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Theorem istrg 24067
Description: Express the predicate "𝑅 is a topological ring". (Contributed by Mario Carneiro, 5-Oct-2015.)
Hypothesis
Ref Expression
istrg.1 𝑀 = (mulGrp‘𝑅)
Assertion
Ref Expression
istrg (𝑅 ∈ TopRing ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ TopMnd))

Proof of Theorem istrg
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 elin 3963 . . 3 (𝑅 ∈ (TopGrp ∩ Ring) ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring))
21anbi1i 623 . 2 ((𝑅 ∈ (TopGrp ∩ Ring) ∧ 𝑀 ∈ TopMnd) ↔ ((𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring) ∧ 𝑀 ∈ TopMnd))
3 fveq2 6897 . . . . 5 (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
4 istrg.1 . . . . 5 𝑀 = (mulGrp‘𝑅)
53, 4eqtr4di 2786 . . . 4 (𝑟 = 𝑅 → (mulGrp‘𝑟) = 𝑀)
65eleq1d 2814 . . 3 (𝑟 = 𝑅 → ((mulGrp‘𝑟) ∈ TopMnd ↔ 𝑀 ∈ TopMnd))
7 df-trg 24063 . . 3 TopRing = {𝑟 ∈ (TopGrp ∩ Ring) ∣ (mulGrp‘𝑟) ∈ TopMnd}
86, 7elrab2 3685 . 2 (𝑅 ∈ TopRing ↔ (𝑅 ∈ (TopGrp ∩ Ring) ∧ 𝑀 ∈ TopMnd))
9 df-3an 1087 . 2 ((𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ TopMnd) ↔ ((𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring) ∧ 𝑀 ∈ TopMnd))
102, 8, 93bitr4i 303 1 (𝑅 ∈ TopRing ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ TopMnd))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  w3a 1085   = wceq 1534  wcel 2099  cin 3946  cfv 6548  mulGrpcmgp 20073  Ringcrg 20172  TopMndctmd 23973  TopGrpctgp 23974  TopRingctrg 24059
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-trg 24063
This theorem is referenced by:  trgtmd  24068  trgtgp  24071  trgring  24074  nrgtrg  24606
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