Theorem tdrgtrg 24492

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 2760	. . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
2		eqid 2760	. . 3 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
3	1, 2	istdrg 24485	. 2 ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) ∈ TopGrp))
4	3	simp1bi 1163	1 ⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ TopRing)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144  ‘cfv 6538  (class class class)co 7420   ↾_s cress 17408  mulGrpcmgp 20360  Unitcui 20585  DivRingcdr 20980  TopGrpctgp 24390  TopRingctrg 24475  TopDRingctdrg 24476

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6546  df-ov 7423  df-tdrg 24480

This theorem is used by:  tdrgring  24494  tdrgtmd  24495  tdrgtps  24496  dvrcn  24503