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Theorem trgtps 22773
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 22771 . 2 (𝑅 ∈ TopRing → 𝑅 ∈ TopGrp)
2 tgptps 22683 . 2 (𝑅 ∈ TopGrp → 𝑅 ∈ TopSp)
31, 2syl 17 1 (𝑅 ∈ TopRing → 𝑅 ∈ TopSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2114  TopSpctps 21535  TopGrpctgp 22674  TopRingctrg 22759
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-nul 5186
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 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-iota 6293  df-fv 6342  df-ov 7143  df-tmd 22675  df-tgp 22676  df-trg 22763
This theorem is referenced by:  tdrgtps  22780  tlmscatps  22794
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