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Theorem istdrg 22296
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 3995 . . 3 (𝑅 ∈ (TopRing ∩ DivRing) ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing))
21anbi1i 618 . 2 ((𝑅 ∈ (TopRing ∩ DivRing) ∧ (𝑀s 𝑈) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀s 𝑈) ∈ TopGrp))
3 fveq2 6412 . . . . . 6 (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
4 istrg.1 . . . . . 6 𝑀 = (mulGrp‘𝑅)
53, 4syl6eqr 2852 . . . . 5 (𝑟 = 𝑅 → (mulGrp‘𝑟) = 𝑀)
6 fveq2 6412 . . . . . 6 (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
7 istdrg.1 . . . . . 6 𝑈 = (Unit‘𝑅)
86, 7syl6eqr 2852 . . . . 5 (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
95, 8oveq12d 6897 . . . 4 (𝑟 = 𝑅 → ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) = (𝑀s 𝑈))
109eleq1d 2864 . . 3 (𝑟 = 𝑅 → (((mulGrp‘𝑟) ↾s (Unit‘𝑟)) ∈ TopGrp ↔ (𝑀s 𝑈) ∈ TopGrp))
11 df-tdrg 22291 . . 3 TopDRing = {𝑟 ∈ (TopRing ∩ DivRing) ∣ ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) ∈ TopGrp}
1210, 11elrab2 3561 . 2 (𝑅 ∈ TopDRing ↔ (𝑅 ∈ (TopRing ∩ DivRing) ∧ (𝑀s 𝑈) ∈ TopGrp))
13 df-3an 1110 . 2 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s 𝑈) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀s 𝑈) ∈ TopGrp))
142, 12, 133bitr4i 295 1 (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s 𝑈) ∈ TopGrp))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  cin 3769  cfv 6102  (class class class)co 6879  s cress 16184  mulGrpcmgp 18804  Unitcui 18954  DivRingcdr 19064  TopGrpctgp 22202  TopRingctrg 22286  TopDRingctdrg 22287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-rex 3096  df-rab 3099  df-v 3388  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-iota 6065  df-fv 6110  df-ov 6882  df-tdrg 22291
This theorem is referenced by:  tdrgunit  22297  tdrgtrg  22303  tdrgdrng  22304  istdrg2  22308  nrgtdrg  22824
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