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Theorem prmidl 32820
Description: The main property of a prime ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
Hypotheses
Ref Expression
prmidlval.1 𝐡 = (Baseβ€˜π‘…)
prmidlval.2 Β· = (.rβ€˜π‘…)
Assertion
Ref Expression
prmidl ((((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdealβ€˜π‘…)) ∧ (𝐼 ∈ (LIdealβ€˜π‘…) ∧ 𝐽 ∈ (LIdealβ€˜π‘…))) ∧ βˆ€π‘₯ ∈ 𝐼 βˆ€π‘¦ ∈ 𝐽 (π‘₯ Β· 𝑦) ∈ 𝑃) β†’ (𝐼 βŠ† 𝑃 ∨ 𝐽 βŠ† 𝑃))
Distinct variable groups:   π‘₯,𝑅,𝑦   π‘₯,𝑃,𝑦   π‘₯,𝐼   π‘₯,𝐽,𝑦
Allowed substitution hints:   𝐡(π‘₯,𝑦)   Β· (π‘₯,𝑦)   𝐼(𝑦)

Proof of Theorem prmidl
Dummy variables π‘Ž 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 raleq 3322 . . . . 5 (𝑏 = 𝐽 β†’ (βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑃 ↔ βˆ€π‘¦ ∈ 𝐽 (π‘₯ Β· 𝑦) ∈ 𝑃))
21ralbidv 3177 . . . 4 (𝑏 = 𝐽 β†’ (βˆ€π‘₯ ∈ 𝐼 βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑃 ↔ βˆ€π‘₯ ∈ 𝐼 βˆ€π‘¦ ∈ 𝐽 (π‘₯ Β· 𝑦) ∈ 𝑃))
3 sseq1 4007 . . . . 5 (𝑏 = 𝐽 β†’ (𝑏 βŠ† 𝑃 ↔ 𝐽 βŠ† 𝑃))
43orbi2d 914 . . . 4 (𝑏 = 𝐽 β†’ ((𝐼 βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃) ↔ (𝐼 βŠ† 𝑃 ∨ 𝐽 βŠ† 𝑃)))
52, 4imbi12d 344 . . 3 (𝑏 = 𝐽 β†’ ((βˆ€π‘₯ ∈ 𝐼 βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑃 β†’ (𝐼 βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃)) ↔ (βˆ€π‘₯ ∈ 𝐼 βˆ€π‘¦ ∈ 𝐽 (π‘₯ Β· 𝑦) ∈ 𝑃 β†’ (𝐼 βŠ† 𝑃 ∨ 𝐽 βŠ† 𝑃))))
6 raleq 3322 . . . . . 6 (π‘Ž = 𝐼 β†’ (βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑃 ↔ βˆ€π‘₯ ∈ 𝐼 βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑃))
7 sseq1 4007 . . . . . . 7 (π‘Ž = 𝐼 β†’ (π‘Ž βŠ† 𝑃 ↔ 𝐼 βŠ† 𝑃))
87orbi1d 915 . . . . . 6 (π‘Ž = 𝐼 β†’ ((π‘Ž βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃) ↔ (𝐼 βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃)))
96, 8imbi12d 344 . . . . 5 (π‘Ž = 𝐼 β†’ ((βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑃 β†’ (π‘Ž βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃)) ↔ (βˆ€π‘₯ ∈ 𝐼 βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑃 β†’ (𝐼 βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃))))
109ralbidv 3177 . . . 4 (π‘Ž = 𝐼 β†’ (βˆ€π‘ ∈ (LIdealβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑃 β†’ (π‘Ž βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃)) ↔ βˆ€π‘ ∈ (LIdealβ€˜π‘…)(βˆ€π‘₯ ∈ 𝐼 βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑃 β†’ (𝐼 βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃))))
11 prmidlval.1 . . . . . . . 8 𝐡 = (Baseβ€˜π‘…)
12 prmidlval.2 . . . . . . . 8 Β· = (.rβ€˜π‘…)
1311, 12isprmidl 32818 . . . . . . 7 (𝑅 ∈ Ring β†’ (𝑃 ∈ (PrmIdealβ€˜π‘…) ↔ (𝑃 ∈ (LIdealβ€˜π‘…) ∧ 𝑃 β‰  𝐡 ∧ βˆ€π‘Ž ∈ (LIdealβ€˜π‘…)βˆ€π‘ ∈ (LIdealβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑃 β†’ (π‘Ž βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃)))))
1413biimpa 477 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdealβ€˜π‘…)) β†’ (𝑃 ∈ (LIdealβ€˜π‘…) ∧ 𝑃 β‰  𝐡 ∧ βˆ€π‘Ž ∈ (LIdealβ€˜π‘…)βˆ€π‘ ∈ (LIdealβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑃 β†’ (π‘Ž βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃))))
1514simp3d 1144 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdealβ€˜π‘…)) β†’ βˆ€π‘Ž ∈ (LIdealβ€˜π‘…)βˆ€π‘ ∈ (LIdealβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑃 β†’ (π‘Ž βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃)))
1615adantr 481 . . . 4 (((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdealβ€˜π‘…)) ∧ (𝐼 ∈ (LIdealβ€˜π‘…) ∧ 𝐽 ∈ (LIdealβ€˜π‘…))) β†’ βˆ€π‘Ž ∈ (LIdealβ€˜π‘…)βˆ€π‘ ∈ (LIdealβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑃 β†’ (π‘Ž βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃)))
17 simprl 769 . . . 4 (((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdealβ€˜π‘…)) ∧ (𝐼 ∈ (LIdealβ€˜π‘…) ∧ 𝐽 ∈ (LIdealβ€˜π‘…))) β†’ 𝐼 ∈ (LIdealβ€˜π‘…))
1810, 16, 17rspcdva 3613 . . 3 (((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdealβ€˜π‘…)) ∧ (𝐼 ∈ (LIdealβ€˜π‘…) ∧ 𝐽 ∈ (LIdealβ€˜π‘…))) β†’ βˆ€π‘ ∈ (LIdealβ€˜π‘…)(βˆ€π‘₯ ∈ 𝐼 βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑃 β†’ (𝐼 βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃)))
19 simprr 771 . . 3 (((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdealβ€˜π‘…)) ∧ (𝐼 ∈ (LIdealβ€˜π‘…) ∧ 𝐽 ∈ (LIdealβ€˜π‘…))) β†’ 𝐽 ∈ (LIdealβ€˜π‘…))
205, 18, 19rspcdva 3613 . 2 (((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdealβ€˜π‘…)) ∧ (𝐼 ∈ (LIdealβ€˜π‘…) ∧ 𝐽 ∈ (LIdealβ€˜π‘…))) β†’ (βˆ€π‘₯ ∈ 𝐼 βˆ€π‘¦ ∈ 𝐽 (π‘₯ Β· 𝑦) ∈ 𝑃 β†’ (𝐼 βŠ† 𝑃 ∨ 𝐽 βŠ† 𝑃)))
2120imp 407 1 ((((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdealβ€˜π‘…)) ∧ (𝐼 ∈ (LIdealβ€˜π‘…) ∧ 𝐽 ∈ (LIdealβ€˜π‘…))) ∧ βˆ€π‘₯ ∈ 𝐼 βˆ€π‘¦ ∈ 𝐽 (π‘₯ Β· 𝑦) ∈ 𝑃) β†’ (𝐼 βŠ† 𝑃 ∨ 𝐽 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 3948  β€˜cfv 6543  (class class class)co 7411  Basecbs 17148  .rcmulr 17202  Ringcrg 20127  LIdealclidl 20928  PrmIdealcprmidl 32815
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 5299  ax-nul 5306  ax-pr 5427
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 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7414  df-prmidl 32816
This theorem is referenced by:  idlmulssprm  32822  isprmidlc  32828
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