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Theorem pridlnr 37996
Description: A prime ideal is a proper ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
pridlnr.1 𝐺 = (1st𝑅)
prdilnr.2 𝑋 = ran 𝐺
Assertion
Ref Expression
pridlnr ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → 𝑃𝑋)

Proof of Theorem pridlnr
Dummy variables 𝑥 𝑦 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pridlnr.1 . . . 4 𝐺 = (1st𝑅)
2 eqid 2740 . . . 4 (2nd𝑅) = (2nd𝑅)
3 prdilnr.2 . . . 4 𝑋 = ran 𝐺
41, 2, 3ispridl 37994 . . 3 (𝑅 ∈ RingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥(2nd𝑅)𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
5 3anan12 1096 . . 3 ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥(2nd𝑅)𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))) ↔ (𝑃𝑋 ∧ (𝑃 ∈ (Idl‘𝑅) ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥(2nd𝑅)𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
64, 5bitrdi 287 . 2 (𝑅 ∈ RingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃𝑋 ∧ (𝑃 ∈ (Idl‘𝑅) ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥(2nd𝑅)𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))))
76simprbda 498 1 ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → 𝑃𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 846  w3a 1087   = wceq 1537  wcel 2108  wne 2946  wral 3067  wss 3976  ran crn 5701  cfv 6573  (class class class)co 7448  1st c1st 8028  2nd c2nd 8029  RingOpscrngo 37854  Idlcidl 37967  PrIdlcpridl 37968
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 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
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 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-pridl 37971
This theorem is referenced by: (None)
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