Theorem trgtgp 24467

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 2761	. . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
2	1	istrg 24463	. 2 ⊢ (𝑅 ∈ TopRing ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ (mulGrp‘𝑅) ∈ TopMnd))
3	2	simp1bi 1163	1 ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopGrp)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  ‘cfv 6531  mulGrpcmgp 20340  Ringcrg 20439  TopMndctmd 24369  TopGrpctgp 24370  TopRingctrg 24455

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6487  df-fv 6539  df-trg 24459

This theorem is used by:  trgtmd2  24468  trgtps  24469  pl1cn  34569