Theorem trgtmd2 24503

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 24502	. 2 ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopGrp)
2		tgptmd 24413	. 2 ⊢ (𝑅 ∈ TopGrp → 𝑅 ∈ TopMnd)
3	1, 2	syl 18	1 ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopMnd)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146  TopMndctmd 24404  TopGrpctgp 24405  TopRingctrg 24490

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-nul 5263

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6501  df-fv 6555  df-ov 7433  df-tgp 24407  df-trg 24494

This theorem is used by:  tdrgtmd  24510  qqhcn  34679