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Theorem ispridl 37414
Description: The predicate "is a prime ideal". (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
pridlval.1 𝐺 = (1st β€˜π‘…)
pridlval.2 𝐻 = (2nd β€˜π‘…)
pridlval.3 𝑋 = ran 𝐺
Assertion
Ref Expression
ispridl (𝑅 ∈ RingOps β†’ (𝑃 ∈ (PrIdlβ€˜π‘…) ↔ (𝑃 ∈ (Idlβ€˜π‘…) ∧ 𝑃 β‰  𝑋 ∧ βˆ€π‘Ž ∈ (Idlβ€˜π‘…)βˆ€π‘ ∈ (Idlβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑃 β†’ (π‘Ž βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃)))))
Distinct variable groups:   π‘₯,𝑅,𝑦,π‘Ž,𝑏   π‘₯,𝑃,𝑦,π‘Ž,𝑏
Allowed substitution hints:   𝐺(π‘₯,𝑦,π‘Ž,𝑏)   𝐻(π‘₯,𝑦,π‘Ž,𝑏)   𝑋(π‘₯,𝑦,π‘Ž,𝑏)

Proof of Theorem ispridl
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 pridlval.1 . . . 4 𝐺 = (1st β€˜π‘…)
2 pridlval.2 . . . 4 𝐻 = (2nd β€˜π‘…)
3 pridlval.3 . . . 4 𝑋 = ran 𝐺
41, 2, 3pridlval 37413 . . 3 (𝑅 ∈ RingOps β†’ (PrIdlβ€˜π‘…) = {𝑖 ∈ (Idlβ€˜π‘…) ∣ (𝑖 β‰  𝑋 ∧ βˆ€π‘Ž ∈ (Idlβ€˜π‘…)βˆ€π‘ ∈ (Idlβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖)))})
54eleq2d 2813 . 2 (𝑅 ∈ RingOps β†’ (𝑃 ∈ (PrIdlβ€˜π‘…) ↔ 𝑃 ∈ {𝑖 ∈ (Idlβ€˜π‘…) ∣ (𝑖 β‰  𝑋 ∧ βˆ€π‘Ž ∈ (Idlβ€˜π‘…)βˆ€π‘ ∈ (Idlβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖)))}))
6 neeq1 2997 . . . . 5 (𝑖 = 𝑃 β†’ (𝑖 β‰  𝑋 ↔ 𝑃 β‰  𝑋))
7 eleq2 2816 . . . . . . . 8 (𝑖 = 𝑃 β†’ ((π‘₯𝐻𝑦) ∈ 𝑖 ↔ (π‘₯𝐻𝑦) ∈ 𝑃))
872ralbidv 3212 . . . . . . 7 (𝑖 = 𝑃 β†’ (βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑖 ↔ βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑃))
9 sseq2 4003 . . . . . . . 8 (𝑖 = 𝑃 β†’ (π‘Ž βŠ† 𝑖 ↔ π‘Ž βŠ† 𝑃))
10 sseq2 4003 . . . . . . . 8 (𝑖 = 𝑃 β†’ (𝑏 βŠ† 𝑖 ↔ 𝑏 βŠ† 𝑃))
119, 10orbi12d 915 . . . . . . 7 (𝑖 = 𝑃 β†’ ((π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖) ↔ (π‘Ž βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃)))
128, 11imbi12d 344 . . . . . 6 (𝑖 = 𝑃 β†’ ((βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖)) ↔ (βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑃 β†’ (π‘Ž βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃))))
13122ralbidv 3212 . . . . 5 (𝑖 = 𝑃 β†’ (βˆ€π‘Ž ∈ (Idlβ€˜π‘…)βˆ€π‘ ∈ (Idlβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖)) ↔ βˆ€π‘Ž ∈ (Idlβ€˜π‘…)βˆ€π‘ ∈ (Idlβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑃 β†’ (π‘Ž βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃))))
146, 13anbi12d 630 . . . 4 (𝑖 = 𝑃 β†’ ((𝑖 β‰  𝑋 ∧ βˆ€π‘Ž ∈ (Idlβ€˜π‘…)βˆ€π‘ ∈ (Idlβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖))) ↔ (𝑃 β‰  𝑋 ∧ βˆ€π‘Ž ∈ (Idlβ€˜π‘…)βˆ€π‘ ∈ (Idlβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑃 β†’ (π‘Ž βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃)))))
1514elrab 3678 . . 3 (𝑃 ∈ {𝑖 ∈ (Idlβ€˜π‘…) ∣ (𝑖 β‰  𝑋 ∧ βˆ€π‘Ž ∈ (Idlβ€˜π‘…)βˆ€π‘ ∈ (Idlβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖)))} ↔ (𝑃 ∈ (Idlβ€˜π‘…) ∧ (𝑃 β‰  𝑋 ∧ βˆ€π‘Ž ∈ (Idlβ€˜π‘…)βˆ€π‘ ∈ (Idlβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑃 β†’ (π‘Ž βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃)))))
16 3anass 1092 . . 3 ((𝑃 ∈ (Idlβ€˜π‘…) ∧ 𝑃 β‰  𝑋 ∧ βˆ€π‘Ž ∈ (Idlβ€˜π‘…)βˆ€π‘ ∈ (Idlβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑃 β†’ (π‘Ž βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃))) ↔ (𝑃 ∈ (Idlβ€˜π‘…) ∧ (𝑃 β‰  𝑋 ∧ βˆ€π‘Ž ∈ (Idlβ€˜π‘…)βˆ€π‘ ∈ (Idlβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑃 β†’ (π‘Ž βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃)))))
1715, 16bitr4i 278 . 2 (𝑃 ∈ {𝑖 ∈ (Idlβ€˜π‘…) ∣ (𝑖 β‰  𝑋 ∧ βˆ€π‘Ž ∈ (Idlβ€˜π‘…)βˆ€π‘ ∈ (Idlβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖)))} ↔ (𝑃 ∈ (Idlβ€˜π‘…) ∧ 𝑃 β‰  𝑋 ∧ βˆ€π‘Ž ∈ (Idlβ€˜π‘…)βˆ€π‘ ∈ (Idlβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑃 β†’ (π‘Ž βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃))))
185, 17bitrdi 287 1 (𝑅 ∈ RingOps β†’ (𝑃 ∈ (PrIdlβ€˜π‘…) ↔ (𝑃 ∈ (Idlβ€˜π‘…) ∧ 𝑃 β‰  𝑋 ∧ βˆ€π‘Ž ∈ (Idlβ€˜π‘…)βˆ€π‘ ∈ (Idlβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑃 β†’ (π‘Ž βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 844   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  βˆ€wral 3055  {crab 3426   βŠ† wss 3943  ran crn 5670  β€˜cfv 6536  (class class class)co 7404  1st c1st 7969  2nd c2nd 7970  RingOpscrngo 37274  Idlcidl 37387  PrIdlcpridl 37388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fv 6544  df-ov 7407  df-pridl 37391
This theorem is referenced by:  pridlidl  37415  pridlnr  37416  pridl  37417  ispridl2  37418  smprngopr  37432  ispridlc  37450
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