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Theorem pridlidl 36841
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 2733 . . . 4 (1st𝑅) = (1st𝑅)
2 eqid 2733 . . . 4 (2nd𝑅) = (2nd𝑅)
3 eqid 2733 . . . 4 ran (1st𝑅) = ran (1st𝑅)
41, 2, 3ispridl 36840 . . 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 1096 . . 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 500 1 ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → 𝑃 ∈ (Idl‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wo 846  w3a 1088  wcel 2107  wne 2941  wral 3062  wss 3947  ran crn 5676  cfv 6540  (class class class)co 7404  1st c1st 7968  2nd c2nd 7969  RingOpscrngo 36700  Idlcidl 36813  PrIdlcpridl 36814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  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-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7407  df-pridl 36817
This theorem is referenced by: (None)
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