Theorem tdrgtrg 24119

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 2736	. . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
2		eqid 2736	. . 3 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
3	1, 2	istdrg 24112	. 2 ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ ((mulGrp‘𝑅) ↾s (Unit‘𝑅)) ∈ TopGrp))
4	3	simp1bi 1145	1 ⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ TopRing)

Syntax hints:   → wi 4   ∈ wcel 2113  ‘cfv 6492  (class class class)co 7358   ↾_s cress 17159  mulGrpcmgp 20077  Unitcui 20293  DivRingcdr 20664  TopGrpctgp 24017  TopRingctrg 24102  TopDRingctdrg 24103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-iota 6448  df-fv 6500  df-ov 7361  df-tdrg 24107

This theorem is referenced by:  tdrgring  24121  tdrgtmd  24122  tdrgtps  24123  dvrcn  24130