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Theorem pridlnr 35747
 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 2759 . . . 4 (2nd𝑅) = (2nd𝑅)
3 prdilnr.2 . . . 4 𝑋 = ran 𝐺
41, 2, 3ispridl 35745 . . 3 (𝑅 ∈ RingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥(2nd𝑅)𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
5 3anan12 1094 . . 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 290 . 2 (𝑅 ∈ RingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃𝑋 ∧ (𝑃 ∈ (Idl‘𝑅) ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥(2nd𝑅)𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))))
76simprbda 503 1 ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → 𝑃𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 845   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112   ≠ wne 2952  ∀wral 3071   ⊆ wss 3859  ran crn 5526  ‘cfv 6336  (class class class)co 7151  1st c1st 7692  2nd c2nd 7693  RingOpscrngo 35605  Idlcidl 35718  PrIdlcpridl 35719 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-iota 6295  df-fun 6338  df-fv 6344  df-ov 7154  df-pridl 35722 This theorem is referenced by: (None)
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