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Theorem istrg 22187
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 3947 . . 3 (𝑅 ∈ (TopGrp ∩ Ring) ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring))
21anbi1i 610 . 2 ((𝑅 ∈ (TopGrp ∩ Ring) ∧ 𝑀 ∈ TopMnd) ↔ ((𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring) ∧ 𝑀 ∈ TopMnd))
3 fveq2 6332 . . . . 5 (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
4 istrg.1 . . . . 5 𝑀 = (mulGrp‘𝑅)
53, 4syl6eqr 2823 . . . 4 (𝑟 = 𝑅 → (mulGrp‘𝑟) = 𝑀)
65eleq1d 2835 . . 3 (𝑟 = 𝑅 → ((mulGrp‘𝑟) ∈ TopMnd ↔ 𝑀 ∈ TopMnd))
7 df-trg 22183 . . 3 TopRing = {𝑟 ∈ (TopGrp ∩ Ring) ∣ (mulGrp‘𝑟) ∈ TopMnd}
86, 7elrab2 3518 . 2 (𝑅 ∈ TopRing ↔ (𝑅 ∈ (TopGrp ∩ Ring) ∧ 𝑀 ∈ TopMnd))
9 df-3an 1073 . 2 ((𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ TopMnd) ↔ ((𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring) ∧ 𝑀 ∈ TopMnd))
102, 8, 93bitr4i 292 1 (𝑅 ∈ TopRing ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ TopMnd))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  cin 3722  cfv 6031  mulGrpcmgp 18697  Ringcrg 18755  TopMndctmd 22094  TopGrpctgp 22095  TopRingctrg 22179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039  df-trg 22183
This theorem is referenced by:  trgtmd  22188  trgtgp  22191  trgring  22194  nrgtrg  22714
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