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Theorem istdrg 23533
Description: Express the predicate "𝑅 is a topological ring". (Contributed by Mario Carneiro, 5-Oct-2015.)
Hypotheses
Ref Expression
istrg.1 𝑀 = (mulGrpβ€˜π‘…)
istdrg.1 π‘ˆ = (Unitβ€˜π‘…)
Assertion
Ref Expression
istdrg (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 β†Ύs π‘ˆ) ∈ TopGrp))

Proof of Theorem istdrg
Dummy variable π‘Ÿ is distinct from all other variables.
StepHypRef Expression
1 elin 3927 . . 3 (𝑅 ∈ (TopRing ∩ DivRing) ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing))
21anbi1i 625 . 2 ((𝑅 ∈ (TopRing ∩ DivRing) ∧ (𝑀 β†Ύs π‘ˆ) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 β†Ύs π‘ˆ) ∈ TopGrp))
3 fveq2 6843 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (mulGrpβ€˜π‘Ÿ) = (mulGrpβ€˜π‘…))
4 istrg.1 . . . . . 6 𝑀 = (mulGrpβ€˜π‘…)
53, 4eqtr4di 2791 . . . . 5 (π‘Ÿ = 𝑅 β†’ (mulGrpβ€˜π‘Ÿ) = 𝑀)
6 fveq2 6843 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (Unitβ€˜π‘Ÿ) = (Unitβ€˜π‘…))
7 istdrg.1 . . . . . 6 π‘ˆ = (Unitβ€˜π‘…)
86, 7eqtr4di 2791 . . . . 5 (π‘Ÿ = 𝑅 β†’ (Unitβ€˜π‘Ÿ) = π‘ˆ)
95, 8oveq12d 7376 . . . 4 (π‘Ÿ = 𝑅 β†’ ((mulGrpβ€˜π‘Ÿ) β†Ύs (Unitβ€˜π‘Ÿ)) = (𝑀 β†Ύs π‘ˆ))
109eleq1d 2819 . . 3 (π‘Ÿ = 𝑅 β†’ (((mulGrpβ€˜π‘Ÿ) β†Ύs (Unitβ€˜π‘Ÿ)) ∈ TopGrp ↔ (𝑀 β†Ύs π‘ˆ) ∈ TopGrp))
11 df-tdrg 23528 . . 3 TopDRing = {π‘Ÿ ∈ (TopRing ∩ DivRing) ∣ ((mulGrpβ€˜π‘Ÿ) β†Ύs (Unitβ€˜π‘Ÿ)) ∈ TopGrp}
1210, 11elrab2 3649 . 2 (𝑅 ∈ TopDRing ↔ (𝑅 ∈ (TopRing ∩ DivRing) ∧ (𝑀 β†Ύs π‘ˆ) ∈ TopGrp))
13 df-3an 1090 . 2 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 β†Ύs π‘ˆ) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 β†Ύs π‘ˆ) ∈ TopGrp))
142, 12, 133bitr4i 303 1 (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 β†Ύs π‘ˆ) ∈ TopGrp))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   ∩ cin 3910  β€˜cfv 6497  (class class class)co 7358   β†Ύs cress 17117  mulGrpcmgp 19901  Unitcui 20073  DivRingcdr 20197  TopGrpctgp 23438  TopRingctrg 23523  TopDRingctdrg 23524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-ov 7361  df-tdrg 23528
This theorem is referenced by:  tdrgunit  23534  tdrgtrg  23540  tdrgdrng  23541  istdrg2  23545  nrgtdrg  24073
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