Theorem trgtgp 23671

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 2732	. . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
2	1	istrg 23667	. 2 ⊢ (𝑅 ∈ TopRing ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ (mulGrp‘𝑅) ∈ TopMnd))
3	2	simp1bi 1145	1 ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopGrp)

Syntax hints:   → wi 4   ∈ wcel 2106  ‘cfv 6543  mulGrpcmgp 19986  Ringcrg 20055  TopMndctmd 23573  TopGrpctgp 23574  TopRingctrg 23659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-trg 23663

This theorem is referenced by:  trgtmd2  23672  trgtps  23673  pl1cn  32930