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Theorem istdrg 23661
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 624 . 2 ((𝑅 ∈ (TopRing ∩ DivRing) ∧ (𝑀 β†Ύs π‘ˆ) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 β†Ύs π‘ˆ) ∈ TopGrp))
3 fveq2 6888 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (mulGrpβ€˜π‘Ÿ) = (mulGrpβ€˜π‘…))
4 istrg.1 . . . . . 6 𝑀 = (mulGrpβ€˜π‘…)
53, 4eqtr4di 2790 . . . . 5 (π‘Ÿ = 𝑅 β†’ (mulGrpβ€˜π‘Ÿ) = 𝑀)
6 fveq2 6888 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (Unitβ€˜π‘Ÿ) = (Unitβ€˜π‘…))
7 istdrg.1 . . . . . 6 π‘ˆ = (Unitβ€˜π‘…)
86, 7eqtr4di 2790 . . . . 5 (π‘Ÿ = 𝑅 β†’ (Unitβ€˜π‘Ÿ) = π‘ˆ)
95, 8oveq12d 7423 . . . 4 (π‘Ÿ = 𝑅 β†’ ((mulGrpβ€˜π‘Ÿ) β†Ύs (Unitβ€˜π‘Ÿ)) = (𝑀 β†Ύs π‘ˆ))
109eleq1d 2818 . . 3 (π‘Ÿ = 𝑅 β†’ (((mulGrpβ€˜π‘Ÿ) β†Ύs (Unitβ€˜π‘Ÿ)) ∈ TopGrp ↔ (𝑀 β†Ύs π‘ˆ) ∈ TopGrp))
11 df-tdrg 23656 . . 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 1089 . 2 ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 β†Ύs π‘ˆ) ∈ TopGrp) ↔ ((𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing) ∧ (𝑀 β†Ύs π‘ˆ) ∈ TopGrp))
142, 12, 133bitr4i 302 1 (𝑅 ∈ TopDRing ↔ (𝑅 ∈ TopRing ∧ 𝑅 ∈ DivRing ∧ (𝑀 β†Ύs π‘ˆ) ∈ TopGrp))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ∩ cin 3946  β€˜cfv 6540  (class class class)co 7405   β†Ύs cress 17169  mulGrpcmgp 19981  Unitcui 20161  DivRingcdr 20307  TopGrpctgp 23566  TopRingctrg 23651  TopDRingctdrg 23652
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6492  df-fv 6548  df-ov 7408  df-tdrg 23656
This theorem is referenced by:  tdrgunit  23662  tdrgtrg  23668  tdrgdrng  23669  istdrg2  23673  nrgtdrg  24201
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