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Theorem pridlidl 37943
Description: A prime ideal is an ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)
Assertion
Ref Expression
pridlidl ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → 𝑃 ∈ (Idl‘𝑅))

Proof of Theorem pridlidl
Dummy variables 𝑥 𝑦 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2734 . . . 4 (1st𝑅) = (1st𝑅)
2 eqid 2734 . . . 4 (2nd𝑅) = (2nd𝑅)
3 eqid 2734 . . . 4 ran (1st𝑅) = ran (1st𝑅)
41, 2, 3ispridl 37942 . . 3 (𝑅 ∈ RingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ ran (1st𝑅) ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥(2nd𝑅)𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
5 3anass 1095 . . 3 ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ ran (1st𝑅) ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥(2nd𝑅)𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ (𝑃 ≠ ran (1st𝑅) ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥(2nd𝑅)𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
64, 5bitrdi 287 . 2 (𝑅 ∈ RingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ (𝑃 ≠ ran (1st𝑅) ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥(2nd𝑅)𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))))
76simprbda 498 1 ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → 𝑃 ∈ (Idl‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 846  w3a 1087  wcel 2103  wne 2942  wral 3063  wss 3970  ran crn 5700  cfv 6572  (class class class)co 7445  1st c1st 8024  2nd c2nd 8025  RingOpscrngo 37802  Idlcidl 37915  PrIdlcpridl 37916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-mpt 5253  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-iota 6524  df-fun 6574  df-fv 6580  df-ov 7448  df-pridl 37919
This theorem is referenced by: (None)
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