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Theorem pridlval 34157
Description: The class of prime ideals of a ring 𝑅. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
pridlval.1 𝐺 = (1st𝑅)
pridlval.2 𝐻 = (2nd𝑅)
pridlval.3 𝑋 = ran 𝐺
Assertion
Ref Expression
pridlval (𝑅 ∈ RingOps → (PrIdl‘𝑅) = {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))})
Distinct variable groups:   𝑅,𝑖,𝑥,𝑦,𝑎,𝑏   𝑖,𝑋   𝑖,𝐻
Allowed substitution hints:   𝐺(𝑥,𝑦,𝑖,𝑎,𝑏)   𝐻(𝑥,𝑦,𝑎,𝑏)   𝑋(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem pridlval
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6332 . . 3 (𝑟 = 𝑅 → (Idl‘𝑟) = (Idl‘𝑅))
2 fveq2 6332 . . . . . . . 8 (𝑟 = 𝑅 → (1st𝑟) = (1st𝑅))
3 pridlval.1 . . . . . . . 8 𝐺 = (1st𝑅)
42, 3syl6eqr 2822 . . . . . . 7 (𝑟 = 𝑅 → (1st𝑟) = 𝐺)
54rneqd 5491 . . . . . 6 (𝑟 = 𝑅 → ran (1st𝑟) = ran 𝐺)
6 pridlval.3 . . . . . 6 𝑋 = ran 𝐺
75, 6syl6eqr 2822 . . . . 5 (𝑟 = 𝑅 → ran (1st𝑟) = 𝑋)
87neeq2d 3002 . . . 4 (𝑟 = 𝑅 → (𝑖 ≠ ran (1st𝑟) ↔ 𝑖𝑋))
9 fveq2 6332 . . . . . . . . . . 11 (𝑟 = 𝑅 → (2nd𝑟) = (2nd𝑅))
10 pridlval.2 . . . . . . . . . . 11 𝐻 = (2nd𝑅)
119, 10syl6eqr 2822 . . . . . . . . . 10 (𝑟 = 𝑅 → (2nd𝑟) = 𝐻)
1211oveqd 6809 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑥(2nd𝑟)𝑦) = (𝑥𝐻𝑦))
1312eleq1d 2834 . . . . . . . 8 (𝑟 = 𝑅 → ((𝑥(2nd𝑟)𝑦) ∈ 𝑖 ↔ (𝑥𝐻𝑦) ∈ 𝑖))
14132ralbidv 3137 . . . . . . 7 (𝑟 = 𝑅 → (∀𝑥𝑎𝑦𝑏 (𝑥(2nd𝑟)𝑦) ∈ 𝑖 ↔ ∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑖))
1514imbi1d 330 . . . . . 6 (𝑟 = 𝑅 → ((∀𝑥𝑎𝑦𝑏 (𝑥(2nd𝑟)𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)) ↔ (∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖))))
161, 15raleqbidv 3300 . . . . 5 (𝑟 = 𝑅 → (∀𝑏 ∈ (Idl‘𝑟)(∀𝑥𝑎𝑦𝑏 (𝑥(2nd𝑟)𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)) ↔ ∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖))))
171, 16raleqbidv 3300 . . . 4 (𝑟 = 𝑅 → (∀𝑎 ∈ (Idl‘𝑟)∀𝑏 ∈ (Idl‘𝑟)(∀𝑥𝑎𝑦𝑏 (𝑥(2nd𝑟)𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)) ↔ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖))))
188, 17anbi12d 608 . . 3 (𝑟 = 𝑅 → ((𝑖 ≠ ran (1st𝑟) ∧ ∀𝑎 ∈ (Idl‘𝑟)∀𝑏 ∈ (Idl‘𝑟)(∀𝑥𝑎𝑦𝑏 (𝑥(2nd𝑟)𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖))) ↔ (𝑖𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))))
191, 18rabeqbidv 3344 . 2 (𝑟 = 𝑅 → {𝑖 ∈ (Idl‘𝑟) ∣ (𝑖 ≠ ran (1st𝑟) ∧ ∀𝑎 ∈ (Idl‘𝑟)∀𝑏 ∈ (Idl‘𝑟)(∀𝑥𝑎𝑦𝑏 (𝑥(2nd𝑟)𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))} = {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))})
20 df-pridl 34135 . 2 PrIdl = (𝑟 ∈ RingOps ↦ {𝑖 ∈ (Idl‘𝑟) ∣ (𝑖 ≠ ran (1st𝑟) ∧ ∀𝑎 ∈ (Idl‘𝑟)∀𝑏 ∈ (Idl‘𝑟)(∀𝑥𝑎𝑦𝑏 (𝑥(2nd𝑟)𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))})
21 fvex 6342 . . 3 (Idl‘𝑅) ∈ V
2221rabex 4943 . 2 {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))} ∈ V
2319, 20, 22fvmpt 6424 1 (𝑅 ∈ RingOps → (PrIdl‘𝑅) = {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wo 826   = wceq 1630  wcel 2144  wne 2942  wral 3060  {crab 3064  wss 3721  ran crn 5250  cfv 6031  (class class class)co 6792  1st c1st 7312  2nd c2nd 7313  RingOpscrngo 34018  Idlcidl 34131  PrIdlcpridl 34132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6795  df-pridl 34135
This theorem is referenced by:  ispridl  34158
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