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Theorem isidl 38021
Description: The predicate "is an ideal of the ring 𝑅". (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
isidl (𝑅 ∈ RingOps → (𝐼 ∈ (Idl‘𝑅) ↔ (𝐼𝑋𝑍𝐼 ∧ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))))
Distinct variable groups:   𝑥,𝑅,𝑦,𝑧   𝑧,𝑋   𝑥,𝐼,𝑦,𝑧
Allowed substitution hints:   𝐺(𝑥,𝑦,𝑧)   𝐻(𝑥,𝑦,𝑧)   𝑋(𝑥,𝑦)   𝑍(𝑥,𝑦,𝑧)

Proof of Theorem isidl
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 idlval.1 . . . 4 𝐺 = (1st𝑅)
2 idlval.2 . . . 4 𝐻 = (2nd𝑅)
3 idlval.3 . . . 4 𝑋 = ran 𝐺
4 idlval.4 . . . 4 𝑍 = (GId‘𝐺)
51, 2, 3, 4idlval 38020 . . 3 (𝑅 ∈ RingOps → (Idl‘𝑅) = {𝑖 ∈ 𝒫 𝑋 ∣ (𝑍𝑖 ∧ ∀𝑥𝑖 (∀𝑦𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))})
65eleq2d 2827 . 2 (𝑅 ∈ RingOps → (𝐼 ∈ (Idl‘𝑅) ↔ 𝐼 ∈ {𝑖 ∈ 𝒫 𝑋 ∣ (𝑍𝑖 ∧ ∀𝑥𝑖 (∀𝑦𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))}))
71fvexi 6920 . . . . . . 7 𝐺 ∈ V
87rnex 7932 . . . . . 6 ran 𝐺 ∈ V
93, 8eqeltri 2837 . . . . 5 𝑋 ∈ V
109elpw2 5334 . . . 4 (𝐼 ∈ 𝒫 𝑋𝐼𝑋)
1110anbi1i 624 . . 3 ((𝐼 ∈ 𝒫 𝑋 ∧ (𝑍𝐼 ∧ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))) ↔ (𝐼𝑋 ∧ (𝑍𝐼 ∧ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))))
12 eleq2 2830 . . . . 5 (𝑖 = 𝐼 → (𝑍𝑖𝑍𝐼))
13 eleq2 2830 . . . . . . . 8 (𝑖 = 𝐼 → ((𝑥𝐺𝑦) ∈ 𝑖 ↔ (𝑥𝐺𝑦) ∈ 𝐼))
1413raleqbi1dv 3338 . . . . . . 7 (𝑖 = 𝐼 → (∀𝑦𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ↔ ∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼))
15 eleq2 2830 . . . . . . . . 9 (𝑖 = 𝐼 → ((𝑧𝐻𝑥) ∈ 𝑖 ↔ (𝑧𝐻𝑥) ∈ 𝐼))
16 eleq2 2830 . . . . . . . . 9 (𝑖 = 𝐼 → ((𝑥𝐻𝑧) ∈ 𝑖 ↔ (𝑥𝐻𝑧) ∈ 𝐼))
1715, 16anbi12d 632 . . . . . . . 8 (𝑖 = 𝐼 → (((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖) ↔ ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))
1817ralbidv 3178 . . . . . . 7 (𝑖 = 𝐼 → (∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖) ↔ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))
1914, 18anbi12d 632 . . . . . 6 (𝑖 = 𝐼 → ((∀𝑦𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)) ↔ (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼))))
2019raleqbi1dv 3338 . . . . 5 (𝑖 = 𝐼 → (∀𝑥𝑖 (∀𝑦𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)) ↔ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼))))
2112, 20anbi12d 632 . . . 4 (𝑖 = 𝐼 → ((𝑍𝑖 ∧ ∀𝑥𝑖 (∀𝑦𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖))) ↔ (𝑍𝐼 ∧ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))))
2221elrab 3692 . . 3 (𝐼 ∈ {𝑖 ∈ 𝒫 𝑋 ∣ (𝑍𝑖 ∧ ∀𝑥𝑖 (∀𝑦𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))} ↔ (𝐼 ∈ 𝒫 𝑋 ∧ (𝑍𝐼 ∧ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))))
23 3anass 1095 . . 3 ((𝐼𝑋𝑍𝐼 ∧ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼))) ↔ (𝐼𝑋 ∧ (𝑍𝐼 ∧ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))))
2411, 22, 233bitr4i 303 . 2 (𝐼 ∈ {𝑖 ∈ 𝒫 𝑋 ∣ (𝑍𝑖 ∧ ∀𝑥𝑖 (∀𝑦𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))} ↔ (𝐼𝑋𝑍𝐼 ∧ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼))))
256, 24bitrdi 287 1 (𝑅 ∈ RingOps → (𝐼 ∈ (Idl‘𝑅) ↔ (𝐼𝑋𝑍𝐼 ∧ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  {crab 3436  Vcvv 3480  wss 3951  𝒫 cpw 4600  ran crn 5686  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  GIdcgi 30509  RingOpscrngo 37901  Idlcidl 38014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-idl 38017
This theorem is referenced by:  isidlc  38022  idlss  38023  idl0cl  38025  idladdcl  38026  idllmulcl  38027  idlrmulcl  38028  rngoidl  38031  0idl  38032  intidl  38036  unichnidl  38038  keridl  38039
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