idlval - Mathbox for Jeff Madsen

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Theorem idlval 38032

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 6817	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (1^st ‘𝑟) = (1^st ‘𝑅))
2		idlval.1	. . . . . . 7 ⊢ 𝐺 = (1^st ‘𝑅)
3	1, 2	eqtr4di 2783	. . . . . 6 ⊢ (𝑟 = 𝑅 → (1^st ‘𝑟) = 𝐺)
4	3	rneqd 5875	. . . . 5 ⊢ (𝑟 = 𝑅 → ran (1^st ‘𝑟) = ran 𝐺)
5		idlval.3	. . . . 5 ⊢ 𝑋 = ran 𝐺
6	4, 5	eqtr4di 2783	. . . 4 ⊢ (𝑟 = 𝑅 → ran (1^st ‘𝑟) = 𝑋)
7	6	pweqd 4565	. . 3 ⊢ (𝑟 = 𝑅 → 𝒫 ran (1^st ‘𝑟) = 𝒫 𝑋)
8	3	fveq2d 6821	. . . . . 6 ⊢ (𝑟 = 𝑅 → (GId‘(1^st ‘𝑟)) = (GId‘𝐺))
9		idlval.4	. . . . . 6 ⊢ 𝑍 = (GId‘𝐺)
10	8, 9	eqtr4di 2783	. . . . 5 ⊢ (𝑟 = 𝑅 → (GId‘(1^st ‘𝑟)) = 𝑍)
11	10	eleq1d 2814	. . . 4 ⊢ (𝑟 = 𝑅 → ((GId‘(1^st ‘𝑟)) ∈ 𝑖 ↔ 𝑍 ∈ 𝑖))
12	3	oveqd 7358	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑥(1^st ‘𝑟)𝑦) = (𝑥𝐺𝑦))
13	12	eleq1d 2814	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((𝑥(1^st ‘𝑟)𝑦) ∈ 𝑖 ↔ (𝑥𝐺𝑦) ∈ 𝑖))
14	13	ralbidv 3153	. . . . . 6 ⊢ (𝑟 = 𝑅 → (∀𝑦 ∈ 𝑖 (𝑥(1^st ‘𝑟)𝑦) ∈ 𝑖 ↔ ∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖))
15		fveq2 6817	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (2^nd ‘𝑟) = (2^nd ‘𝑅))
16		idlval.2	. . . . . . . . . . 11 ⊢ 𝐻 = (2^nd ‘𝑅)
17	15, 16	eqtr4di 2783	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (2^nd ‘𝑟) = 𝐻)
18	17	oveqd 7358	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑧(2^nd ‘𝑟)𝑥) = (𝑧𝐻𝑥))
19	18	eleq1d 2814	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((𝑧(2^nd ‘𝑟)𝑥) ∈ 𝑖 ↔ (𝑧𝐻𝑥) ∈ 𝑖))
20	17	oveqd 7358	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑥(2^nd ‘𝑟)𝑧) = (𝑥𝐻𝑧))
21	20	eleq1d 2814	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((𝑥(2^nd ‘𝑟)𝑧) ∈ 𝑖 ↔ (𝑥𝐻𝑧) ∈ 𝑖))
22	19, 21	anbi12d 632	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (((𝑧(2^nd ‘𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2^nd ‘𝑟)𝑧) ∈ 𝑖) ↔ ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))
23	6, 22	raleqbidv 3310	. . . . . 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 632	. . . . 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 3153	. . . 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 632	. . 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 3411	. 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 38029	. 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 6831	. . . . . 6 ⊢ 𝐺 ∈ V
30	29	rnex 7835	. . . . 5 ⊢ ran 𝐺 ∈ V
31	5, 30	eqeltri 2825	. . . 4 ⊢ 𝑋 ∈ V
32	31	pwex 5316	. . 3 ⊢ 𝒫 𝑋 ∈ V
33	32	rabex 5275	. 2 ⊢ {𝑖 ∈ 𝒫 𝑋 ∣ (𝑍 ∈ 𝑖 ∧ ∀𝑥 ∈ 𝑖 (∀𝑦 ∈ 𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))} ∈ V
34	27, 28, 33	fvmpt 6924	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 1541 ∈ wcel 2110 ∀wral 3045 {crab 3393 Vcvv 3434 𝒫 cpw 4548 ran crn 5615 ‘cfv 6477 (class class class)co 7341 1^st c1st 7914 2^nd c2nd 7915 GIdcgi 30460 RingOpscrngo 37913 Idlcidl 38026

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7663

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rab 3394 df-v 3436 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4858 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-iota 6433 df-fun 6479 df-fv 6485 df-ov 7344 df-idl 38029

This theorem is referenced by: isidl 38033

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