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Theorem pridl 34444
Description: The main property of a prime ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypothesis
Ref Expression
pridl.1 𝐻 = (2nd𝑅)
Assertion
Ref Expression
pridl (((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (Idl‘𝑅) ∧ 𝐵 ∈ (Idl‘𝑅) ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝐻𝑦) ∈ 𝑃)) → (𝐴𝑃𝐵𝑃))
Distinct variable groups:   𝑥,𝑅,𝑦   𝑥,𝑃,𝑦   𝑥,𝐴   𝑥,𝐵,𝑦
Allowed substitution hints:   𝐴(𝑦)   𝐻(𝑥,𝑦)

Proof of Theorem pridl
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2777 . . . . . . 7 (1st𝑅) = (1st𝑅)
2 pridl.1 . . . . . . 7 𝐻 = (2nd𝑅)
3 eqid 2777 . . . . . . 7 ran (1st𝑅) = ran (1st𝑅)
41, 2, 3ispridl 34441 . . . . . 6 (𝑅 ∈ RingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ ran (1st𝑅) ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
5 df-3an 1073 . . . . . 6 ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ ran (1st𝑅) ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))) ↔ ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ ran (1st𝑅)) ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
64, 5syl6bb 279 . . . . 5 (𝑅 ∈ RingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ ran (1st𝑅)) ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
76simplbda 495 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))
8 raleq 3329 . . . . . 6 (𝑎 = 𝐴 → (∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 ↔ ∀𝑥𝐴𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃))
9 sseq1 3844 . . . . . . 7 (𝑎 = 𝐴 → (𝑎𝑃𝐴𝑃))
109orbi1d 903 . . . . . 6 (𝑎 = 𝐴 → ((𝑎𝑃𝑏𝑃) ↔ (𝐴𝑃𝑏𝑃)))
118, 10imbi12d 336 . . . . 5 (𝑎 = 𝐴 → ((∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) ↔ (∀𝑥𝐴𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝐴𝑃𝑏𝑃))))
12 raleq 3329 . . . . . . 7 (𝑏 = 𝐵 → (∀𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 ↔ ∀𝑦𝐵 (𝑥𝐻𝑦) ∈ 𝑃))
1312ralbidv 3167 . . . . . 6 (𝑏 = 𝐵 → (∀𝑥𝐴𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝐻𝑦) ∈ 𝑃))
14 sseq1 3844 . . . . . . 7 (𝑏 = 𝐵 → (𝑏𝑃𝐵𝑃))
1514orbi2d 902 . . . . . 6 (𝑏 = 𝐵 → ((𝐴𝑃𝑏𝑃) ↔ (𝐴𝑃𝐵𝑃)))
1613, 15imbi12d 336 . . . . 5 (𝑏 = 𝐵 → ((∀𝑥𝐴𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝐴𝑃𝑏𝑃)) ↔ (∀𝑥𝐴𝑦𝐵 (𝑥𝐻𝑦) ∈ 𝑃 → (𝐴𝑃𝐵𝑃))))
1711, 16rspc2v 3523 . . . 4 ((𝐴 ∈ (Idl‘𝑅) ∧ 𝐵 ∈ (Idl‘𝑅)) → (∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) → (∀𝑥𝐴𝑦𝐵 (𝑥𝐻𝑦) ∈ 𝑃 → (𝐴𝑃𝐵𝑃))))
187, 17syl5com 31 . . 3 ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → ((𝐴 ∈ (Idl‘𝑅) ∧ 𝐵 ∈ (Idl‘𝑅)) → (∀𝑥𝐴𝑦𝐵 (𝑥𝐻𝑦) ∈ 𝑃 → (𝐴𝑃𝐵𝑃))))
1918expd 406 . 2 ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → (𝐴 ∈ (Idl‘𝑅) → (𝐵 ∈ (Idl‘𝑅) → (∀𝑥𝐴𝑦𝐵 (𝑥𝐻𝑦) ∈ 𝑃 → (𝐴𝑃𝐵𝑃)))))
20193imp2 1411 1 (((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (Idl‘𝑅) ∧ 𝐵 ∈ (Idl‘𝑅) ∧ ∀𝑥𝐴𝑦𝐵 (𝑥𝐻𝑦) ∈ 𝑃)) → (𝐴𝑃𝐵𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wo 836  w3a 1071   = wceq 1601  wcel 2106  wne 2968  wral 3089  wss 3791  ran crn 5356  cfv 6135  (class class class)co 6922  1st c1st 7443  2nd c2nd 7444  RingOpscrngo 34301  Idlcidl 34414  PrIdlcpridl 34415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-iota 6099  df-fun 6137  df-fv 6143  df-ov 6925  df-pridl 34418
This theorem is referenced by: (None)
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