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Theorem istdrg 22769
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 3924 . . 3 (𝑅 ∈ (TopRing ∩ DivRing) ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing))
21anbi1i 626 . 2 ((𝑅 ∈ (TopRing ∩ DivRing) ∧ (𝑀s 𝑈) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀s 𝑈) ∈ TopGrp))
3 fveq2 6652 . . . . . 6 (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
4 istrg.1 . . . . . 6 𝑀 = (mulGrp‘𝑅)
53, 4eqtr4di 2875 . . . . 5 (𝑟 = 𝑅 → (mulGrp‘𝑟) = 𝑀)
6 fveq2 6652 . . . . . 6 (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
7 istdrg.1 . . . . . 6 𝑈 = (Unit‘𝑅)
86, 7eqtr4di 2875 . . . . 5 (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
95, 8oveq12d 7158 . . . 4 (𝑟 = 𝑅 → ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) = (𝑀s 𝑈))
109eleq1d 2898 . . 3 (𝑟 = 𝑅 → (((mulGrp‘𝑟) ↾s (Unit‘𝑟)) ∈ TopGrp ↔ (𝑀s 𝑈) ∈ TopGrp))
11 df-tdrg 22764 . . 3 TopDRing = {𝑟 ∈ (TopRing ∩ DivRing) ∣ ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) ∈ TopGrp}
1210, 11elrab2 3658 . 2 (𝑅 ∈ TopDRing ↔ (𝑅 ∈ (TopRing ∩ DivRing) ∧ (𝑀s 𝑈) ∈ TopGrp))
13 df-3an 1086 . 2 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s 𝑈) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀s 𝑈) ∈ TopGrp))
142, 12, 133bitr4i 306 1 (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀s 𝑈) ∈ TopGrp))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2114  cin 3907  cfv 6334  (class class class)co 7140  s cress 16475  mulGrpcmgp 19230  Unitcui 19383  DivRingcdr 19493  TopGrpctgp 22674  TopRingctrg 22759  TopDRingctdrg 22760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-rab 3139  df-v 3471  df-un 3913  df-in 3915  df-ss 3925  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-iota 6293  df-fv 6342  df-ov 7143  df-tdrg 22764
This theorem is referenced by:  tdrgunit  22770  tdrgtrg  22776  tdrgdrng  22777  istdrg2  22781  nrgtdrg  23297
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