Theorem trgtps 24118

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 24116	. 2 ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopGrp)
2		tgptps 24028	. 2 ⊢ (𝑅 ∈ TopGrp → 𝑅 ∈ TopSp)
3	1, 2	syl 17	1 ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopSp)

Syntax hints:   → wi 4   ∈ wcel 2114  TopSpctps 22880  TopGrpctgp 24019  TopRingctrg 24104

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5252

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-rab 3401  df-v 3443  df-sbc 3742  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-iota 6449  df-fv 6501  df-ov 7363  df-tmd 24020  df-tgp 24021  df-trg 24108

This theorem is referenced by:  tdrgtps  24125  tlmscatps  24139