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Theorem istdrg 24083
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 3963 . . 3 (𝑅 ∈ (TopRing ∩ DivRing) ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing))
21anbi1i 623 . 2 ((𝑅 ∈ (TopRing ∩ DivRing) ∧ (𝑀 β†Ύs π‘ˆ) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 β†Ύs π‘ˆ) ∈ TopGrp))
3 fveq2 6897 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (mulGrpβ€˜π‘Ÿ) = (mulGrpβ€˜π‘…))
4 istrg.1 . . . . . 6 𝑀 = (mulGrpβ€˜π‘…)
53, 4eqtr4di 2786 . . . . 5 (π‘Ÿ = 𝑅 β†’ (mulGrpβ€˜π‘Ÿ) = 𝑀)
6 fveq2 6897 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (Unitβ€˜π‘Ÿ) = (Unitβ€˜π‘…))
7 istdrg.1 . . . . . 6 π‘ˆ = (Unitβ€˜π‘…)
86, 7eqtr4di 2786 . . . . 5 (π‘Ÿ = 𝑅 β†’ (Unitβ€˜π‘Ÿ) = π‘ˆ)
95, 8oveq12d 7438 . . . 4 (π‘Ÿ = 𝑅 β†’ ((mulGrpβ€˜π‘Ÿ) β†Ύs (Unitβ€˜π‘Ÿ)) = (𝑀 β†Ύs π‘ˆ))
109eleq1d 2814 . . 3 (π‘Ÿ = 𝑅 β†’ (((mulGrpβ€˜π‘Ÿ) β†Ύs (Unitβ€˜π‘Ÿ)) ∈ TopGrp ↔ (𝑀 β†Ύs π‘ˆ) ∈ TopGrp))
11 df-tdrg 24078 . . 3 TopDRing = {π‘Ÿ ∈ (TopRing ∩ DivRing) ∣ ((mulGrpβ€˜π‘Ÿ) β†Ύs (Unitβ€˜π‘Ÿ)) ∈ TopGrp}
1210, 11elrab2 3685 . 2 (𝑅 ∈ TopDRing ↔ (𝑅 ∈ (TopRing ∩ DivRing) ∧ (𝑀 β†Ύs π‘ˆ) ∈ TopGrp))
13 df-3an 1087 . 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 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   ∩ cin 3946  β€˜cfv 6548  (class class class)co 7420   β†Ύs cress 17209  mulGrpcmgp 20074  Unitcui 20294  DivRingcdr 20624  TopGrpctgp 23988  TopRingctrg 24073  TopDRingctdrg 24074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-iota 6500  df-fv 6556  df-ov 7423  df-tdrg 24078
This theorem is referenced by:  tdrgunit  24084  tdrgtrg  24090  tdrgdrng  24091  istdrg2  24095  nrgtdrg  24623
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