Theorem trgtmd2 24112

Description: A topological ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)

Assertion

Ref	Expression
trgtmd2	⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopMnd)

Proof of Theorem trgtmd2

Step	Hyp	Ref	Expression
1		trgtgp 24111	. 2 ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopGrp)
2		tgptmd 24022	. 2 ⊢ (𝑅 ∈ TopGrp → 𝑅 ∈ TopMnd)
3	1, 2	syl 17	1 ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopMnd)

Syntax hints:   → wi 4   ∈ wcel 2109  TopMndctmd 24013  TopGrpctgp 24014  TopRingctrg 24099

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 2708  ax-nul 5281

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 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-rab 3421  df-v 3466  df-sbc 3771  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-iota 6489  df-fv 6544  df-ov 7413  df-tgp 24016  df-trg 24103

This theorem is referenced by:  tdrgtmd  24119  qqhcn  34027