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Theorem istdrg 24174
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 ∧ (𝑀s 𝑈) ∈ TopGrp))

Proof of Theorem istdrg
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 elin 3967 . . 3 (𝑅 ∈ (TopRing ∩ DivRing) ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing))
21anbi1i 624 . 2 ((𝑅 ∈ (TopRing ∩ DivRing) ∧ (𝑀s 𝑈) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀s 𝑈) ∈ TopGrp))
3 fveq2 6906 . . . . . 6 (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
4 istrg.1 . . . . . 6 𝑀 = (mulGrp‘𝑅)
53, 4eqtr4di 2795 . . . . 5 (𝑟 = 𝑅 → (mulGrp‘𝑟) = 𝑀)
6 fveq2 6906 . . . . . 6 (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
7 istdrg.1 . . . . . 6 𝑈 = (Unit‘𝑅)
86, 7eqtr4di 2795 . . . . 5 (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
95, 8oveq12d 7449 . . . 4 (𝑟 = 𝑅 → ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) = (𝑀s 𝑈))
109eleq1d 2826 . . 3 (𝑟 = 𝑅 → (((mulGrp‘𝑟) ↾s (Unit‘𝑟)) ∈ TopGrp ↔ (𝑀s 𝑈) ∈ TopGrp))
11 df-tdrg 24169 . . 3 TopDRing = {𝑟 ∈ (TopRing ∩ DivRing) ∣ ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) ∈ TopGrp}
1210, 11elrab2 3695 . 2 (𝑅 ∈ TopDRing ↔ (𝑅 ∈ (TopRing ∩ DivRing) ∧ (𝑀s 𝑈) ∈ TopGrp))
13 df-3an 1089 . 2 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s 𝑈) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀s 𝑈) ∈ TopGrp))
142, 12, 133bitr4i 303 1 (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s 𝑈) ∈ TopGrp))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  cin 3950  cfv 6561  (class class class)co 7431  s cress 17274  mulGrpcmgp 20137  Unitcui 20355  DivRingcdr 20729  TopGrpctgp 24079  TopRingctrg 24164  TopDRingctdrg 24165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-tdrg 24169
This theorem is referenced by:  tdrgunit  24175  tdrgtrg  24181  tdrgdrng  24182  istdrg2  24186  nrgtdrg  24714
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