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Theorem pridlnr 34257
Description: A prime ideal is a proper ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
pridlnr.1 𝐺 = (1st𝑅)
prdilnr.2 𝑋 = ran 𝐺
Assertion
Ref Expression
pridlnr ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → 𝑃𝑋)

Proof of Theorem pridlnr
Dummy variables 𝑥 𝑦 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pridlnr.1 . . . 4 𝐺 = (1st𝑅)
2 eqid 2765 . . . 4 (2nd𝑅) = (2nd𝑅)
3 prdilnr.2 . . . 4 𝑋 = ran 𝐺
41, 2, 3ispridl 34255 . . 3 (𝑅 ∈ RingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥(2nd𝑅)𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
5 3anan12 1117 . . 3 ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥(2nd𝑅)𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))) ↔ (𝑃𝑋 ∧ (𝑃 ∈ (Idl‘𝑅) ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥(2nd𝑅)𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
64, 5syl6bb 278 . 2 (𝑅 ∈ RingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃𝑋 ∧ (𝑃 ∈ (Idl‘𝑅) ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥(2nd𝑅)𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃))))))
76simprbda 492 1 ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → 𝑃𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wo 873  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wral 3055  wss 3732  ran crn 5278  cfv 6068  (class class class)co 6842  1st c1st 7364  2nd c2nd 7365  RingOpscrngo 34115  Idlcidl 34228  PrIdlcpridl 34229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-iota 6031  df-fun 6070  df-fv 6076  df-ov 6845  df-pridl 34232
This theorem is referenced by: (None)
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