Theorem trgtps 24087

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

Step	Hyp	Ref	Expression
1		trgtgp 24085	. 2 ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopGrp)
2		tgptps 23997	. 2 ⊢ (𝑅 ∈ TopGrp → 𝑅 ∈ TopSp)
3	1, 2	syl 17	1 ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopSp)

Syntax hints:   → wi 4   ∈ wcel 2099  TopSpctps 22847  TopGrpctgp 23988  TopRingctrg 24073

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-nul 5306

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-iota 6500  df-fv 6556  df-ov 7423  df-tmd 23989  df-tgp 23990  df-trg 24077

This theorem is referenced by:  tdrgtps  24094  tlmscatps  24108