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Theorem pridl 38044
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 2737 . . . . . . 7 (1st𝑅) = (1st𝑅)
2 pridl.1 . . . . . . 7 𝐻 = (2nd𝑅)
3 eqid 2737 . . . . . . 7 ran (1st𝑅) = ran (1st𝑅)
41, 2, 3ispridl 38041 . . . . . 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 1089 . . . . . 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, 5bitrdi 287 . . . . 5 (𝑅 ∈ RingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ ran (1st𝑅)) ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
76simplbda 499 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))
8 raleq 3323 . . . . . 6 (𝑎 = 𝐴 → (∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 ↔ ∀𝑥𝐴𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃))
9 sseq1 4009 . . . . . . 7 (𝑎 = 𝐴 → (𝑎𝑃𝐴𝑃))
109orbi1d 917 . . . . . 6 (𝑎 = 𝐴 → ((𝑎𝑃𝑏𝑃) ↔ (𝐴𝑃𝑏𝑃)))
118, 10imbi12d 344 . . . . 5 (𝑎 = 𝐴 → ((∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) ↔ (∀𝑥𝐴𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝐴𝑃𝑏𝑃))))
12 raleq 3323 . . . . . . 7 (𝑏 = 𝐵 → (∀𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 ↔ ∀𝑦𝐵 (𝑥𝐻𝑦) ∈ 𝑃))
1312ralbidv 3178 . . . . . 6 (𝑏 = 𝐵 → (∀𝑥𝐴𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝐻𝑦) ∈ 𝑃))
14 sseq1 4009 . . . . . . 7 (𝑏 = 𝐵 → (𝑏𝑃𝐵𝑃))
1514orbi2d 916 . . . . . 6 (𝑏 = 𝐵 → ((𝐴𝑃𝑏𝑃) ↔ (𝐴𝑃𝐵𝑃)))
1613, 15imbi12d 344 . . . . 5 (𝑏 = 𝐵 → ((∀𝑥𝐴𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝐴𝑃𝑏𝑃)) ↔ (∀𝑥𝐴𝑦𝐵 (𝑥𝐻𝑦) ∈ 𝑃 → (𝐴𝑃𝐵𝑃))))
1711, 16rspc2v 3633 . . . 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 415 . 2 ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → (𝐴 ∈ (Idl‘𝑅) → (𝐵 ∈ (Idl‘𝑅) → (∀𝑥𝐴𝑦𝐵 (𝑥𝐻𝑦) ∈ 𝑃 → (𝐴𝑃𝐵𝑃)))))
20193imp2 1350 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 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wss 3951  ran crn 5686  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  RingOpscrngo 37901  Idlcidl 38014  PrIdlcpridl 38015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-pridl 38018
This theorem is referenced by: (None)
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