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Theorem istrg 24012
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 3957 . . 3 (𝑅 ∈ (TopGrp ∩ Ring) ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring))
21anbi1i 623 . 2 ((𝑅 ∈ (TopGrp ∩ Ring) ∧ 𝑀 ∈ TopMnd) ↔ ((𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring) ∧ 𝑀 ∈ TopMnd))
3 fveq2 6882 . . . . 5 (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
4 istrg.1 . . . . 5 𝑀 = (mulGrp‘𝑅)
53, 4eqtr4di 2782 . . . 4 (𝑟 = 𝑅 → (mulGrp‘𝑟) = 𝑀)
65eleq1d 2810 . . 3 (𝑟 = 𝑅 → ((mulGrp‘𝑟) ∈ TopMnd ↔ 𝑀 ∈ TopMnd))
7 df-trg 24008 . . 3 TopRing = {𝑟 ∈ (TopGrp ∩ Ring) ∣ (mulGrp‘𝑟) ∈ TopMnd}
86, 7elrab2 3679 . 2 (𝑅 ∈ TopRing ↔ (𝑅 ∈ (TopGrp ∩ Ring) ∧ 𝑀 ∈ TopMnd))
9 df-3an 1086 . 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 1084   = wceq 1533  wcel 2098  cin 3940  cfv 6534  mulGrpcmgp 20035  Ringcrg 20134  TopMndctmd 23918  TopGrpctgp 23919  TopRingctrg 24004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-iota 6486  df-fv 6542  df-trg 24008
This theorem is referenced by:  trgtmd  24013  trgtgp  24016  trgring  24019  nrgtrg  24551
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