Theorem trgring 24106

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 2733	. . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
2	1	istrg 24099	. 2 ⊢ (𝑅 ∈ TopRing ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ (mulGrp‘𝑅) ∈ TopMnd))
3	2	simp2bi 1146	1 ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∈ wcel 2113  ‘cfv 6489  mulGrpcmgp 20066  Ringcrg 20159  TopMndctmd 24005  TopGrpctgp 24006  TopRingctrg 24091

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-iota 6445  df-fv 6497  df-trg 24095

This theorem is referenced by:  trggrp  24107  tdrgring  24110