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Theorem ngptps 23217
 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 23215 . 2 (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)
2 mstps 23071 . 2 (𝐺 ∈ MetSp → 𝐺 ∈ TopSp)
31, 2syl 17 1 (𝐺 ∈ NrmGrp → 𝐺 ∈ TopSp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2115  TopSpctps 21546  MetSpcms 22934  NrmGrpcngp 23193 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3142  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5054  df-opab 5116  df-xp 5549  df-co 5552  df-res 5555  df-iota 6304  df-fv 6353  df-xms 22936  df-ms 22937  df-ngp 23199 This theorem is referenced by:  nmcn  23458  cnmpt1ip  23860  cnmpt2ip  23861  csscld  23862  clsocv  23863  rrxtps  42881
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