (dist‘𝑔)}
1513, 14elrab2 3333 . 2 (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ (Grp ∩ MetSp) ∧ (𝑁 ) ⊆ 𝐷))
16 df-3an 1033 . 2 ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ) ⊆ 𝐷) ↔ ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp) ∧ (𝑁 ) ⊆ 𝐷))
172, 15, 163bitr4i 291 1 (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ) ⊆ 𝐷))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ∩ cin 3539   ⊆ wss 3540   ∘ ccom 5042  ‘cfv 5804  distcds 15777  Grpcgrp 17245  -gcsg 17247  MetSpcmt 21933  normcnm 22191  NrmGrpcngp 22192 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-co 5047  df-iota 5768  df-fv 5812  df-ngp 22198 This theorem is referenced by:  isngp2  22211  ngpgrp  22213  ngpms  22214  tngngp2  22266  cnngp  22393  zhmnrg  29339
 Copyright terms: Public domain W3C validator