Theorem isidl 37974

Description: The predicate "is an ideal of the ring 𝑅". (Contributed by Jeff Madsen, 10-Jun-2010.)

Hypotheses

Ref	Expression
idlval.1	⊢ 𝐺 = (1st ‘𝑅)
idlval.2	⊢ 𝐻 = (2nd ‘𝑅)
idlval.3	⊢ 𝑋 = ran 𝐺
idlval.4	⊢ 𝑍 = (GId‘𝐺)

Assertion

Ref	Expression
isidl	⊢ (𝑅 ∈ RingOps → (𝐼 ∈ (Idl‘𝑅) ↔ (𝐼 ⊆ 𝑋 ∧ 𝑍 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((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.

Step	Hyp	Ref	Expression
1		idlval.1	. . . 4 ⊢ 𝐺 = (1st ‘𝑅)
2		idlval.2	. . . 4 ⊢ 𝐻 = (2nd ‘𝑅)
3		idlval.3	. . . 4 ⊢ 𝑋 = ran 𝐺
4		idlval.4	. . . 4 ⊢ 𝑍 = (GId‘𝐺)
5	1, 2, 3, 4	idlval 37973	. . 3 ⊢ (𝑅 ∈ RingOps → (Idl‘𝑅) = {𝑖 ∈ 𝒫 𝑋 ∣ (𝑍 ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))})
6	5	eleq2d 2830	. 2 ⊢ (𝑅 ∈ RingOps → (𝐼 ∈ (Idl‘𝑅) ↔ 𝐼 ∈ {𝑖 ∈ 𝒫 𝑋 ∣ (𝑍 ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))}))
7	1	fvexi 6934	. . . . . . 7 ⊢ 𝐺 ∈ V
8	7	rnex 7950	. . . . . 6 ⊢ ran 𝐺 ∈ V
9	3, 8	eqeltri 2840	. . . . 5 ⊢ 𝑋 ∈ V
10	9	elpw2 5352	. . . 4 ⊢ (𝐼 ∈ 𝒫 𝑋 ↔ 𝐼 ⊆ 𝑋)
11	10	anbi1i 623	. . 3 ⊢ ((𝐼 ∈ 𝒫 𝑋 ∧ (𝑍 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))) ↔ (𝐼 ⊆ 𝑋 ∧ (𝑍 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))))
12		eleq2 2833	. . . . 5 ⊢ (𝑖 = 𝐼 → (𝑍 ∈ 𝑖 ↔ 𝑍 ∈ 𝐼))
13		eleq2 2833	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → ((𝑥𝐺𝑦) ∈ 𝑖 ↔ (𝑥𝐺𝑦) ∈ 𝐼))
14	13	raleqbi1dv 3346	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ↔ ∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼))
15		eleq2 2833	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → ((𝑧𝐻𝑥) ∈ 𝑖 ↔ (𝑧𝐻𝑥) ∈ 𝐼))
16		eleq2 2833	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → ((𝑥𝐻𝑧) ∈ 𝑖 ↔ (𝑥𝐻𝑧) ∈ 𝐼))
17	15, 16	anbi12d 631	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → (((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖) ↔ ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))
18	17	ralbidv 3184	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖) ↔ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))
19	14, 18	anbi12d 631	. . . . . 6 ⊢ (𝑖 = 𝐼 → ((∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)) ↔ (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼))))
20	19	raleqbi1dv 3346	. . . . 5 ⊢ (𝑖 = 𝐼 → (∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)) ↔ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼))))
21	12, 20	anbi12d 631	. . . 4 ⊢ (𝑖 = 𝐼 → ((𝑍 ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖))) ↔ (𝑍 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))))
22	21	elrab 3708	. . 3 ⊢ (𝐼 ∈ {𝑖 ∈ 𝒫 𝑋 ∣ (𝑍 ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))} ↔ (𝐼 ∈ 𝒫 𝑋 ∧ (𝑍 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))))
23		3anass 1095	. . 3 ⊢ ((𝐼 ⊆ 𝑋 ∧ 𝑍 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼))) ↔ (𝐼 ⊆ 𝑋 ∧ (𝑍 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))))
24	11, 22, 23	3bitr4i 303	. 2 ⊢ (𝐼 ∈ {𝑖 ∈ 𝒫 𝑋 ∣ (𝑍 ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))} ↔ (𝐼 ⊆ 𝑋 ∧ 𝑍 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼))))
25	6, 24	bitrdi 287	1 ⊢ (𝑅 ∈ RingOps → (𝐼 ∈ (Idl‘𝑅) ↔ (𝐼 ⊆ 𝑋 ∧ 𝑍 ∈ 𝐼 ∧ ∀𝑥 ∈ 𝐼 (∀𝑦 ∈ 𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067  {crab 3443  Vcvv 3488   ⊆ wss 3976  𝒫 cpw 4622  ran crn 5701  ‘cfv 6573  (class class class)co 7448  1^st c1st 8028  2^nd c2nd 8029  GIdcgi 30522  RingOpscrngo 37854  Idlcidl 37967

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-idl 37970

This theorem is referenced by:  isidlc  37975  idlss  37976  idl0cl  37978  idladdcl  37979  idllmulcl  37980  idlrmulcl  37981  rngoidl  37984  0idl  37985  intidl  37989  unichnidl  37991  keridl  37992