Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pridlval Structured version   Visualization version   GIF version

Theorem pridlval 35198
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 6669 . . 3 (𝑟 = 𝑅 → (Idl‘𝑟) = (Idl‘𝑅))
2 fveq2 6669 . . . . . . . 8 (𝑟 = 𝑅 → (1st𝑟) = (1st𝑅))
3 pridlval.1 . . . . . . . 8 𝐺 = (1st𝑅)
42, 3syl6eqr 2879 . . . . . . 7 (𝑟 = 𝑅 → (1st𝑟) = 𝐺)
54rneqd 5807 . . . . . 6 (𝑟 = 𝑅 → ran (1st𝑟) = ran 𝐺)
6 pridlval.3 . . . . . 6 𝑋 = ran 𝐺
75, 6syl6eqr 2879 . . . . 5 (𝑟 = 𝑅 → ran (1st𝑟) = 𝑋)
87neeq2d 3081 . . . 4 (𝑟 = 𝑅 → (𝑖 ≠ ran (1st𝑟) ↔ 𝑖𝑋))
9 fveq2 6669 . . . . . . . . . . 11 (𝑟 = 𝑅 → (2nd𝑟) = (2nd𝑅))
10 pridlval.2 . . . . . . . . . . 11 𝐻 = (2nd𝑅)
119, 10syl6eqr 2879 . . . . . . . . . 10 (𝑟 = 𝑅 → (2nd𝑟) = 𝐻)
1211oveqd 7167 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑥(2nd𝑟)𝑦) = (𝑥𝐻𝑦))
1312eleq1d 2902 . . . . . . . 8 (𝑟 = 𝑅 → ((𝑥(2nd𝑟)𝑦) ∈ 𝑖 ↔ (𝑥𝐻𝑦) ∈ 𝑖))
14132ralbidv 3204 . . . . . . 7 (𝑟 = 𝑅 → (∀𝑥𝑎𝑦𝑏 (𝑥(2nd𝑟)𝑦) ∈ 𝑖 ↔ ∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑖))
1514imbi1d 343 . . . . . 6 (𝑟 = 𝑅 → ((∀𝑥𝑎𝑦𝑏 (𝑥(2nd𝑟)𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)) ↔ (∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖))))
161, 15raleqbidv 3407 . . . . 5 (𝑟 = 𝑅 → (∀𝑏 ∈ (Idl‘𝑟)(∀𝑥𝑎𝑦𝑏 (𝑥(2nd𝑟)𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)) ↔ ∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖))))
171, 16raleqbidv 3407 . . . 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 630 . . 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 3491 . 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 35176 . 2 PrIdl = (𝑟 ∈ RingOps ↦ {𝑖 ∈ (Idl‘𝑟) ∣ (𝑖 ≠ ran (1st𝑟) ∧ ∀𝑎 ∈ (Idl‘𝑟)∀𝑏 ∈ (Idl‘𝑟)(∀𝑥𝑎𝑦𝑏 (𝑥(2nd𝑟)𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))})
21 fvex 6682 . . 3 (Idl‘𝑅) ∈ V
2221rabex 5232 . 2 {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))} ∈ V
2319, 20, 22fvmpt 6767 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 396  wo 843   = wceq 1530  wcel 2107  wne 3021  wral 3143  {crab 3147  wss 3940  ran crn 5555  cfv 6354  (class class class)co 7150  1st c1st 7683  2nd c2nd 7684  RingOpscrngo 35059  Idlcidl 35172  PrIdlcpridl 35173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6313  df-fun 6356  df-fv 6362  df-ov 7153  df-pridl 35176
This theorem is referenced by:  ispridl  35199
  Copyright terms: Public domain W3C validator