idlval - Mathbox for Jeff Madsen

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Theorem idlval 35783

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 6668	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (1^st ‘𝑟) = (1^st ‘𝑅))
2		idlval.1	. . . . . . 7 ⊢ 𝐺 = (1^st ‘𝑅)
3	1, 2	eqtr4di 2791	. . . . . 6 ⊢ (𝑟 = 𝑅 → (1^st ‘𝑟) = 𝐺)
4	3	rneqd 5775	. . . . 5 ⊢ (𝑟 = 𝑅 → ran (1^st ‘𝑟) = ran 𝐺)
5		idlval.3	. . . . 5 ⊢ 𝑋 = ran 𝐺
6	4, 5	eqtr4di 2791	. . . 4 ⊢ (𝑟 = 𝑅 → ran (1^st ‘𝑟) = 𝑋)
7	6	pweqd 4504	. . 3 ⊢ (𝑟 = 𝑅 → 𝒫 ran (1^st ‘𝑟) = 𝒫 𝑋)
8	3	fveq2d 6672	. . . . . 6 ⊢ (𝑟 = 𝑅 → (GId‘(1^st ‘𝑟)) = (GId‘𝐺))
9		idlval.4	. . . . . 6 ⊢ 𝑍 = (GId‘𝐺)
10	8, 9	eqtr4di 2791	. . . . 5 ⊢ (𝑟 = 𝑅 → (GId‘(1^st ‘𝑟)) = 𝑍)
11	10	eleq1d 2817	. . . 4 ⊢ (𝑟 = 𝑅 → ((GId‘(1^st ‘𝑟)) ∈ 𝑖 ↔ 𝑍 ∈ 𝑖))
12	3	oveqd 7181	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑥(1^st ‘𝑟)𝑦) = (𝑥𝐺𝑦))
13	12	eleq1d 2817	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((𝑥(1^st ‘𝑟)𝑦) ∈ 𝑖 ↔ (𝑥𝐺𝑦) ∈ 𝑖))
14	13	ralbidv 3109	. . . . . 6 ⊢ (𝑟 = 𝑅 → (∀𝑦 ∈ 𝑖 (𝑥(1^st ‘𝑟)𝑦) ∈ 𝑖 ↔ ∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖))
15		fveq2 6668	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (2^nd ‘𝑟) = (2^nd ‘𝑅))
16		idlval.2	. . . . . . . . . . 11 ⊢ 𝐻 = (2^nd ‘𝑅)
17	15, 16	eqtr4di 2791	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (2^nd ‘𝑟) = 𝐻)
18	17	oveqd 7181	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑧(2^nd ‘𝑟)𝑥) = (𝑧𝐻𝑥))
19	18	eleq1d 2817	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((𝑧(2^nd ‘𝑟)𝑥) ∈ 𝑖 ↔ (𝑧𝐻𝑥) ∈ 𝑖))
20	17	oveqd 7181	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑥(2^nd ‘𝑟)𝑧) = (𝑥𝐻𝑧))
21	20	eleq1d 2817	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((𝑥(2^nd ‘𝑟)𝑧) ∈ 𝑖 ↔ (𝑥𝐻𝑧) ∈ 𝑖))
22	19, 21	anbi12d 634	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (((𝑧(2^nd ‘𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2^nd ‘𝑟)𝑧) ∈ 𝑖) ↔ ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))
23	6, 22	raleqbidv 3303	. . . . . 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 634	. . . . 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 3109	. . . 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 634	. . 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 3386	. 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 35780	. 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 6682	. . . . . 6 ⊢ 𝐺 ∈ V
30	29	rnex 7636	. . . . 5 ⊢ ran 𝐺 ∈ V
31	5, 30	eqeltri 2829	. . . 4 ⊢ 𝑋 ∈ V
32	31	pwex 5244	. . 3 ⊢ 𝒫 𝑋 ∈ V
33	32	rabex 5197	. 2 ⊢ {𝑖 ∈ 𝒫 𝑋 ∣ (𝑍 ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))} ∈ V
34	27, 28, 33	fvmpt 6769	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 399 = wceq 1542 ∈ wcel 2113 ∀wral 3053 {crab 3057 Vcvv 3397 𝒫 cpw 4485 ran crn 5520 ‘cfv 6333 (class class class)co 7164 1^st c1st 7705 2^nd c2nd 7706 GIdcgi 28417 RingOpscrngo 35664 Idlcidl 35777

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-sbc 3680 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6291 df-fun 6335 df-fv 6341 df-ov 7167 df-idl 35780

This theorem is referenced by: isidl 35784

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