Theorem istdrg 23225

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 3899	. . 3 ⊢ (𝑅 ∈ (TopRing ∩ DivRing) ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing))
2	1	anbi1i 623	. 2 ⊢ ((𝑅 ∈ (TopRing ∩ DivRing) ∧ (𝑀 ↾s 𝑈) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 ↾s 𝑈) ∈ TopGrp))
3		fveq2 6756	. . . . . 6 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
4		istrg.1	. . . . . 6 ⊢ 𝑀 = (mulGrp‘𝑅)
5	3, 4	eqtr4di 2797	. . . . 5 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = 𝑀)
6		fveq2 6756	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
7		istdrg.1	. . . . . 6 ⊢ 𝑈 = (Unit‘𝑅)
8	6, 7	eqtr4di 2797	. . . . 5 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
9	5, 8	oveq12d 7273	. . . 4 ⊢ (𝑟 = 𝑅 → ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) = (𝑀 ↾s 𝑈))
10	9	eleq1d 2823	. . 3 ⊢ (𝑟 = 𝑅 → (((mulGrp‘𝑟) ↾s (Unit‘𝑟)) ∈ TopGrp ↔ (𝑀 ↾s 𝑈) ∈ TopGrp))
11		df-tdrg 23220	. . 3 ⊢ TopDRing = {𝑟 ∈ (TopRing ∩ DivRing) ∣ ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) ∈ TopGrp}
12	10, 11	elrab2 3620	. 2 ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ (TopRing ∩ DivRing) ∧ (𝑀 ↾s 𝑈) ∈ TopGrp))
13		df-3an 1087	. 2 ⊢ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s 𝑈) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 ↾s 𝑈) ∈ TopGrp))
14	2, 12, 13	3bitr4i 302	1 ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s 𝑈) ∈ TopGrp))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ∩ cin 3882  ‘cfv 6418  (class class class)co 7255   ↾_s cress 16867  mulGrpcmgp 19635  Unitcui 19796  DivRingcdr 19906  TopGrpctgp 23130  TopRingctrg 23215  TopDRingctdrg 23216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426  df-ov 7258  df-tdrg 23220

This theorem is referenced by:  tdrgunit  23226  tdrgtrg  23232  tdrgdrng  23233  istdrg2  23237  nrgtdrg  23763