Theorem idlval 36881

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 6892	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = (1st ‘𝑅))
2		idlval.1	. . . . . . 7 ⊢ 𝐺 = (1st ‘𝑅)
3	1, 2	eqtr4di 2791	. . . . . 6 ⊢ (𝑟 = 𝑅 → (1st ‘𝑟) = 𝐺)
4	3	rneqd 5938	. . . . 5 ⊢ (𝑟 = 𝑅 → ran (1st ‘𝑟) = ran 𝐺)
5		idlval.3	. . . . 5 ⊢ 𝑋 = ran 𝐺
6	4, 5	eqtr4di 2791	. . . 4 ⊢ (𝑟 = 𝑅 → ran (1st ‘𝑟) = 𝑋)
7	6	pweqd 4620	. . 3 ⊢ (𝑟 = 𝑅 → 𝒫 ran (1st ‘𝑟) = 𝒫 𝑋)
8	3	fveq2d 6896	. . . . . 6 ⊢ (𝑟 = 𝑅 → (GId‘(1st ‘𝑟)) = (GId‘𝐺))
9		idlval.4	. . . . . 6 ⊢ 𝑍 = (GId‘𝐺)
10	8, 9	eqtr4di 2791	. . . . 5 ⊢ (𝑟 = 𝑅 → (GId‘(1st ‘𝑟)) = 𝑍)
11	10	eleq1d 2819	. . . 4 ⊢ (𝑟 = 𝑅 → ((GId‘(1st ‘𝑟)) ∈ 𝑖 ↔ 𝑍 ∈ 𝑖))
12	3	oveqd 7426	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑥(1st ‘𝑟)𝑦) = (𝑥𝐺𝑦))
13	12	eleq1d 2819	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((𝑥(1st ‘𝑟)𝑦) ∈ 𝑖 ↔ (𝑥𝐺𝑦) ∈ 𝑖))
14	13	ralbidv 3178	. . . . . 6 ⊢ (𝑟 = 𝑅 → (∀𝑦 ∈ 𝑖 (𝑥(1st ‘𝑟)𝑦) ∈ 𝑖 ↔ ∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖))
15		fveq2 6892	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (2nd ‘𝑟) = (2nd ‘𝑅))
16		idlval.2	. . . . . . . . . . 11 ⊢ 𝐻 = (2nd ‘𝑅)
17	15, 16	eqtr4di 2791	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (2nd ‘𝑟) = 𝐻)
18	17	oveqd 7426	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑧(2nd ‘𝑟)𝑥) = (𝑧𝐻𝑥))
19	18	eleq1d 2819	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((𝑧(2nd ‘𝑟)𝑥) ∈ 𝑖 ↔ (𝑧𝐻𝑥) ∈ 𝑖))
20	17	oveqd 7426	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑥(2nd ‘𝑟)𝑧) = (𝑥𝐻𝑧))
21	20	eleq1d 2819	. . . . . . . 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 3343	. . . . . 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 3178	. . . 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 3450	. 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 36878	. 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 6906	. . . . . 6 ⊢ 𝐺 ∈ V
30	29	rnex 7903	. . . . 5 ⊢ ran 𝐺 ∈ V
31	5, 30	eqeltri 2830	. . . 4 ⊢ 𝑋 ∈ V
32	31	pwex 5379	. . 3 ⊢ 𝒫 𝑋 ∈ V
33	32	rabex 5333	. 2 ⊢ {𝑖 ∈ 𝒫 𝑋 ∣ (𝑍 ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))} ∈ V
34	27, 28, 33	fvmpt 6999	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 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  {crab 3433  Vcvv 3475  𝒫 cpw 4603  ran crn 5678  ‘cfv 6544  (class class class)co 7409  1^st c1st 7973  2^nd c2nd 7974  GIdcgi 29743  RingOpscrngo 36762  Idlcidl 36875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-idl 36878

This theorem is referenced by:  isidl  36882