Theorem trgtmd 24101

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

Hypothesis

Ref	Expression
istrg.1	⊢ 𝑀 = (mulGrp‘𝑅)

Assertion

Ref	Expression
trgtmd	⊢ (𝑅 ∈ TopRing → 𝑀 ∈ TopMnd)

Proof of Theorem trgtmd

Step	Hyp	Ref	Expression
1		istrg.1	. . 3 ⊢ 𝑀 = (mulGrp‘𝑅)
2	1	istrg 24100	. 2 ⊢ (𝑅 ∈ TopRing ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ TopMnd))
3	2	simp3bi 1147	1 ⊢ (𝑅 ∈ TopRing → 𝑀 ∈ TopMnd)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  ‘cfv 6530  mulGrpcmgp 20098  Ringcrg 20191  TopMndctmd 24006  TopGrpctgp 24007  TopRingctrg 24092

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707

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 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-iota 6483  df-fv 6538  df-trg 24096

This theorem is referenced by:  mulrcn  24115  cnmpt1mulr  24118  cnmpt2mulr  24119  nrgtdrg  24630  iistmd  33879