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Theorem pridlnr 36498
Description: A prime ideal is a proper ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
pridlnr.1 𝐺 = (1st β€˜π‘…)
prdilnr.2 𝑋 = ran 𝐺
Assertion
Ref Expression
pridlnr ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) β†’ 𝑃 β‰  𝑋)

Proof of Theorem pridlnr
Dummy variables π‘₯ 𝑦 π‘Ž 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pridlnr.1 . . . 4 𝐺 = (1st β€˜π‘…)
2 eqid 2737 . . . 4 (2nd β€˜π‘…) = (2nd β€˜π‘…)
3 prdilnr.2 . . . 4 𝑋 = ran 𝐺
41, 2, 3ispridl 36496 . . 3 (𝑅 ∈ RingOps β†’ (𝑃 ∈ (PrIdlβ€˜π‘…) ↔ (𝑃 ∈ (Idlβ€˜π‘…) ∧ 𝑃 β‰  𝑋 ∧ 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 3anan12 1097 . . 3 ((𝑃 ∈ (Idlβ€˜π‘…) ∧ 𝑃 β‰  𝑋 ∧ 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 β‰  𝑋 ∧ (𝑃 ∈ (Idlβ€˜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 287 . 2 (𝑅 ∈ RingOps β†’ (𝑃 ∈ (PrIdlβ€˜π‘…) ↔ (𝑃 β‰  𝑋 ∧ (𝑃 ∈ (Idlβ€˜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 500 1 ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) β†’ 𝑃 β‰  𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆ€wral 3065   βŠ† wss 3911  ran crn 5635  β€˜cfv 6497  (class class class)co 7358  1st c1st 7920  2nd c2nd 7921  RingOpscrngo 36356  Idlcidl 36469  PrIdlcpridl 36470
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 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-pridl 36473
This theorem is referenced by: (None)
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