Theorem tdrgring 24062

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

Assertion

Ref	Expression
tdrgring	⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ Ring)

Proof of Theorem tdrgring

Step	Hyp	Ref	Expression
1		tdrgtrg 24060	. 2 ⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ TopRing)
2		trgring 24058	. 2 ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ Ring)
3	1, 2	syl 17	1 ⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∈ wcel 2109  Ringcrg 20142  TopRingctrg 24043  TopDRingctdrg 24044

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-iota 6464  df-fv 6519  df-ov 7390  df-trg 24047  df-tdrg 24048

This theorem is referenced by: (None)