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Theorem ngptps 24631
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 24629 . 2 (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)
2 mstps 24481 . 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 22954  MetSpcms 24344  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-xp 5695  df-co 5698  df-res 5701  df-iota 6516  df-fv 6571  df-xms 24346  df-ms 24347  df-ngp 24612
This theorem is referenced by:  nmcn  24880  cnmpt1ip  25295  cnmpt2ip  25296  csscld  25297  clsocv  25298  rrxtps  46242
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