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Theorem ispridl 34146
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 34145 . . 3 (𝑅 ∈ RingOps → (PrIdl‘𝑅) = {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))})
54eleq2d 2825 . 2 (𝑅 ∈ RingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ 𝑃 ∈ {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))}))
6 neeq1 2994 . . . . 5 (𝑖 = 𝑃 → (𝑖𝑋𝑃𝑋))
7 eleq2 2828 . . . . . . . 8 (𝑖 = 𝑃 → ((𝑥𝐻𝑦) ∈ 𝑖 ↔ (𝑥𝐻𝑦) ∈ 𝑃))
872ralbidv 3127 . . . . . . 7 (𝑖 = 𝑃 → (∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑖 ↔ ∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃))
9 sseq2 3768 . . . . . . . 8 (𝑖 = 𝑃 → (𝑎𝑖𝑎𝑃))
10 sseq2 3768 . . . . . . . 8 (𝑖 = 𝑃 → (𝑏𝑖𝑏𝑃))
119, 10orbi12d 748 . . . . . . 7 (𝑖 = 𝑃 → ((𝑎𝑖𝑏𝑖) ↔ (𝑎𝑃𝑏𝑃)))
128, 11imbi12d 333 . . . . . 6 (𝑖 = 𝑃 → ((∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)) ↔ (∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))
13122ralbidv 3127 . . . . 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 749 . . . 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 3504 . . 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 1081 . . 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 267 . 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 276 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 196  wo 382  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  wral 3050  {crab 3054  wss 3715  ran crn 5267  cfv 6049  (class class class)co 6813  1st c1st 7331  2nd c2nd 7332  RingOpscrngo 34006  Idlcidl 34119  PrIdlcpridl 34120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-pridl 34123
This theorem is referenced by:  pridlidl  34147  pridlnr  34148  pridl  34149  ispridl2  34150  smprngopr  34164  ispridlc  34182
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