Theorem idlval 38705

Description: Obsolete theorem, use 2idlval 21427 instead. The class of ideals of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)

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 6888	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = (1st ‘𝑅))
2		idlval.1	. . . . . . 7 ⊢ 𝐺 = (1st ‘𝑅)
3	1, 2	eqtr4di 2819	. . . . . 6 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = 𝐺)
4	3	rneqd 5933	. . . . 5 ⊢ (𝑟 = 𝑅 → ran (1st ‘𝑟) = ran 𝐺)
5		idlval.3	. . . . 5 ⊢ 𝑋 = ran 𝐺
6	4, 5	eqtr4di 2819	. . . 4 ⊢ (𝑟 = 𝑅 → ran (1st ‘𝑟) = 𝑋)
7	6	pweqd 4584	. . 3 ⊢ (𝑟 = 𝑅 → 𝒫 ran (1st ‘𝑟) = 𝒫 𝑋)
8	3	fveq2d 6892	. . . . . 6 ⊢ (𝑟 = 𝑅 → (GId‘(1st ‘𝑟)) = (GId‘𝐺))
9		idlval.4	. . . . . 6 ⊢ 𝑍 = (GId‘𝐺)
10	8, 9	eqtr4di 2819	. . . . 5 ⊢ (𝑟 = 𝑅 → (GId‘(1st ‘𝑟)) = 𝑍)
11	10	eleq1d 2851	. . . 4 ⊢ (𝑟 = 𝑅 → ((GId‘(1st ‘𝑟)) ∈ 𝑖 ↔ 𝑍 ∈ 𝑖))
12	3	oveqd 7440	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑥(1st ‘𝑟)𝑦) = (𝑥𝐺𝑦))
13	12	eleq1d 2851	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((𝑥(1st ‘𝑟)𝑦) ∈ 𝑖 ↔ (𝑥𝐺𝑦) ∈ 𝑖))
14	13	ralbidv 3191	. . . . . 6 ⊢ (𝑟 = 𝑅 → (∀𝑦 ∈ 𝑖 (𝑥(1st ‘𝑟)𝑦) ∈ 𝑖 ↔ ∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖))
15		fveq2 6888	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (2nd ‘𝑟) = (2nd ‘𝑅))
16		idlval.2	. . . . . . . . . . 11 ⊢ 𝐻 = (2nd ‘𝑅)
17	15, 16	eqtr4di 2819	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (2nd ‘𝑟) = 𝐻)
18	17	oveqd 7440	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑧(2nd ‘𝑟)𝑥) = (𝑧𝐻𝑥))
19	18	eleq1d 2851	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((𝑧(2nd ‘𝑟)𝑥) ∈ 𝑖 ↔ (𝑧𝐻𝑥) ∈ 𝑖))
20	17	oveqd 7440	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑥(2nd ‘𝑟)𝑧) = (𝑥𝐻𝑧))
21	20	eleq1d 2851	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((𝑥(2nd ‘𝑟)𝑧) ∈ 𝑖 ↔ (𝑥𝐻𝑧) ∈ 𝑖))
22	19, 21	anbi12d 644	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (((𝑧(2nd ‘𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2nd ‘𝑟)𝑧) ∈ 𝑖) ↔ ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))
23	6, 22	raleqbidv 3341	. . . . . 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 644	. . . . 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 3191	. . . 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 644	. . 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 3437	. 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 38702	. 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 6902	. . . . . 6 ⊢ 𝐺 ∈ V
30	29	rnex 7916	. . . . 5 ⊢ ran 𝐺 ∈ V
31	5, 30	eqeltri 2862	. . . 4 ⊢ 𝑋 ∈ V
32	31	pwex 5356	. . 3 ⊢ 𝒫 𝑋 ∈ V
33	32	rabex 5314	. 2 ⊢ {𝑖 ∈ 𝒫 𝑋 ∣ (𝑍 ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))} ∈ V
34	27, 28, 33	fvmpt 6996	1 ⊢ (𝑅 ∈ RingOps → (Idl‘𝑅) = {𝑖 ∈ 𝒫 𝑋 ∣ (𝑍 ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3082  {crab 3419  Vcvv 3458  𝒫 cpw 4567  ran crn 5667  ‘cfv 6543  (class class class)co 7423  1^st c1st 7993  2^nd c2nd 7994  GIdcgi 30879  RingOpscrngo 38586  Idlcidl 38699

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-iota 6499  df-fun 6545  df-fv 6551  df-ov 7426  df-idl 38702

This theorem is used by:  isidl  38706