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Theorem pridl 37399
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 2724 . . . . . . 7 (1st β€˜π‘…) = (1st β€˜π‘…)
2 pridl.1 . . . . . . 7 𝐻 = (2nd β€˜π‘…)
3 eqid 2724 . . . . . . 7 ran (1st β€˜π‘…) = ran (1st β€˜π‘…)
41, 2, 3ispridl 37396 . . . . . 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 1086 . . . . . 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 499 . . . 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 3314 . . . . . 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 4000 . . . . . . 7 (π‘Ž = 𝐴 β†’ (π‘Ž βŠ† 𝑃 ↔ 𝐴 βŠ† 𝑃))
109orbi1d 913 . . . . . 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 3314 . . . . . . 7 (𝑏 = 𝐡 β†’ (βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑃 ↔ βˆ€π‘¦ ∈ 𝐡 (π‘₯𝐻𝑦) ∈ 𝑃))
1312ralbidv 3169 . . . . . 6 (𝑏 = 𝐡 β†’ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑃 ↔ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 (π‘₯𝐻𝑦) ∈ 𝑃))
14 sseq1 4000 . . . . . . 7 (𝑏 = 𝐡 β†’ (𝑏 βŠ† 𝑃 ↔ 𝐡 βŠ† 𝑃))
1514orbi2d 912 . . . . . 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 3615 . . . 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 415 . 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 1346 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 395   ∨ wo 844   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2932  βˆ€wral 3053   βŠ† wss 3941  ran crn 5668  β€˜cfv 6534  (class class class)co 7402  1st c1st 7967  2nd c2nd 7968  RingOpscrngo 37256  Idlcidl 37369  PrIdlcpridl 37370
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 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
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 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6486  df-fun 6536  df-fv 6542  df-ov 7405  df-pridl 37373
This theorem is referenced by: (None)
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