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Theorem pridl 36893
Description: The main property of a prime ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypothesis
Ref Expression
pridl.1 𝐻 = (2nd β€˜π‘…)
Assertion
Ref Expression
pridl (((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) ∧ (𝐴 ∈ (Idlβ€˜π‘…) ∧ 𝐡 ∈ (Idlβ€˜π‘…) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 (π‘₯𝐻𝑦) ∈ 𝑃)) β†’ (𝐴 βŠ† 𝑃 ∨ 𝐡 βŠ† 𝑃))
Distinct variable groups:   π‘₯,𝑅,𝑦   π‘₯,𝑃,𝑦   π‘₯,𝐴   π‘₯,𝐡,𝑦
Allowed substitution hints:   𝐴(𝑦)   𝐻(π‘₯,𝑦)
Proof of Theorem pridl
Dummy variables π‘Ž 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2732 . . . . . . 7 (1st β€˜π‘…) = (1st β€˜π‘…)
2 pridl.1 . . . . . . 7 𝐻 = (2nd β€˜π‘…)
3 eqid 2732 . . . . . . 7 ran (1st β€˜π‘…) = ran (1st β€˜π‘…)
41, 2, 3ispridl 36890 . . . . . 6 (𝑅 ∈ RingOps β†’ (𝑃 ∈ (PrIdlβ€˜π‘…) ↔ (𝑃 ∈ (Idlβ€˜π‘…) ∧ 𝑃 β‰  ran (1st β€˜π‘…) ∧ 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)))))
5 df-3an 1089 . . . . . 6 ((𝑃 ∈ (Idlβ€˜π‘…) ∧ 𝑃 β‰  ran (1st β€˜π‘…) ∧ 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‰  ran (1st β€˜π‘…)) ∧ 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))))
64, 5bitrdi 286 . . . . 5 (𝑅 ∈ RingOps β†’ (𝑃 ∈ (PrIdlβ€˜π‘…) ↔ ((𝑃 ∈ (Idlβ€˜π‘…) ∧ 𝑃 β‰  ran (1st β€˜π‘…)) ∧ 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)))))
76simplbda 500 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) β†’ 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)))
8 raleq 3322 . . . . . 6 (π‘Ž = 𝐴 β†’ (βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑃 ↔ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑃))
9 sseq1 4006 . . . . . . 7 (π‘Ž = 𝐴 β†’ (π‘Ž βŠ† 𝑃 ↔ 𝐴 βŠ† 𝑃))
109orbi1d 915 . . . . . 6 (π‘Ž = 𝐴 β†’ ((π‘Ž βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃) ↔ (𝐴 βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃)))
118, 10imbi12d 344 . . . . 5 (π‘Ž = 𝐴 β†’ ((βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑃 β†’ (π‘Ž βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃)) ↔ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑃 β†’ (𝐴 βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃))))
12 raleq 3322 . . . . . . 7 (𝑏 = 𝐡 β†’ (βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑃 ↔ βˆ€π‘¦ ∈ 𝐡 (π‘₯𝐻𝑦) ∈ 𝑃))
1312ralbidv 3177 . . . . . 6 (𝑏 = 𝐡 β†’ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑃 ↔ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 (π‘₯𝐻𝑦) ∈ 𝑃))
14 sseq1 4006 . . . . . . 7 (𝑏 = 𝐡 β†’ (𝑏 βŠ† 𝑃 ↔ 𝐡 βŠ† 𝑃))
1514orbi2d 914 . . . . . 6 (𝑏 = 𝐡 β†’ ((𝐴 βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃) ↔ (𝐴 βŠ† 𝑃 ∨ 𝐡 βŠ† 𝑃)))
1613, 15imbi12d 344 . . . . 5 (𝑏 = 𝐡 β†’ ((βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑃 β†’ (𝐴 βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃)) ↔ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 (π‘₯𝐻𝑦) ∈ 𝑃 β†’ (𝐴 βŠ† 𝑃 ∨ 𝐡 βŠ† 𝑃))))
1711, 16rspc2v 3621 . . . 4 ((𝐴 ∈ (Idlβ€˜π‘…) ∧ 𝐡 ∈ (Idlβ€˜π‘…)) β†’ (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)) β†’ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 (π‘₯𝐻𝑦) ∈ 𝑃 β†’ (𝐴 βŠ† 𝑃 ∨ 𝐡 βŠ† 𝑃))))
187, 17syl5com 31 . . 3 ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) β†’ ((𝐴 ∈ (Idlβ€˜π‘…) ∧ 𝐡 ∈ (Idlβ€˜π‘…)) β†’ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 (π‘₯𝐻𝑦) ∈ 𝑃 β†’ (𝐴 βŠ† 𝑃 ∨ 𝐡 βŠ† 𝑃))))
1918expd 416 . 2 ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) β†’ (𝐴 ∈ (Idlβ€˜π‘…) β†’ (𝐡 ∈ (Idlβ€˜π‘…) β†’ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 (π‘₯𝐻𝑦) ∈ 𝑃 β†’ (𝐴 βŠ† 𝑃 ∨ 𝐡 βŠ† 𝑃)))))
20193imp2 1349 1 (((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) ∧ (𝐴 ∈ (Idlβ€˜π‘…) ∧ 𝐡 ∈ (Idlβ€˜π‘…) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 (π‘₯𝐻𝑦) ∈ 𝑃)) β†’ (𝐴 βŠ† 𝑃 ∨ 𝐡 βŠ† 𝑃))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061   βŠ† wss 3947  ran crn 5676  β€˜cfv 6540  (class class class)co 7405  1st c1st 7969  2nd c2nd 7970  RingOpscrngo 36750  Idlcidl 36863  PrIdlcpridl 36864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7408  df-pridl 36867
This theorem is referenced by: (None)
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