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Theorem ngptps 22453
Description: A normed group is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
Assertion
Ref Expression
ngptps (𝐺 ∈ NrmGrp → 𝐺 ∈ TopSp)

Proof of Theorem ngptps
StepHypRef Expression
1 ngpms 22451 . 2 (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)
2 mstps 22307 . 2 (𝐺 ∈ MetSp → 𝐺 ∈ TopSp)
31, 2syl 17 1 (𝐺 ∈ NrmGrp → 𝐺 ∈ TopSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2030  TopSpctps 20784  MetSpcmt 22170  NrmGrpcngp 22429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-co 5152  df-res 5155  df-iota 5889  df-fv 5934  df-xms 22172  df-ms 22173  df-ngp 22435
This theorem is referenced by:  nmcn  22694  cnmpt1ip  23092  cnmpt2ip  23093  csscld  23094  clsocv  23095  rrxtps  40822
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