Theorem tdrgdrng 24183

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 2736	. . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
2		eqid 2736	. . 3 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
3	1, 2	istdrg 24175	. 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 6560  (class class class)co 7432   ↾_s cress 17275  mulGrpcmgp 20138  Unitcui 20356  DivRingcdr 20730  TopGrpctgp 24080  TopRingctrg 24165  TopDRingctdrg 24166

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 2707

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 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-ov 7435  df-tdrg 24170

This theorem is referenced by:  tvclvec  24208