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Theorem idlval 36475
Description: The class of ideals of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
idlval.1 𝐺 = (1st𝑅)
idlval.2 𝐻 = (2nd𝑅)
idlval.3 𝑋 = ran 𝐺
idlval.4 𝑍 = (GId‘𝐺)
Assertion
Ref Expression
idlval (𝑅 ∈ RingOps → (Idl‘𝑅) = {𝑖 ∈ 𝒫 𝑋 ∣ (𝑍𝑖 ∧ ∀𝑥𝑖 (∀𝑦𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))})
Distinct variable groups:   𝑥,𝑅,𝑦,𝑧,𝑖   𝑧,𝑋,𝑖   𝑖,𝑍   𝑖,𝐺   𝑖,𝐻
Allowed substitution hints:   𝐺(𝑥,𝑦,𝑧)   𝐻(𝑥,𝑦,𝑧)   𝑋(𝑥,𝑦)   𝑍(𝑥,𝑦,𝑧)

Proof of Theorem idlval
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6843 . . . . . . 7 (𝑟 = 𝑅 → (1st𝑟) = (1st𝑅))
2 idlval.1 . . . . . . 7 𝐺 = (1st𝑅)
31, 2eqtr4di 2795 . . . . . 6 (𝑟 = 𝑅 → (1st𝑟) = 𝐺)
43rneqd 5894 . . . . 5 (𝑟 = 𝑅 → ran (1st𝑟) = ran 𝐺)
5 idlval.3 . . . . 5 𝑋 = ran 𝐺
64, 5eqtr4di 2795 . . . 4 (𝑟 = 𝑅 → ran (1st𝑟) = 𝑋)
76pweqd 4578 . . 3 (𝑟 = 𝑅 → 𝒫 ran (1st𝑟) = 𝒫 𝑋)
83fveq2d 6847 . . . . . 6 (𝑟 = 𝑅 → (GId‘(1st𝑟)) = (GId‘𝐺))
9 idlval.4 . . . . . 6 𝑍 = (GId‘𝐺)
108, 9eqtr4di 2795 . . . . 5 (𝑟 = 𝑅 → (GId‘(1st𝑟)) = 𝑍)
1110eleq1d 2823 . . . 4 (𝑟 = 𝑅 → ((GId‘(1st𝑟)) ∈ 𝑖𝑍𝑖))
123oveqd 7375 . . . . . . . 8 (𝑟 = 𝑅 → (𝑥(1st𝑟)𝑦) = (𝑥𝐺𝑦))
1312eleq1d 2823 . . . . . . 7 (𝑟 = 𝑅 → ((𝑥(1st𝑟)𝑦) ∈ 𝑖 ↔ (𝑥𝐺𝑦) ∈ 𝑖))
1413ralbidv 3175 . . . . . 6 (𝑟 = 𝑅 → (∀𝑦𝑖 (𝑥(1st𝑟)𝑦) ∈ 𝑖 ↔ ∀𝑦𝑖 (𝑥𝐺𝑦) ∈ 𝑖))
15 fveq2 6843 . . . . . . . . . . 11 (𝑟 = 𝑅 → (2nd𝑟) = (2nd𝑅))
16 idlval.2 . . . . . . . . . . 11 𝐻 = (2nd𝑅)
1715, 16eqtr4di 2795 . . . . . . . . . 10 (𝑟 = 𝑅 → (2nd𝑟) = 𝐻)
1817oveqd 7375 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑧(2nd𝑟)𝑥) = (𝑧𝐻𝑥))
1918eleq1d 2823 . . . . . . . 8 (𝑟 = 𝑅 → ((𝑧(2nd𝑟)𝑥) ∈ 𝑖 ↔ (𝑧𝐻𝑥) ∈ 𝑖))
2017oveqd 7375 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑥(2nd𝑟)𝑧) = (𝑥𝐻𝑧))
2120eleq1d 2823 . . . . . . . 8 (𝑟 = 𝑅 → ((𝑥(2nd𝑟)𝑧) ∈ 𝑖 ↔ (𝑥𝐻𝑧) ∈ 𝑖))
2219, 21anbi12d 632 . . . . . . 7 (𝑟 = 𝑅 → (((𝑧(2nd𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2nd𝑟)𝑧) ∈ 𝑖) ↔ ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))
236, 22raleqbidv 3320 . . . . . 6 (𝑟 = 𝑅 → (∀𝑧 ∈ ran (1st𝑟)((𝑧(2nd𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2nd𝑟)𝑧) ∈ 𝑖) ↔ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))
2414, 23anbi12d 632 . . . . 5 (𝑟 = 𝑅 → ((∀𝑦𝑖 (𝑥(1st𝑟)𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ ran (1st𝑟)((𝑧(2nd𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2nd𝑟)𝑧) ∈ 𝑖)) ↔ (∀𝑦𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖))))
2524ralbidv 3175 . . . 4 (𝑟 = 𝑅 → (∀𝑥𝑖 (∀𝑦𝑖 (𝑥(1st𝑟)𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ ran (1st𝑟)((𝑧(2nd𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2nd𝑟)𝑧) ∈ 𝑖)) ↔ ∀𝑥𝑖 (∀𝑦𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖))))
2611, 25anbi12d 632 . . 3 (𝑟 = 𝑅 → (((GId‘(1st𝑟)) ∈ 𝑖 ∧ ∀𝑥𝑖 (∀𝑦𝑖 (𝑥(1st𝑟)𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ ran (1st𝑟)((𝑧(2nd𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2nd𝑟)𝑧) ∈ 𝑖))) ↔ (𝑍𝑖 ∧ ∀𝑥𝑖 (∀𝑦𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))))
277, 26rabeqbidv 3425 . 2 (𝑟 = 𝑅 → {𝑖 ∈ 𝒫 ran (1st𝑟) ∣ ((GId‘(1st𝑟)) ∈ 𝑖 ∧ ∀𝑥𝑖 (∀𝑦𝑖 (𝑥(1st𝑟)𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ ran (1st𝑟)((𝑧(2nd𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2nd𝑟)𝑧) ∈ 𝑖)))} = {𝑖 ∈ 𝒫 𝑋 ∣ (𝑍𝑖 ∧ ∀𝑥𝑖 (∀𝑦𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))})
28 df-idl 36472 . 2 Idl = (𝑟 ∈ RingOps ↦ {𝑖 ∈ 𝒫 ran (1st𝑟) ∣ ((GId‘(1st𝑟)) ∈ 𝑖 ∧ ∀𝑥𝑖 (∀𝑦𝑖 (𝑥(1st𝑟)𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ ran (1st𝑟)((𝑧(2nd𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2nd𝑟)𝑧) ∈ 𝑖)))})
292fvexi 6857 . . . . . 6 𝐺 ∈ V
3029rnex 7850 . . . . 5 ran 𝐺 ∈ V
315, 30eqeltri 2834 . . . 4 𝑋 ∈ V
3231pwex 5336 . . 3 𝒫 𝑋 ∈ V
3332rabex 5290 . 2 {𝑖 ∈ 𝒫 𝑋 ∣ (𝑍𝑖 ∧ ∀𝑥𝑖 (∀𝑦𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))} ∈ V
3427, 28, 33fvmpt 6949 1 (𝑅 ∈ RingOps → (Idl‘𝑅) = {𝑖 ∈ 𝒫 𝑋 ∣ (𝑍𝑖 ∧ ∀𝑥𝑖 (∀𝑦𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3065  {crab 3408  Vcvv 3446  𝒫 cpw 4561  ran crn 5635  cfv 6497  (class class class)co 7358  1st c1st 7920  2nd c2nd 7921  GIdcgi 29435  RingOpscrngo 36356  Idlcidl 36469
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-pow 5321  ax-pr 5385  ax-un 7673
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-pw 4563  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-idl 36472
This theorem is referenced by:  isidl  36476
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