Theorem trgring 24180

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 2736	. . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
2	1	istrg 24173	. 2 ⊢ (𝑅 ∈ TopRing ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ (mulGrp‘𝑅) ∈ TopMnd))
3	2	simp2bi 1146	1 ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∈ wcel 2107  ‘cfv 6560  mulGrpcmgp 20138  Ringcrg 20231  TopMndctmd 24079  TopGrpctgp 24080  TopRingctrg 24165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-trg 24169

This theorem is referenced by:  trggrp  24181  tdrgring  24184