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Theorem pridl 33465
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 2621 . . . . . . 7 (1st𝑅) = (1st𝑅)
2 pridl.1 . . . . . . 7 𝐻 = (2nd𝑅)
3 eqid 2621 . . . . . . 7 ran (1st𝑅) = ran (1st𝑅)
41, 2, 3ispridl 33462 . . . . . 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 1038 . . . . . 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 276 . . . . 5 (𝑅 ∈ RingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ ran (1st𝑅)) ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
76simplbda 653 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))
8 raleq 3127 . . . . . 6 (𝑎 = 𝐴 → (∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 ↔ ∀𝑥𝐴𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃))
9 sseq1 3605 . . . . . . 7 (𝑎 = 𝐴 → (𝑎𝑃𝐴𝑃))
109orbi1d 738 . . . . . 6 (𝑎 = 𝐴 → ((𝑎𝑃𝑏𝑃) ↔ (𝐴𝑃𝑏𝑃)))
118, 10imbi12d 334 . . . . 5 (𝑎 = 𝐴 → ((∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) ↔ (∀𝑥𝐴𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝐴𝑃𝑏𝑃))))
12 raleq 3127 . . . . . . 7 (𝑏 = 𝐵 → (∀𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 ↔ ∀𝑦𝐵 (𝑥𝐻𝑦) ∈ 𝑃))
1312ralbidv 2980 . . . . . 6 (𝑏 = 𝐵 → (∀𝑥𝐴𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝐻𝑦) ∈ 𝑃))
14 sseq1 3605 . . . . . . 7 (𝑏 = 𝐵 → (𝑏𝑃𝐵𝑃))
1514orbi2d 737 . . . . . 6 (𝑏 = 𝐵 → ((𝐴𝑃𝑏𝑃) ↔ (𝐴𝑃𝐵𝑃)))
1613, 15imbi12d 334 . . . . 5 (𝑏 = 𝐵 → ((∀𝑥𝐴𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝐴𝑃𝑏𝑃)) ↔ (∀𝑥𝐴𝑦𝐵 (𝑥𝐻𝑦) ∈ 𝑃 → (𝐴𝑃𝐵𝑃))))
1711, 16rspc2v 3306 . . . 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 452 . 2 ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → (𝐴 ∈ (Idl‘𝑅) → (𝐵 ∈ (Idl‘𝑅) → (∀𝑥𝐴𝑦𝐵 (𝑥𝐻𝑦) ∈ 𝑃 → (𝐴𝑃𝐵𝑃)))))
20193imp2 1279 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  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  wss 3555  ran crn 5075  cfv 5847  (class class class)co 6604  1st c1st 7111  2nd c2nd 7112  RingOpscrngo 33322  Idlcidl 33435  PrIdlcpridl 33436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-pridl 33439
This theorem is referenced by: (None)
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