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Theorem pridl 36499
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 2737 . . . . . . 7 (1st β€˜π‘…) = (1st β€˜π‘…)
2 pridl.1 . . . . . . 7 𝐻 = (2nd β€˜π‘…)
3 eqid 2737 . . . . . . 7 ran (1st β€˜π‘…) = ran (1st β€˜π‘…)
41, 2, 3ispridl 36496 . . . . . 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 1090 . . . . . 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 287 . . . . 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 501 . . . 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 3310 . . . . . 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 3970 . . . . . . 7 (π‘Ž = 𝐴 β†’ (π‘Ž βŠ† 𝑃 ↔ 𝐴 βŠ† 𝑃))
109orbi1d 916 . . . . . 6 (π‘Ž = 𝐴 β†’ ((π‘Ž βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃) ↔ (𝐴 βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃)))
118, 10imbi12d 345 . . . . 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 3310 . . . . . . 7 (𝑏 = 𝐡 β†’ (βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑃 ↔ βˆ€π‘¦ ∈ 𝐡 (π‘₯𝐻𝑦) ∈ 𝑃))
1312ralbidv 3175 . . . . . 6 (𝑏 = 𝐡 β†’ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑃 ↔ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 (π‘₯𝐻𝑦) ∈ 𝑃))
14 sseq1 3970 . . . . . . 7 (𝑏 = 𝐡 β†’ (𝑏 βŠ† 𝑃 ↔ 𝐡 βŠ† 𝑃))
1514orbi2d 915 . . . . . 6 (𝑏 = 𝐡 β†’ ((𝐴 βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃) ↔ (𝐴 βŠ† 𝑃 ∨ 𝐡 βŠ† 𝑃)))
1613, 15imbi12d 345 . . . . 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 3591 . . . 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 417 . 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 1350 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 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆ€wral 3065   βŠ† wss 3911  ran crn 5635  β€˜cfv 6497  (class class class)co 7358  1st c1st 7920  2nd c2nd 7921  RingOpscrngo 36356  Idlcidl 36469  PrIdlcpridl 36470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-pridl 36473
This theorem is referenced by: (None)
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