idlval - Mathbox for Jeff Madsen

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Theorem idlval 37973

Description: The class of ideals of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)

**Hypotheses**
Ref	Expression
idlval.1	⊢ 𝐺 = (1^st ‘𝑅)
idlval.2	⊢ 𝐻 = (2^nd ‘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 6920	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (1^st ‘𝑟) = (1^st ‘𝑅))
2		idlval.1	. . . . . . 7 ⊢ 𝐺 = (1^st ‘𝑅)
3	1, 2	eqtr4di 2798	. . . . . 6 ⊢ (𝑟 = 𝑅 → (1^st ‘𝑟) = 𝐺)
4	3	rneqd 5963	. . . . 5 ⊢ (𝑟 = 𝑅 → ran (1^st ‘𝑟) = ran 𝐺)
5		idlval.3	. . . . 5 ⊢ 𝑋 = ran 𝐺
6	4, 5	eqtr4di 2798	. . . 4 ⊢ (𝑟 = 𝑅 → ran (1^st ‘𝑟) = 𝑋)
7	6	pweqd 4639	. . 3 ⊢ (𝑟 = 𝑅 → 𝒫 ran (1^st ‘𝑟) = 𝒫 𝑋)
8	3	fveq2d 6924	. . . . . 6 ⊢ (𝑟 = 𝑅 → (GId‘(1^st ‘𝑟)) = (GId‘𝐺))
9		idlval.4	. . . . . 6 ⊢ 𝑍 = (GId‘𝐺)
10	8, 9	eqtr4di 2798	. . . . 5 ⊢ (𝑟 = 𝑅 → (GId‘(1^st ‘𝑟)) = 𝑍)
11	10	eleq1d 2829	. . . 4 ⊢ (𝑟 = 𝑅 → ((GId‘(1^st ‘𝑟)) ∈ 𝑖 ↔ 𝑍 ∈ 𝑖))
12	3	oveqd 7465	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑥(1^st ‘𝑟)𝑦) = (𝑥𝐺𝑦))
13	12	eleq1d 2829	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((𝑥(1^st ‘𝑟)𝑦) ∈ 𝑖 ↔ (𝑥𝐺𝑦) ∈ 𝑖))
14	13	ralbidv 3184	. . . . . 6 ⊢ (𝑟 = 𝑅 → (∀𝑦 ∈ 𝑖 (𝑥(1^st ‘𝑟)𝑦) ∈ 𝑖 ↔ ∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖))
15		fveq2 6920	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (2^nd ‘𝑟) = (2^nd ‘𝑅))
16		idlval.2	. . . . . . . . . . 11 ⊢ 𝐻 = (2^nd ‘𝑅)
17	15, 16	eqtr4di 2798	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (2^nd ‘𝑟) = 𝐻)
18	17	oveqd 7465	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑧(2^nd ‘𝑟)𝑥) = (𝑧𝐻𝑥))
19	18	eleq1d 2829	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((𝑧(2^nd ‘𝑟)𝑥) ∈ 𝑖 ↔ (𝑧𝐻𝑥) ∈ 𝑖))
20	17	oveqd 7465	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑥(2^nd ‘𝑟)𝑧) = (𝑥𝐻𝑧))
21	20	eleq1d 2829	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((𝑥(2^nd ‘𝑟)𝑧) ∈ 𝑖 ↔ (𝑥𝐻𝑧) ∈ 𝑖))
22	19, 21	anbi12d 631	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (((𝑧(2^nd ‘𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2^nd ‘𝑟)𝑧) ∈ 𝑖) ↔ ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))
23	6, 22	raleqbidv 3354	. . . . . 6 ⊢ (𝑟 = 𝑅 → (∀𝑧 ∈ ran (1^st ‘𝑟)((𝑧(2^nd ‘𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2^nd ‘𝑟)𝑧) ∈ 𝑖) ↔ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))
24	14, 23	anbi12d 631	. . . . 5 ⊢ (𝑟 = 𝑅 → ((∀𝑦 ∈ 𝑖 (𝑥(1^st ‘𝑟)𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ ran (1^st ‘𝑟)((𝑧(2^nd ‘𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2^nd ‘𝑟)𝑧) ∈ 𝑖)) ↔ (∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖))))
25	24	ralbidv 3184	. . . 4 ⊢ (𝑟 = 𝑅 → (∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥(1^st ‘𝑟)𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ ran (1^st ‘𝑟)((𝑧(2^nd ‘𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2^nd ‘𝑟)𝑧) ∈ 𝑖)) ↔ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖))))
26	11, 25	anbi12d 631	. . 3 ⊢ (𝑟 = 𝑅 → (((GId‘(1^st ‘𝑟)) ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥(1^st ‘𝑟)𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ ran (1^st ‘𝑟)((𝑧(2^nd ‘𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2^nd ‘𝑟)𝑧) ∈ 𝑖))) ↔ (𝑍 ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))))
27	7, 26	rabeqbidv 3462	. 2 ⊢ (𝑟 = 𝑅 → {𝑖 ∈ 𝒫 ran (1^st ‘𝑟) ∣ ((GId‘(1^st ‘𝑟)) ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥(1^st ‘𝑟)𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ ran (1^st ‘𝑟)((𝑧(2^nd ‘𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2^nd ‘𝑟)𝑧) ∈ 𝑖)))} = {𝑖 ∈ 𝒫 𝑋 ∣ (𝑍 ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))})
28		df-idl 37970	. 2 ⊢ Idl = (𝑟 ∈ RingOps ↦ {𝑖 ∈ 𝒫 ran (1^st ‘𝑟) ∣ ((GId‘(1^st ‘𝑟)) ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥(1^st ‘𝑟)𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ ran (1^st ‘𝑟)((𝑧(2^nd ‘𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2^nd ‘𝑟)𝑧) ∈ 𝑖)))})
29	2	fvexi 6934	. . . . . 6 ⊢ 𝐺 ∈ V
30	29	rnex 7950	. . . . 5 ⊢ ran 𝐺 ∈ V
31	5, 30	eqeltri 2840	. . . 4 ⊢ 𝑋 ∈ V
32	31	pwex 5398	. . 3 ⊢ 𝒫 𝑋 ∈ V
33	32	rabex 5357	. 2 ⊢ {𝑖 ∈ 𝒫 𝑋 ∣ (𝑍 ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))} ∈ V
34	27, 28, 33	fvmpt 7029	1 ⊢ (𝑅 ∈ RingOps → (Idl‘𝑅) = {𝑖 ∈ 𝒫 𝑋 ∣ (𝑍 ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∀wral 3067 {crab 3443 Vcvv 3488 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: isidl 37974

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