Theorem idlval 38007

Description: The class of ideals of a ring. (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
idlval	⊢ (𝑅 ∈ RingOps → (Idl‘𝑅) = {𝑖 ∈ 𝒫 𝑋 ∣ (𝑍 ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ 𝑋 ((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.

Step	Hyp	Ref	Expression
1		fveq2 6858	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = (1st ‘𝑅))
2		idlval.1	. . . . . . 7 ⊢ 𝐺 = (1st ‘𝑅)
3	1, 2	eqtr4di 2782	. . . . . 6 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = 𝐺)
4	3	rneqd 5902	. . . . 5 ⊢ (𝑟 = 𝑅 → ran (1st ‘𝑟) = ran 𝐺)
5		idlval.3	. . . . 5 ⊢ 𝑋 = ran 𝐺
6	4, 5	eqtr4di 2782	. . . 4 ⊢ (𝑟 = 𝑅 → ran (1st ‘𝑟) = 𝑋)
7	6	pweqd 4580	. . 3 ⊢ (𝑟 = 𝑅 → 𝒫 ran (1st ‘𝑟) = 𝒫 𝑋)
8	3	fveq2d 6862	. . . . . 6 ⊢ (𝑟 = 𝑅 → (GId‘(1st ‘𝑟)) = (GId‘𝐺))
9		idlval.4	. . . . . 6 ⊢ 𝑍 = (GId‘𝐺)
10	8, 9	eqtr4di 2782	. . . . 5 ⊢ (𝑟 = 𝑅 → (GId‘(1st ‘𝑟)) = 𝑍)
11	10	eleq1d 2813	. . . 4 ⊢ (𝑟 = 𝑅 → ((GId‘(1st ‘𝑟)) ∈ 𝑖 ↔ 𝑍 ∈ 𝑖))
12	3	oveqd 7404	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑥(1st ‘𝑟)𝑦) = (𝑥𝐺𝑦))
13	12	eleq1d 2813	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((𝑥(1st ‘𝑟)𝑦) ∈ 𝑖 ↔ (𝑥𝐺𝑦) ∈ 𝑖))
14	13	ralbidv 3156	. . . . . 6 ⊢ (𝑟 = 𝑅 → (∀𝑦 ∈ 𝑖 (𝑥(1st ‘𝑟)𝑦) ∈ 𝑖 ↔ ∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖))
15		fveq2 6858	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (2nd ‘𝑟) = (2nd ‘𝑅))
16		idlval.2	. . . . . . . . . . 11 ⊢ 𝐻 = (2nd ‘𝑅)
17	15, 16	eqtr4di 2782	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (2nd ‘𝑟) = 𝐻)
18	17	oveqd 7404	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑧(2nd ‘𝑟)𝑥) = (𝑧𝐻𝑥))
19	18	eleq1d 2813	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((𝑧(2nd ‘𝑟)𝑥) ∈ 𝑖 ↔ (𝑧𝐻𝑥) ∈ 𝑖))
20	17	oveqd 7404	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑥(2nd ‘𝑟)𝑧) = (𝑥𝐻𝑧))
21	20	eleq1d 2813	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((𝑥(2nd ‘𝑟)𝑧) ∈ 𝑖 ↔ (𝑥𝐻𝑧) ∈ 𝑖))
22	19, 21	anbi12d 632	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (((𝑧(2nd ‘𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2nd ‘𝑟)𝑧) ∈ 𝑖) ↔ ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))
23	6, 22	raleqbidv 3319	. . . . . 6 ⊢ (𝑟 = 𝑅 → (∀𝑧 ∈ ran (1st ‘𝑟)((𝑧(2nd ‘𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2nd ‘𝑟)𝑧) ∈ 𝑖) ↔ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))
24	14, 23	anbi12d 632	. . . . 5 ⊢ (𝑟 = 𝑅 → ((∀𝑦 ∈ 𝑖 (𝑥(1st ‘𝑟)𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ ran (1st ‘𝑟)((𝑧(2nd ‘𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2nd ‘𝑟)𝑧) ∈ 𝑖)) ↔ (∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖))))
25	24	ralbidv 3156	. . . 4 ⊢ (𝑟 = 𝑅 → (∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥(1st ‘𝑟)𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ ran (1st ‘𝑟)((𝑧(2nd ‘𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2nd ‘𝑟)𝑧) ∈ 𝑖)) ↔ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖))))
26	11, 25	anbi12d 632	. . 3 ⊢ (𝑟 = 𝑅 → (((GId‘(1st ‘𝑟)) ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥(1st ‘𝑟)𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ ran (1st ‘𝑟)((𝑧(2nd ‘𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2nd ‘𝑟)𝑧) ∈ 𝑖))) ↔ (𝑍 ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))))
27	7, 26	rabeqbidv 3424	. 2 ⊢ (𝑟 = 𝑅 → {𝑖 ∈ 𝒫 ran (1st ‘𝑟) ∣ ((GId‘(1st ‘𝑟)) ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥(1st ‘𝑟)𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ ran (1st ‘𝑟)((𝑧(2nd ‘𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2nd ‘𝑟)𝑧) ∈ 𝑖)))} = {𝑖 ∈ 𝒫 𝑋 ∣ (𝑍 ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))})
28		df-idl 38004	. 2 ⊢ Idl = (𝑟 ∈ RingOps ↦ {𝑖 ∈ 𝒫 ran (1st ‘𝑟) ∣ ((GId‘(1st ‘𝑟)) ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥(1st ‘𝑟)𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ ran (1st ‘𝑟)((𝑧(2nd ‘𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2nd ‘𝑟)𝑧) ∈ 𝑖)))})
29	2	fvexi 6872	. . . . . 6 ⊢ 𝐺 ∈ V
30	29	rnex 7886	. . . . 5 ⊢ ran 𝐺 ∈ V
31	5, 30	eqeltri 2824	. . . 4 ⊢ 𝑋 ∈ V
32	31	pwex 5335	. . 3 ⊢ 𝒫 𝑋 ∈ V
33	32	rabex 5294	. 2 ⊢ {𝑖 ∈ 𝒫 𝑋 ∣ (𝑍 ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))} ∈ V
34	27, 28, 33	fvmpt 6968	1 ⊢ (𝑅 ∈ RingOps → (Idl‘𝑅) = {𝑖 ∈ 𝒫 𝑋 ∣ (𝑍 ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3405  Vcvv 3447  𝒫 cpw 4563  ran crn 5639  ‘cfv 6511  (class class class)co 7387  1^st c1st 7966  2^nd c2nd 7967  GIdcgi 30419  RingOpscrngo 37888  Idlcidl 38001

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-idl 38004

This theorem is referenced by:  isidl  38008