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Theorem trgtps 22350
 Description: A topological ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
Assertion
Ref Expression
trgtps (𝑅 ∈ TopRing → 𝑅 ∈ TopSp)

Proof of Theorem trgtps
StepHypRef Expression
1 trgtgp 22348 . 2 (𝑅 ∈ TopRing → 𝑅 ∈ TopGrp)
2 tgptps 22261 . 2 (𝑅 ∈ TopGrp → 𝑅 ∈ TopSp)
31, 2syl 17 1 (𝑅 ∈ TopRing → 𝑅 ∈ TopSp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2164  TopSpctps 21114  TopGrpctgp 22252  TopRingctrg 22336 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-nul 5015 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-iota 6090  df-fv 6135  df-ov 6913  df-tmd 22253  df-tgp 22254  df-trg 22340 This theorem is referenced by:  tdrgtps  22357  tlmscatps  22371
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