Theorem tdrgtrg 22783

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 2823	. . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
2		eqid 2823	. . 3 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
3	1, 2	istdrg 22776	. 2 ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) ∈ TopGrp))
4	3	simp1bi 1141	1 ⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ TopRing)

Syntax hints:   → wi 4   ∈ wcel 2114  ‘cfv 6357  (class class class)co 7158   ↾_s cress 16486  mulGrpcmgp 19241  Unitcui 19391  DivRingcdr 19504  TopGrpctgp 22681  TopRingctrg 22766  TopDRingctdrg 22767

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-iota 6316  df-fv 6365  df-ov 7161  df-tdrg 22771

This theorem is referenced by:  tdrgring  22785  tdrgtmd  22786  tdrgtps  22787  dvrcn  22794