Theorem tdrgtrg 24088

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

Assertion

Ref	Expression
tdrgtrg	⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ TopRing)

Proof of Theorem tdrgtrg

Step	Hyp	Ref	Expression
1		eqid 2731	. . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
2		eqid 2731	. . 3 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
3	1, 2	istdrg 24081	. 2 ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) ∈ TopGrp))
4	3	simp1bi 1145	1 ⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ TopRing)

Syntax hints:   → wi 4   ∈ wcel 2111  ‘cfv 6481  (class class class)co 7346   ↾_s cress 17141  mulGrpcmgp 20058  Unitcui 20273  DivRingcdr 20644  TopGrpctgp 23986  TopRingctrg 24071  TopDRingctdrg 24072

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 2113  ax-9 2121  ax-ext 2703

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 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fv 6489  df-ov 7349  df-tdrg 24076

This theorem is referenced by:  tdrgring  24090  tdrgtmd  24091  tdrgtps  24092  dvrcn  24099