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Theorem ngptps 22814
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 22812 . 2 (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)
2 mstps 22668 . 2 (𝐺 ∈ MetSp → 𝐺 ∈ TopSp)
31, 2syl 17 1 (𝐺 ∈ NrmGrp → 𝐺 ∈ TopSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  TopSpctps 21144  MetSpcms 22531  NrmGrpcngp 22790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-rex 3095  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-xp 5361  df-co 5364  df-res 5367  df-iota 6099  df-fv 6143  df-xms 22533  df-ms 22534  df-ngp 22796
This theorem is referenced by:  nmcn  23055  cnmpt1ip  23453  cnmpt2ip  23454  csscld  23455  clsocv  23456  rrxtps  41423
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