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Theorem pridlidl 37207
Description: A prime ideal is an ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)
Assertion
Ref Expression
pridlidl ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) β†’ 𝑃 ∈ (Idlβ€˜π‘…))

Proof of Theorem pridlidl
Dummy variables π‘₯ 𝑦 π‘Ž 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2731 . . . 4 (1st β€˜π‘…) = (1st β€˜π‘…)
2 eqid 2731 . . . 4 (2nd β€˜π‘…) = (2nd β€˜π‘…)
3 eqid 2731 . . . 4 ran (1st β€˜π‘…) = ran (1st β€˜π‘…)
41, 2, 3ispridl 37206 . . 3 (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‘₯(2nd β€˜π‘…)𝑦) ∈ 𝑃 β†’ (π‘Ž βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃)))))
5 3anass 1094 . . 3 ((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‘₯(2nd β€˜π‘…)𝑦) ∈ 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‘₯(2nd β€˜π‘…)𝑦) ∈ 𝑃 β†’ (π‘Ž βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃)))))
64, 5bitrdi 286 . 2 (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‘₯(2nd β€˜π‘…)𝑦) ∈ 𝑃 β†’ (π‘Ž βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃))))))
76simprbda 498 1 ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) β†’ 𝑃 ∈ (Idlβ€˜π‘…))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∨ wo 844   ∧ w3a 1086   ∈ wcel 2105   β‰  wne 2939  βˆ€wral 3060   βŠ† wss 3949  ran crn 5678  β€˜cfv 6544  (class class class)co 7412  1st c1st 7976  2nd c2nd 7977  RingOpscrngo 37066  Idlcidl 37179  PrIdlcpridl 37180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  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-rn 5688  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7415  df-pridl 37183
This theorem is referenced by: (None)
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