Theorem tdrgdrng 24097

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

Assertion

Ref	Expression
tdrgdrng	⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ DivRing)

Proof of Theorem tdrgdrng

Step	Hyp	Ref	Expression
1		eqid 2734	. . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
2		eqid 2734	. . 3 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
3	1, 2	istdrg 24089	. 2 ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) ∈ TopGrp))
4	3	simp2bi 1146	1 ⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ DivRing)

Syntax hints:   → wi 4   ∈ wcel 2107  ‘cfv 6527  (class class class)co 7399   ↾_s cress 17236  mulGrpcmgp 20085  Unitcui 20300  DivRingcdr 20674  TopGrpctgp 23994  TopRingctrg 24079  TopDRingctdrg 24080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-rab 3414  df-v 3459  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4881  df-br 5117  df-iota 6480  df-fv 6535  df-ov 7402  df-tdrg 24084

This theorem is referenced by:  tvclvec  24122