Theorem istdrg 24398

Description: Express the predicate "𝑅 is a topological ring". (Contributed by Mario Carneiro, 5-Oct-2015.)

Hypotheses

Ref	Expression
istrg.1	⊢ 𝑀 = (mulGrp‘𝑅)
istdrg.1	⊢ 𝑈 = (Unit‘𝑅)

Assertion

Ref	Expression
istdrg	⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s 𝑈) ∈ TopGrp))

Proof of Theorem istdrg

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3918	. . 3 ⊢ (𝑅 ∈ (TopRing ∩ DivRing) ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing))
2	1	anbi1i 636	. 2 ⊢ ((𝑅 ∈ (TopRing ∩ DivRing) ∧ (𝑀 ↾s 𝑈) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 ↾s 𝑈) ∈ TopGrp))
3		fveq2 6882	. . . . . 6 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
4		istrg.1	. . . . . 6 ⊢ 𝑀 = (mulGrp‘𝑅)
5	3, 4	eqtr4di 2815	. . . . 5 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = 𝑀)
6		fveq2 6882	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
7		istdrg.1	. . . . . 6 ⊢ 𝑈 = (Unit‘𝑅)
8	6, 7	eqtr4di 2815	. . . . 5 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
9	5, 8	oveq12d 7435	. . . 4 ⊢ (𝑟 = 𝑅 → ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) = (𝑀 ↾s 𝑈))
10	9	eleq1d 2847	. . 3 ⊢ (𝑟 = 𝑅 → (((mulGrp‘𝑟) ↾s (Unit‘𝑟)) ∈ TopGrp ↔ (𝑀 ↾s 𝑈) ∈ TopGrp))
11		df-tdrg 24393	. . 3 ⊢ TopDRing = {𝑟 ∈ (TopRing ∩ DivRing) ∣ ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) ∈ TopGrp}
12	10, 11	elrab2 3652	. 2 ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ (TopRing ∩ DivRing) ∧ (𝑀 ↾s 𝑈) ∈ TopGrp))
13		df-3an 1105	. 2 ⊢ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s 𝑈) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 ↾s 𝑈) ∈ TopGrp))
14	2, 12, 13	3bitr4i 306	1 ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s 𝑈) ∈ TopGrp))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ∩ cin 3901  ‘cfv 6537  (class class class)co 7417   ↾_s cress 17328  mulGrpcmgp 20279  Unitcui 20502  DivRingcdr 20896  TopGrpctgp 24303  TopRingctrg 24388  TopDRingctdrg 24389

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 2734

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 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-iota 6493  df-fv 6545  df-ov 7420  df-tdrg 24393

This theorem is used by:  tdrgunit  24399  tdrgtrg  24405  tdrgdrng  24406  istdrg2  24410  nrgtdrg  24925