Theorem trgtps 24063

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 24061	. 2 ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopGrp)
2		tgptps 23973	. 2 ⊢ (𝑅 ∈ TopGrp → 𝑅 ∈ TopSp)
3	1, 2	syl 17	1 ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopSp)

Syntax hints:   → wi 4   ∈ wcel 2109  TopSpctps 22825  TopGrpctgp 23964  TopRingctrg 24049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-nul 5263

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-rab 3409  df-v 3452  df-sbc 3756  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-iota 6466  df-fv 6521  df-ov 7392  df-tmd 23965  df-tgp 23966  df-trg 24053

This theorem is referenced by:  tdrgtps  24070  tlmscatps  24084