Theorem trgtgp 24176

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

Assertion

Ref	Expression
trgtgp	⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopGrp)

Proof of Theorem trgtgp

Step	Hyp	Ref	Expression
1		eqid 2737	. . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
2	1	istrg 24172	. 2 ⊢ (𝑅 ∈ TopRing ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ (mulGrp‘𝑅) ∈ TopMnd))
3	2	simp1bi 1146	1 ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopGrp)

Syntax hints:   → wi 4   ∈ wcel 2108  ‘cfv 6561  mulGrpcmgp 20137  Ringcrg 20230  TopMndctmd 24078  TopGrpctgp 24079  TopRingctrg 24164

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 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-trg 24168

This theorem is referenced by:  trgtmd2  24177  trgtps  24178  pl1cn  33954