Theorem trgtmd2 24093

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 24092	. 2 ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopGrp)
2		tgptmd 24003	. 2 ⊢ (𝑅 ∈ TopGrp → 𝑅 ∈ TopMnd)
3	1, 2	syl 17	1 ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopMnd)

Syntax hints:   → wi 4   ∈ wcel 2098  TopMndctmd 23994  TopGrpctgp 23995  TopRingctrg 24080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-nul 5310

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-ov 7429  df-tgp 23997  df-trg 24084

This theorem is referenced by:  tdrgtmd  24100  qqhcn  33625