Theorem tdrgdrng 24432

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 2760	. . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
2		eqid 2760	. . 3 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
3	1, 2	istdrg 24424	. 2 ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) ∈ TopGrp))
4	3	simp2bi 1164	1 ⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ DivRing)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  ‘cfv 6535  (class class class)co 7416   ↾_s cress 17347  mulGrpcmgp 20299  Unitcui 20524  DivRingcdr 20919  TopGrpctgp 24329  TopRingctrg 24414  TopDRingctdrg 24415

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6491  df-fv 6543  df-ov 7419  df-tdrg 24419

This theorem is used by:  tvclvec  24457