Theorem trgring 24158

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

Assertion

Ref	Expression
trgring	⊢ (𝑅 ∈ TopRing → 𝑅 ∈ Ring)

Proof of Theorem trgring

Step	Hyp	Ref	Expression
1		eqid 2741	. . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
2	1	istrg 24151	. 2 ⊢ (𝑅 ∈ TopRing ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ (mulGrp‘𝑅) ∈ TopMnd))
3	2	simp2bi 1153	1 ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∈ wcel 2121  ‘cfv 6489  mulGrpcmgp 20116  Ringcrg 20209  TopMndctmd 24057  TopGrpctgp 24058  TopRingctrg 24143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-iota 6445  df-fv 6497  df-trg 24147

This theorem is referenced by:  trggrp  24159  tdrgring  24162