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Theorem idlval 35283
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 6663 . . . . . . 7 (𝑟 = 𝑅 → (1st𝑟) = (1st𝑅))
2 idlval.1 . . . . . . 7 𝐺 = (1st𝑅)
31, 2syl6eqr 2872 . . . . . 6 (𝑟 = 𝑅 → (1st𝑟) = 𝐺)
43rneqd 5801 . . . . 5 (𝑟 = 𝑅 → ran (1st𝑟) = ran 𝐺)
5 idlval.3 . . . . 5 𝑋 = ran 𝐺
64, 5syl6eqr 2872 . . . 4 (𝑟 = 𝑅 → ran (1st𝑟) = 𝑋)
76pweqd 4542 . . 3 (𝑟 = 𝑅 → 𝒫 ran (1st𝑟) = 𝒫 𝑋)
83fveq2d 6667 . . . . . 6 (𝑟 = 𝑅 → (GId‘(1st𝑟)) = (GId‘𝐺))
9 idlval.4 . . . . . 6 𝑍 = (GId‘𝐺)
108, 9syl6eqr 2872 . . . . 5 (𝑟 = 𝑅 → (GId‘(1st𝑟)) = 𝑍)
1110eleq1d 2895 . . . 4 (𝑟 = 𝑅 → ((GId‘(1st𝑟)) ∈ 𝑖𝑍𝑖))
123oveqd 7165 . . . . . . . 8 (𝑟 = 𝑅 → (𝑥(1st𝑟)𝑦) = (𝑥𝐺𝑦))
1312eleq1d 2895 . . . . . . 7 (𝑟 = 𝑅 → ((𝑥(1st𝑟)𝑦) ∈ 𝑖 ↔ (𝑥𝐺𝑦) ∈ 𝑖))
1413ralbidv 3195 . . . . . 6 (𝑟 = 𝑅 → (∀𝑦𝑖 (𝑥(1st𝑟)𝑦) ∈ 𝑖 ↔ ∀𝑦𝑖 (𝑥𝐺𝑦) ∈ 𝑖))
15 fveq2 6663 . . . . . . . . . . 11 (𝑟 = 𝑅 → (2nd𝑟) = (2nd𝑅))
16 idlval.2 . . . . . . . . . . 11 𝐻 = (2nd𝑅)
1715, 16syl6eqr 2872 . . . . . . . . . 10 (𝑟 = 𝑅 → (2nd𝑟) = 𝐻)
1817oveqd 7165 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑧(2nd𝑟)𝑥) = (𝑧𝐻𝑥))
1918eleq1d 2895 . . . . . . . 8 (𝑟 = 𝑅 → ((𝑧(2nd𝑟)𝑥) ∈ 𝑖 ↔ (𝑧𝐻𝑥) ∈ 𝑖))
2017oveqd 7165 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑥(2nd𝑟)𝑧) = (𝑥𝐻𝑧))
2120eleq1d 2895 . . . . . . . 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 3400 . . . . . 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 3195 . . . 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 3484 . 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 35280 . 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 6677 . . . . . 6 𝐺 ∈ V
3029rnex 7609 . . . . 5 ran 𝐺 ∈ V
315, 30eqeltri 2907 . . . 4 𝑋 ∈ V
3231pwex 5272 . . 3 𝒫 𝑋 ∈ V
3332rabex 5226 . 2 {𝑖 ∈ 𝒫 𝑋 ∣ (𝑍𝑖 ∧ ∀𝑥𝑖 (∀𝑦𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))} ∈ V
3427, 28, 33fvmpt 6761 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 398   = wceq 1531  wcel 2108  wral 3136  {crab 3140  Vcvv 3493  𝒫 cpw 4537  ran crn 5549  cfv 6348  (class class class)co 7148  1st c1st 7679  2nd c2nd 7680  GIdcgi 28259  RingOpscrngo 35164  Idlcidl 35277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7151  df-idl 35280
This theorem is referenced by:  isidl  35284
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