Theorem trgtgp 23319

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 2738	. . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
2	1	istrg 23315	. 2 ⊢ (𝑅 ∈ TopRing ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ (mulGrp‘𝑅) ∈ TopMnd))
3	2	simp1bi 1144	1 ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopGrp)

Syntax hints:   → wi 4   ∈ wcel 2106  ‘cfv 6433  mulGrpcmgp 19720  Ringcrg 19783  TopMndctmd 23221  TopGrpctgp 23222  TopRingctrg 23307

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-trg 23311

This theorem is referenced by:  trgtmd2  23320  trgtps  23321  pl1cn  31905