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Theorem ispridl 34844
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 34843 . . 3 (𝑅 ∈ RingOps → (PrIdl‘𝑅) = {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))})
54eleq2d 2868 . 2 (𝑅 ∈ RingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ 𝑃 ∈ {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))}))
6 neeq1 3046 . . . . 5 (𝑖 = 𝑃 → (𝑖𝑋𝑃𝑋))
7 eleq2 2871 . . . . . . . 8 (𝑖 = 𝑃 → ((𝑥𝐻𝑦) ∈ 𝑖 ↔ (𝑥𝐻𝑦) ∈ 𝑃))
872ralbidv 3166 . . . . . . 7 (𝑖 = 𝑃 → (∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑖 ↔ ∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃))
9 sseq2 3914 . . . . . . . 8 (𝑖 = 𝑃 → (𝑎𝑖𝑎𝑃))
10 sseq2 3914 . . . . . . . 8 (𝑖 = 𝑃 → (𝑏𝑖𝑏𝑃))
119, 10orbi12d 913 . . . . . . 7 (𝑖 = 𝑃 → ((𝑎𝑖𝑏𝑖) ↔ (𝑎𝑃𝑏𝑃)))
128, 11imbi12d 346 . . . . . 6 (𝑖 = 𝑃 → ((∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)) ↔ (∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
13122ralbidv 3166 . . . . 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 630 . . . 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 3618 . . 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 1088 . . 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 279 . 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 288 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 207  wa 396  wo 842  w3a 1080   = wceq 1522  wcel 2081  wne 2984  wral 3105  {crab 3109  wss 3859  ran crn 5444  cfv 6225  (class class class)co 7016  1st c1st 7543  2nd c2nd 7544  RingOpscrngo 34704  Idlcidl 34817  PrIdlcpridl 34818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-iota 6189  df-fun 6227  df-fv 6233  df-ov 7019  df-pridl 34821
This theorem is referenced by:  pridlidl  34845  pridlnr  34846  pridl  34847  ispridl2  34848  smprngopr  34862  ispridlc  34880
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