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Theorem prmidl 32558
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 3323 . . . . 5 (𝑏 = 𝐽 β†’ (βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑃 ↔ βˆ€π‘¦ ∈ 𝐽 (π‘₯ Β· 𝑦) ∈ 𝑃))
21ralbidv 3178 . . . 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 4008 . . . . 5 (𝑏 = 𝐽 β†’ (𝑏 βŠ† 𝑃 ↔ 𝐽 βŠ† 𝑃))
43orbi2d 915 . . . 4 (𝑏 = 𝐽 β†’ ((𝐼 βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃) ↔ (𝐼 βŠ† 𝑃 ∨ 𝐽 βŠ† 𝑃)))
52, 4imbi12d 345 . . 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 3323 . . . . . 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 4008 . . . . . . 7 (π‘Ž = 𝐼 β†’ (π‘Ž βŠ† 𝑃 ↔ 𝐼 βŠ† 𝑃))
87orbi1d 916 . . . . . 6 (π‘Ž = 𝐼 β†’ ((π‘Ž βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃) ↔ (𝐼 βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃)))
96, 8imbi12d 345 . . . . 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 3178 . . . 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 32556 . . . . . . 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 478 . . . . . 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 1145 . . . . 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 482 . . . 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 770 . . . 4 (((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdealβ€˜π‘…)) ∧ (𝐼 ∈ (LIdealβ€˜π‘…) ∧ 𝐽 ∈ (LIdealβ€˜π‘…))) β†’ 𝐼 ∈ (LIdealβ€˜π‘…))
1810, 16, 17rspcdva 3614 . . 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 772 . . 3 (((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdealβ€˜π‘…)) ∧ (𝐼 ∈ (LIdealβ€˜π‘…) ∧ 𝐽 ∈ (LIdealβ€˜π‘…))) β†’ 𝐽 ∈ (LIdealβ€˜π‘…))
205, 18, 19rspcdva 3614 . 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 408 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 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062   βŠ† wss 3949  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  .rcmulr 17198  Ringcrg 20056  LIdealclidl 20783  PrmIdealcprmidl 32553
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 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-prmidl 32554
This theorem is referenced by:  idlmulssprm  32560  isprmidlc  32566
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