Theorem trgtmd 24490

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 24489	. 2 ⊢ (𝑅 ∈ TopRing ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ TopMnd))
3	2	simp3bi 1165	1 ⊢ (𝑅 ∈ TopRing → 𝑀 ∈ TopMnd)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  ‘cfv 6539  mulGrpcmgp 20366  Ringcrg 20465  TopMndctmd 24395  TopGrpctgp 24396  TopRingctrg 24481

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 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

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 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6547  df-trg 24485

This theorem is used by:  mulrcn  24504  cnmpt1mulr  24507  cnmpt2mulr  24508  nrgtdrg  25018  iistmd  34581