Theorem istdrg 22774

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 4169	. . 3 ⊢ (𝑅 ∈ (TopRing ∩ DivRing) ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing))
2	1	anbi1i 625	. 2 ⊢ ((𝑅 ∈ (TopRing ∩ DivRing) ∧ (𝑀 ↾s 𝑈) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 ↾s 𝑈) ∈ TopGrp))
3		fveq2 6670	. . . . . 6 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
4		istrg.1	. . . . . 6 ⊢ 𝑀 = (mulGrp‘𝑅)
5	3, 4	syl6eqr 2874	. . . . 5 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = 𝑀)
6		fveq2 6670	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
7		istdrg.1	. . . . . 6 ⊢ 𝑈 = (Unit‘𝑅)
8	6, 7	syl6eqr 2874	. . . . 5 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
9	5, 8	oveq12d 7174	. . . 4 ⊢ (𝑟 = 𝑅 → ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) = (𝑀 ↾s 𝑈))
10	9	eleq1d 2897	. . 3 ⊢ (𝑟 = 𝑅 → (((mulGrp‘𝑟) ↾s (Unit‘𝑟)) ∈ TopGrp ↔ (𝑀 ↾s 𝑈) ∈ TopGrp))
11		df-tdrg 22769	. . 3 ⊢ TopDRing = {𝑟 ∈ (TopRing ∩ DivRing) ∣ ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) ∈ TopGrp}
12	10, 11	elrab2 3683	. 2 ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ (TopRing ∩ DivRing) ∧ (𝑀 ↾s 𝑈) ∈ TopGrp))
13		df-3an 1085	. 2 ⊢ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s 𝑈) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 ↾s 𝑈) ∈ TopGrp))
14	2, 12, 13	3bitr4i 305	1 ⊢ (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 ↾s 𝑈) ∈ TopGrp))

Syntax hints:   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ∩ cin 3935  ‘cfv 6355  (class class class)co 7156   ↾_s cress 16484  mulGrpcmgp 19239  Unitcui 19389  DivRingcdr 19502  TopGrpctgp 22679  TopRingctrg 22764  TopDRingctdrg 22765

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-iota 6314  df-fv 6363  df-ov 7159  df-tdrg 22769

This theorem is referenced by:  tdrgunit  22775  tdrgtrg  22781  tdrgdrng  22782  istdrg2  22786  nrgtdrg  23302