Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pridl Structured version   Visualization version   GIF version
Theorem pridl 37504
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 2728 . . . . . . 7 (1st β€˜π‘…) = (1st β€˜π‘…)
2 pridl.1 . . . . . . 7 𝐻 = (2nd β€˜π‘…)
3 eqid 2728 . . . . . . 7 ran (1st β€˜π‘…) = ran (1st β€˜π‘…)
41, 2, 3ispridl 37501 . . . . . 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 1087 . . . . . 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 3318 . . . . . 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 4003 . . . . . . 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 3318 . . . . . . 7 (𝑏 = 𝐡 β†’ (βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑃 ↔ βˆ€π‘¦ ∈ 𝐡 (π‘₯𝐻𝑦) ∈ 𝑃))
1312ralbidv 3173 . . . . . 6 (𝑏 = 𝐡 β†’ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑃 ↔ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 (π‘₯𝐻𝑦) ∈ 𝑃))
14 sseq1 4003 . . . . . . 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 3619 . . . 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 1347 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 846   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   β‰  wne 2936  βˆ€wral 3057   βŠ† wss 3945  ran crn 5673  β€˜cfv 6542  (class class class)co 7414  1st c1st 7985  2nd c2nd 7986  RingOpscrngo 37361  Idlcidl 37474  PrIdlcpridl 37475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7417  df-pridl 37478
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator