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Theorem pridl 34068
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 2724 . . . . . . 7 (1st𝑅) = (1st𝑅)
2 pridl.1 . . . . . . 7 𝐻 = (2nd𝑅)
3 eqid 2724 . . . . . . 7 ran (1st𝑅) = ran (1st𝑅)
41, 2, 3ispridl 34065 . . . . . 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 1074 . . . . . 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 655 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))
8 raleq 3241 . . . . . 6 (𝑎 = 𝐴 → (∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 ↔ ∀𝑥𝐴𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃))
9 sseq1 3732 . . . . . . 7 (𝑎 = 𝐴 → (𝑎𝑃𝐴𝑃))
109orbi1d 741 . . . . . 6 (𝑎 = 𝐴 → ((𝑎𝑃𝑏𝑃) ↔ (𝐴𝑃𝑏𝑃)))
118, 10imbi12d 333 . . . . 5 (𝑎 = 𝐴 → ((∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) ↔ (∀𝑥𝐴𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝐴𝑃𝑏𝑃))))
12 raleq 3241 . . . . . . 7 (𝑏 = 𝐵 → (∀𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 ↔ ∀𝑦𝐵 (𝑥𝐻𝑦) ∈ 𝑃))
1312ralbidv 3088 . . . . . 6 (𝑏 = 𝐵 → (∀𝑥𝐴𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝐻𝑦) ∈ 𝑃))
14 sseq1 3732 . . . . . . 7 (𝑏 = 𝐵 → (𝑏𝑃𝐵𝑃))
1514orbi2d 740 . . . . . 6 (𝑏 = 𝐵 → ((𝐴𝑃𝑏𝑃) ↔ (𝐴𝑃𝐵𝑃)))
1613, 15imbi12d 333 . . . . 5 (𝑏 = 𝐵 → ((∀𝑥𝐴𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝐴𝑃𝑏𝑃)) ↔ (∀𝑥𝐴𝑦𝐵 (𝑥𝐻𝑦) ∈ 𝑃 → (𝐴𝑃𝐵𝑃))))
1711, 16rspc2v 3426 . . . 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 451 . 2 ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → (𝐴 ∈ (Idl‘𝑅) → (𝐵 ∈ (Idl‘𝑅) → (∀𝑥𝐴𝑦𝐵 (𝑥𝐻𝑦) ∈ 𝑃 → (𝐴𝑃𝐵𝑃)))))
20193imp2 1406 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 382  wa 383  w3a 1072   = wceq 1596  wcel 2103  wne 2896  wral 3014  wss 3680  ran crn 5219  cfv 6001  (class class class)co 6765  1st c1st 7283  2nd c2nd 7284  RingOpscrngo 33925  Idlcidl 34038  PrIdlcpridl 34039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-iota 5964  df-fun 6003  df-fv 6009  df-ov 6768  df-pridl 34042
This theorem is referenced by: (None)
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