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Theorem ispridl 34311
Description: The predicate "is a prime ideal". (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
pridlval.1 𝐺 = (1st𝑅)
pridlval.2 𝐻 = (2nd𝑅)
pridlval.3 𝑋 = ran 𝐺
Assertion
Ref Expression
ispridl (𝑅 ∈ RingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
Distinct variable groups:   𝑥,𝑅,𝑦,𝑎,𝑏   𝑥,𝑃,𝑦,𝑎,𝑏
Allowed substitution hints:   𝐺(𝑥,𝑦,𝑎,𝑏)   𝐻(𝑥,𝑦,𝑎,𝑏)   𝑋(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem ispridl
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 pridlval.1 . . . 4 𝐺 = (1st𝑅)
2 pridlval.2 . . . 4 𝐻 = (2nd𝑅)
3 pridlval.3 . . . 4 𝑋 = ran 𝐺
41, 2, 3pridlval 34310 . . 3 (𝑅 ∈ RingOps → (PrIdl‘𝑅) = {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))})
54eleq2d 2862 . 2 (𝑅 ∈ RingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ 𝑃 ∈ {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))}))
6 neeq1 3031 . . . . 5 (𝑖 = 𝑃 → (𝑖𝑋𝑃𝑋))
7 eleq2 2865 . . . . . . . 8 (𝑖 = 𝑃 → ((𝑥𝐻𝑦) ∈ 𝑖 ↔ (𝑥𝐻𝑦) ∈ 𝑃))
872ralbidv 3168 . . . . . . 7 (𝑖 = 𝑃 → (∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑖 ↔ ∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃))
9 sseq2 3821 . . . . . . . 8 (𝑖 = 𝑃 → (𝑎𝑖𝑎𝑃))
10 sseq2 3821 . . . . . . . 8 (𝑖 = 𝑃 → (𝑏𝑖𝑏𝑃))
119, 10orbi12d 943 . . . . . . 7 (𝑖 = 𝑃 → ((𝑎𝑖𝑏𝑖) ↔ (𝑎𝑃𝑏𝑃)))
128, 11imbi12d 336 . . . . . 6 (𝑖 = 𝑃 → ((∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)) ↔ (∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
13122ralbidv 3168 . . . . 5 (𝑖 = 𝑃 → (∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)) ↔ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
146, 13anbi12d 625 . . . 4 (𝑖 = 𝑃 → ((𝑖𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖))) ↔ (𝑃𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
1514elrab 3554 . . 3 (𝑃 ∈ {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))} ↔ (𝑃 ∈ (Idl‘𝑅) ∧ (𝑃𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
16 3anass 1117 . . 3 ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ (𝑃𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
1715, 16bitr4i 270 . 2 (𝑃 ∈ {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))} ↔ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
185, 17syl6bb 279 1 (𝑅 ∈ RingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  wo 874  w3a 1108   = wceq 1653  wcel 2157  wne 2969  wral 3087  {crab 3091  wss 3767  ran crn 5311  cfv 6099  (class class class)co 6876  1st c1st 7397  2nd c2nd 7398  RingOpscrngo 34171  Idlcidl 34284  PrIdlcpridl 34285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2375  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-iota 6062  df-fun 6101  df-fv 6107  df-ov 6879  df-pridl 34288
This theorem is referenced by:  pridlidl  34312  pridlnr  34313  pridl  34314  ispridl2  34315  smprngopr  34329  ispridlc  34347
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