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Theorem pridl 35196
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 2818 . . . . . . 7 (1st𝑅) = (1st𝑅)
2 pridl.1 . . . . . . 7 𝐻 = (2nd𝑅)
3 eqid 2818 . . . . . . 7 ran (1st𝑅) = ran (1st𝑅)
41, 2, 3ispridl 35193 . . . . . 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 1081 . . . . . 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 288 . . . . 5 (𝑅 ∈ RingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ ran (1st𝑅)) ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
76simplbda 500 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))
8 raleq 3403 . . . . . 6 (𝑎 = 𝐴 → (∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 ↔ ∀𝑥𝐴𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃))
9 sseq1 3989 . . . . . . 7 (𝑎 = 𝐴 → (𝑎𝑃𝐴𝑃))
109orbi1d 910 . . . . . 6 (𝑎 = 𝐴 → ((𝑎𝑃𝑏𝑃) ↔ (𝐴𝑃𝑏𝑃)))
118, 10imbi12d 346 . . . . 5 (𝑎 = 𝐴 → ((∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)) ↔ (∀𝑥𝐴𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝐴𝑃𝑏𝑃))))
12 raleq 3403 . . . . . . 7 (𝑏 = 𝐵 → (∀𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 ↔ ∀𝑦𝐵 (𝑥𝐻𝑦) ∈ 𝑃))
1312ralbidv 3194 . . . . . 6 (𝑏 = 𝐵 → (∀𝑥𝐴𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 ↔ ∀𝑥𝐴𝑦𝐵 (𝑥𝐻𝑦) ∈ 𝑃))
14 sseq1 3989 . . . . . . 7 (𝑏 = 𝐵 → (𝑏𝑃𝐵𝑃))
1514orbi2d 909 . . . . . 6 (𝑏 = 𝐵 → ((𝐴𝑃𝑏𝑃) ↔ (𝐴𝑃𝐵𝑃)))
1613, 15imbi12d 346 . . . . 5 (𝑏 = 𝐵 → ((∀𝑥𝐴𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝐴𝑃𝑏𝑃)) ↔ (∀𝑥𝐴𝑦𝐵 (𝑥𝐻𝑦) ∈ 𝑃 → (𝐴𝑃𝐵𝑃))))
1711, 16rspc2v 3630 . . . 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 416 . 2 ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → (𝐴 ∈ (Idl‘𝑅) → (𝐵 ∈ (Idl‘𝑅) → (∀𝑥𝐴𝑦𝐵 (𝑥𝐻𝑦) ∈ 𝑃 → (𝐴𝑃𝐵𝑃)))))
20193imp2 1341 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 396  wo 841  w3a 1079   = wceq 1528  wcel 2105  wne 3013  wral 3135  wss 3933  ran crn 5549  cfv 6348  (class class class)co 7145  1st c1st 7676  2nd c2nd 7677  RingOpscrngo 35053  Idlcidl 35166  PrIdlcpridl 35167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-pridl 35170
This theorem is referenced by: (None)
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