Theorem mxidlprm 33977

Description: Every maximal ideal is prime. Statement in [Lang] p. 92. (Contributed by Thierry Arnoux, 21-Jan-2024.)

Hypothesis

Ref	Expression
mxidlprm.1	⊢ × = (LSSum‘(mulGrp‘𝑅))

Assertion

Ref	Expression
mxidlprm	⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (PrmIdeal‘𝑅))

Proof of Theorem mxidlprm

Dummy variables 𝑎 𝑘 𝑢 𝑏 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		crngring 20452	. . 3 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2	1	adantr 486	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑅 ∈ Ring)
3		eqid 2761	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
4	3	mxidlidl 33970	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))
5	1, 4	sylan 592	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))
6	3	mxidlnr 33971	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ≠ (Base‘𝑅))
7	1, 6	sylan 592	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ≠ (Base‘𝑅))
8		simpllr 788	. . . . . . . . . . . . 13 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) ∧ 𝑢 ∈ 𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r‘𝑅) = (𝑢(+g‘𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r‘𝑅)𝑥)) → (1r‘𝑅) = (𝑢(+g‘𝑅)𝑘))
9		simpr 490	. . . . . . . . . . . . . 14 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) ∧ 𝑢 ∈ 𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r‘𝑅) = (𝑢(+g‘𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r‘𝑅)𝑥)) → 𝑘 = (𝑎(.r‘𝑅)𝑥))
10	9	oveq2d 7428	. . . . . . . . . . . . 13 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) ∧ 𝑢 ∈ 𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r‘𝑅) = (𝑢(+g‘𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r‘𝑅)𝑥)) → (𝑢(+g‘𝑅)𝑘) = (𝑢(+g‘𝑅)(𝑎(.r‘𝑅)𝑥)))
11	8, 10	eqtrd 2796	. . . . . . . . . . . 12 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) ∧ 𝑢 ∈ 𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r‘𝑅) = (𝑢(+g‘𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r‘𝑅)𝑥)) → (1r‘𝑅) = (𝑢(+g‘𝑅)(𝑎(.r‘𝑅)𝑥)))
12	11	oveq1d 7427	. . . . . . . . . . 11 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) ∧ 𝑢 ∈ 𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r‘𝑅) = (𝑢(+g‘𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r‘𝑅)𝑥)) → ((1r‘𝑅)(.r‘𝑅)𝑦) = ((𝑢(+g‘𝑅)(𝑎(.r‘𝑅)𝑥))(.r‘𝑅)𝑦))
13	2	ad4antr 745	. . . . . . . . . . . . 13 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → 𝑅 ∈ Ring)
14	13	ad5antr 747	. . . . . . . . . . . 12 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) ∧ 𝑢 ∈ 𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r‘𝑅) = (𝑢(+g‘𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r‘𝑅)𝑥)) → 𝑅 ∈ Ring)
15		simp-8r 804	. . . . . . . . . . . 12 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) ∧ 𝑢 ∈ 𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r‘𝑅) = (𝑢(+g‘𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r‘𝑅)𝑥)) → 𝑦 ∈ (Base‘𝑅))
16		eqid 2761	. . . . . . . . . . . . 13 ⊢ (.r‘𝑅) = (.r‘𝑅)
17		eqid 2761	. . . . . . . . . . . . 13 ⊢ (1r‘𝑅) = (1r‘𝑅)
18	3, 16, 17	ringlidm 20478	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ (Base‘𝑅)) → ((1r‘𝑅)(.r‘𝑅)𝑦) = 𝑦)
19	14, 15, 18	syl2anc 596	. . . . . . . . . . 11 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) ∧ 𝑢 ∈ 𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r‘𝑅) = (𝑢(+g‘𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r‘𝑅)𝑥)) → ((1r‘𝑅)(.r‘𝑅)𝑦) = 𝑦)
20		eqid 2761	. . . . . . . . . . . . . . . . 17 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
21	3, 20	lidlss 21470	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ (LIdeal‘𝑅) → 𝑀 ⊆ (Base‘𝑅))
22	5, 21	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ⊆ (Base‘𝑅))
23	22	ad4antr 745	. . . . . . . . . . . . . 14 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → 𝑀 ⊆ (Base‘𝑅))
24	23	ad5antr 747	. . . . . . . . . . . . 13 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) ∧ 𝑢 ∈ 𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r‘𝑅) = (𝑢(+g‘𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r‘𝑅)𝑥)) → 𝑀 ⊆ (Base‘𝑅))
25		simp-5r 798	. . . . . . . . . . . . 13 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) ∧ 𝑢 ∈ 𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r‘𝑅) = (𝑢(+g‘𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r‘𝑅)𝑥)) → 𝑢 ∈ 𝑀)
26	24, 25	sseldd 3932	. . . . . . . . . . . 12 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) ∧ 𝑢 ∈ 𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r‘𝑅) = (𝑢(+g‘𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r‘𝑅)𝑥)) → 𝑢 ∈ (Base‘𝑅))
27		simplr 781	. . . . . . . . . . . . 13 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) ∧ 𝑢 ∈ 𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r‘𝑅) = (𝑢(+g‘𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r‘𝑅)𝑥)) → 𝑎 ∈ (Base‘𝑅))
28		simp-4r 796	. . . . . . . . . . . . . 14 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → 𝑥 ∈ (Base‘𝑅))
29	28	ad5antr 747	. . . . . . . . . . . . 13 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) ∧ 𝑢 ∈ 𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r‘𝑅) = (𝑢(+g‘𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r‘𝑅)𝑥)) → 𝑥 ∈ (Base‘𝑅))
30	3, 16	ringcl 20457	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑎(.r‘𝑅)𝑥) ∈ (Base‘𝑅))
31	14, 27, 29, 30	syl3anc 1398	. . . . . . . . . . . 12 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) ∧ 𝑢 ∈ 𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r‘𝑅) = (𝑢(+g‘𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r‘𝑅)𝑥)) → (𝑎(.r‘𝑅)𝑥) ∈ (Base‘𝑅))
32		eqid 2761	. . . . . . . . . . . . 13 ⊢ (+g‘𝑅) = (+g‘𝑅)
33	3, 32, 16	ringdir 20470	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ (𝑢 ∈ (Base‘𝑅) ∧ (𝑎(.r‘𝑅)𝑥) ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((𝑢(+g‘𝑅)(𝑎(.r‘𝑅)𝑥))(.r‘𝑅)𝑦) = ((𝑢(.r‘𝑅)𝑦)(+g‘𝑅)((𝑎(.r‘𝑅)𝑥)(.r‘𝑅)𝑦)))
34	14, 26, 31, 15, 33	syl13anc 1399	. . . . . . . . . . 11 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) ∧ 𝑢 ∈ 𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r‘𝑅) = (𝑢(+g‘𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r‘𝑅)𝑥)) → ((𝑢(+g‘𝑅)(𝑎(.r‘𝑅)𝑥))(.r‘𝑅)𝑦) = ((𝑢(.r‘𝑅)𝑦)(+g‘𝑅)((𝑎(.r‘𝑅)𝑥)(.r‘𝑅)𝑦)))
35	12, 19, 34	3eqtr3d 2804	. . . . . . . . . 10 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) ∧ 𝑢 ∈ 𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r‘𝑅) = (𝑢(+g‘𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r‘𝑅)𝑥)) → 𝑦 = ((𝑢(.r‘𝑅)𝑦)(+g‘𝑅)((𝑎(.r‘𝑅)𝑥)(.r‘𝑅)𝑦)))
36		simp-5r 798	. . . . . . . . . . . . 13 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → 𝑀 ∈ (MaxIdeal‘𝑅))
37	13, 36, 4	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → 𝑀 ∈ (LIdeal‘𝑅))
38	37	ad5antr 747	. . . . . . . . . . 11 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) ∧ 𝑢 ∈ 𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r‘𝑅) = (𝑢(+g‘𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r‘𝑅)𝑥)) → 𝑀 ∈ (LIdeal‘𝑅))
39		simp-10l 807	. . . . . . . . . . . . 13 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) ∧ 𝑢 ∈ 𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r‘𝑅) = (𝑢(+g‘𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r‘𝑅)𝑥)) → 𝑅 ∈ CRing)
40	3, 16	crngcom 20458	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑢 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑅)𝑢) = (𝑢(.r‘𝑅)𝑦))
41	39, 15, 26, 40	syl3anc 1398	. . . . . . . . . . . 12 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) ∧ 𝑢 ∈ 𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r‘𝑅) = (𝑢(+g‘𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r‘𝑅)𝑥)) → (𝑦(.r‘𝑅)𝑢) = (𝑢(.r‘𝑅)𝑦))
42	20, 3, 16	lidlmcl 21484	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ (LIdeal‘𝑅)) ∧ (𝑦 ∈ (Base‘𝑅) ∧ 𝑢 ∈ 𝑀)) → (𝑦(.r‘𝑅)𝑢) ∈ 𝑀)
43	14, 38, 15, 25, 42	syl22anc 852	. . . . . . . . . . . 12 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) ∧ 𝑢 ∈ 𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r‘𝑅) = (𝑢(+g‘𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r‘𝑅)𝑥)) → (𝑦(.r‘𝑅)𝑢) ∈ 𝑀)
44	41, 43	eqeltrrd 2862	. . . . . . . . . . 11 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) ∧ 𝑢 ∈ 𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r‘𝑅) = (𝑢(+g‘𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r‘𝑅)𝑥)) → (𝑢(.r‘𝑅)𝑦) ∈ 𝑀)
45	3, 16	ringass 20460	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((𝑎(.r‘𝑅)𝑥)(.r‘𝑅)𝑦) = (𝑎(.r‘𝑅)(𝑥(.r‘𝑅)𝑦)))
46	14, 27, 29, 15, 45	syl13anc 1399	. . . . . . . . . . . 12 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) ∧ 𝑢 ∈ 𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r‘𝑅) = (𝑢(+g‘𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r‘𝑅)𝑥)) → ((𝑎(.r‘𝑅)𝑥)(.r‘𝑅)𝑦) = (𝑎(.r‘𝑅)(𝑥(.r‘𝑅)𝑦)))
47		simp-7r 802	. . . . . . . . . . . . 13 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) ∧ 𝑢 ∈ 𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r‘𝑅) = (𝑢(+g‘𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r‘𝑅)𝑥)) → (𝑥(.r‘𝑅)𝑦) ∈ 𝑀)
48	20, 3, 16	lidlmcl 21484	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ (LIdeal‘𝑅)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀)) → (𝑎(.r‘𝑅)(𝑥(.r‘𝑅)𝑦)) ∈ 𝑀)
49	14, 38, 27, 47, 48	syl22anc 852	. . . . . . . . . . . 12 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) ∧ 𝑢 ∈ 𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r‘𝑅) = (𝑢(+g‘𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r‘𝑅)𝑥)) → (𝑎(.r‘𝑅)(𝑥(.r‘𝑅)𝑦)) ∈ 𝑀)
50	46, 49	eqeltrd 2861	. . . . . . . . . . 11 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) ∧ 𝑢 ∈ 𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r‘𝑅) = (𝑢(+g‘𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r‘𝑅)𝑥)) → ((𝑎(.r‘𝑅)𝑥)(.r‘𝑅)𝑦) ∈ 𝑀)
51	20, 32	lidlacl 21480	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ (LIdeal‘𝑅)) ∧ ((𝑢(.r‘𝑅)𝑦) ∈ 𝑀 ∧ ((𝑎(.r‘𝑅)𝑥)(.r‘𝑅)𝑦) ∈ 𝑀)) → ((𝑢(.r‘𝑅)𝑦)(+g‘𝑅)((𝑎(.r‘𝑅)𝑥)(.r‘𝑅)𝑦)) ∈ 𝑀)
52	14, 38, 44, 50, 51	syl22anc 852	. . . . . . . . . 10 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) ∧ 𝑢 ∈ 𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r‘𝑅) = (𝑢(+g‘𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r‘𝑅)𝑥)) → ((𝑢(.r‘𝑅)𝑦)(+g‘𝑅)((𝑎(.r‘𝑅)𝑥)(.r‘𝑅)𝑦)) ∈ 𝑀)
53	35, 52	eqeltrd 2861	. . . . . . . . 9 ⊢ (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) ∧ 𝑢 ∈ 𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r‘𝑅) = (𝑢(+g‘𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r‘𝑅)𝑥)) → 𝑦 ∈ 𝑀)
54		simplr 781	. . . . . . . . . 10 ⊢ (((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) ∧ 𝑢 ∈ 𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r‘𝑅) = (𝑢(+g‘𝑅)𝑘)) → 𝑘 ∈ ((Base‘𝑅) × {𝑥}))
55		eqid 2761	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
56	55, 3	mgpbas 20345	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅))
57	55, 16	mgpplusg 20344	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (+g‘(mulGrp‘𝑅))
58		mxidlprm.1	. . . . . . . . . . . 12 ⊢ × = (LSSum‘(mulGrp‘𝑅))
59		fvexd 6892	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → (mulGrp‘𝑅) ∈ V)
60		ssidd 3954	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → (Base‘𝑅) ⊆ (Base‘𝑅))
61	56, 57, 58, 59, 60, 28	elgrplsmsn 33927	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → (𝑘 ∈ ((Base‘𝑅) × {𝑥}) ↔ ∃𝑎 ∈ (Base‘𝑅)𝑘 = (𝑎(.r‘𝑅)𝑥)))
62	61	ad3antrrr 743	. . . . . . . . . 10 ⊢ (((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) ∧ 𝑢 ∈ 𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r‘𝑅) = (𝑢(+g‘𝑅)𝑘)) → (𝑘 ∈ ((Base‘𝑅) × {𝑥}) ↔ ∃𝑎 ∈ (Base‘𝑅)𝑘 = (𝑎(.r‘𝑅)𝑥)))
63	54, 62	mpbid 235	. . . . . . . . 9 ⊢ (((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) ∧ 𝑢 ∈ 𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r‘𝑅) = (𝑢(+g‘𝑅)𝑘)) → ∃𝑎 ∈ (Base‘𝑅)𝑘 = (𝑎(.r‘𝑅)𝑥))
64	53, 63	r19.29a 3171	. . . . . . . 8 ⊢ (((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) ∧ 𝑢 ∈ 𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r‘𝑅) = (𝑢(+g‘𝑅)𝑘)) → 𝑦 ∈ 𝑀)
65	3, 17	ringidcl 20474	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
66	13, 65	syl 18	. . . . . . . . . 10 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → (1r‘𝑅) ∈ (Base‘𝑅))
67		eqid 2761	. . . . . . . . . . . . 13 ⊢ (LSSum‘𝑅) = (LSSum‘𝑅)
68		eqid 2761	. . . . . . . . . . . . 13 ⊢ (RSpan‘𝑅) = (RSpan‘𝑅)
69		eqid 2761	. . . . . . . . . . . . . . . 16 ⊢ (LPIdeal‘𝑅) = (LPIdeal‘𝑅)
70	69, 20	lpiss 21633	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ Ring → (LPIdeal‘𝑅) ⊆ (LIdeal‘𝑅))
71	13, 70	syl 18	. . . . . . . . . . . . . 14 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → (LPIdeal‘𝑅) ⊆ (LIdeal‘𝑅))
72	3, 55, 58, 68, 13, 28	lsmsnidl 33934	. . . . . . . . . . . . . 14 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → ((Base‘𝑅) × {𝑥}) ∈ (LPIdeal‘𝑅))
73	71, 72	sseldd 3932	. . . . . . . . . . . . 13 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → ((Base‘𝑅) × {𝑥}) ∈ (LIdeal‘𝑅))
74	3, 67, 68, 13, 37, 73	lsmidl 21518	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) ∈ (LIdeal‘𝑅))
75		rlmlmod 21458	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod)
76	13, 75	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → (ringLMod‘𝑅) ∈ LMod)
77		rlmbas 21448	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝑅) = (Base‘(ringLMod‘𝑅))
78		rspval 21469	. . . . . . . . . . . . . . . 16 ⊢ (RSpan‘𝑅) = (LSpan‘(ringLMod‘𝑅))
79	77, 78	lspssid 21240	. . . . . . . . . . . . . . 15 ⊢ (((ringLMod‘𝑅) ∈ LMod ∧ 𝑀 ⊆ (Base‘𝑅)) → 𝑀 ⊆ ((RSpan‘𝑅)‘𝑀))
80	76, 23, 79	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → 𝑀 ⊆ ((RSpan‘𝑅)‘𝑀))
81	28	snssd 4747	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → {𝑥} ⊆ (Base‘𝑅))
82	3, 55, 58, 13, 60, 81	ringlsmss 33930	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → ((Base‘𝑅) × {𝑥}) ⊆ (Base‘𝑅))
83	23, 82	unssd 4138	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → (𝑀 ∪ ((Base‘𝑅) × {𝑥})) ⊆ (Base‘𝑅))
84		ssun1 4124	. . . . . . . . . . . . . . . 16 ⊢ 𝑀 ⊆ (𝑀 ∪ ((Base‘𝑅) × {𝑥}))
85	84	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → 𝑀 ⊆ (𝑀 ∪ ((Base‘𝑅) × {𝑥})))
86	77, 78	lspss 21239	. . . . . . . . . . . . . . 15 ⊢ (((ringLMod‘𝑅) ∈ LMod ∧ (𝑀 ∪ ((Base‘𝑅) × {𝑥})) ⊆ (Base‘𝑅) ∧ 𝑀 ⊆ (𝑀 ∪ ((Base‘𝑅) × {𝑥}))) → ((RSpan‘𝑅)‘𝑀) ⊆ ((RSpan‘𝑅)‘(𝑀 ∪ ((Base‘𝑅) × {𝑥}))))
87	76, 83, 85, 86	syl3anc 1398	. . . . . . . . . . . . . 14 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → ((RSpan‘𝑅)‘𝑀) ⊆ ((RSpan‘𝑅)‘(𝑀 ∪ ((Base‘𝑅) × {𝑥}))))
88	80, 87	sstrd 3941	. . . . . . . . . . . . 13 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → 𝑀 ⊆ ((RSpan‘𝑅)‘(𝑀 ∪ ((Base‘𝑅) × {𝑥}))))
89	3, 67, 68, 13, 37, 73	lsmidllsp 21517	. . . . . . . . . . . . 13 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) = ((RSpan‘𝑅)‘(𝑀 ∪ ((Base‘𝑅) × {𝑥}))))
90	88, 89	sseqtrrd 3968	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → 𝑀 ⊆ (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})))
91	3	mxidlmax 33972	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ ((𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) ∈ (LIdeal‘𝑅) ∧ 𝑀 ⊆ (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})))) → ((𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) = 𝑀 ∨ (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) = (Base‘𝑅)))
92	13, 36, 74, 90, 91	syl22anc 852	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → ((𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) = 𝑀 ∨ (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) = (Base‘𝑅)))
93		eqid 2761	. . . . . . . . . . . . . . . . 17 ⊢ (0g‘𝑅) = (0g‘𝑅)
94	20, 93	lidl0cl 21479	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (LIdeal‘𝑅)) → (0g‘𝑅) ∈ 𝑀)
95	13, 37, 94	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → (0g‘𝑅) ∈ 𝑀)
96		oveq1 7419	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = (0g‘𝑅) → (𝑎(+g‘𝑅)𝑏) = ((0g‘𝑅)(+g‘𝑅)𝑏))
97	96	eqeq2d 2772	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = (0g‘𝑅) → (𝑥 = (𝑎(+g‘𝑅)𝑏) ↔ 𝑥 = ((0g‘𝑅)(+g‘𝑅)𝑏)))
98	97	rexbidv 3187	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = (0g‘𝑅) → (∃𝑏 ∈ ((Base‘𝑅) × {𝑥})𝑥 = (𝑎(+g‘𝑅)𝑏) ↔ ∃𝑏 ∈ ((Base‘𝑅) × {𝑥})𝑥 = ((0g‘𝑅)(+g‘𝑅)𝑏)))
99	98	adantl 487	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) ∧ 𝑎 = (0g‘𝑅)) → (∃𝑏 ∈ ((Base‘𝑅) × {𝑥})𝑥 = (𝑎(+g‘𝑅)𝑏) ↔ ∃𝑏 ∈ ((Base‘𝑅) × {𝑥})𝑥 = ((0g‘𝑅)(+g‘𝑅)𝑏)))
100		oveq1 7419	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = (1r‘𝑅) → (𝑎(.r‘𝑅)𝑥) = ((1r‘𝑅)(.r‘𝑅)𝑥))
101	100	eqeq2d 2772	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = (1r‘𝑅) → (𝑥 = (𝑎(.r‘𝑅)𝑥) ↔ 𝑥 = ((1r‘𝑅)(.r‘𝑅)𝑥)))
102	101	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) ∧ 𝑎 = (1r‘𝑅)) → (𝑥 = (𝑎(.r‘𝑅)𝑥) ↔ 𝑥 = ((1r‘𝑅)(.r‘𝑅)𝑥)))
103	3, 16, 17	ringlidm 20478	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → ((1r‘𝑅)(.r‘𝑅)𝑥) = 𝑥)
104	13, 28, 103	syl2anc 596	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → ((1r‘𝑅)(.r‘𝑅)𝑥) = 𝑥)
105	104	eqcomd 2767	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → 𝑥 = ((1r‘𝑅)(.r‘𝑅)𝑥))
106	66, 102, 105	rspcedvd 3579	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → ∃𝑎 ∈ (Base‘𝑅)𝑥 = (𝑎(.r‘𝑅)𝑥))
107	56, 57, 58, 59, 60, 28	elgrplsmsn 33927	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → (𝑥 ∈ ((Base‘𝑅) × {𝑥}) ↔ ∃𝑎 ∈ (Base‘𝑅)𝑥 = (𝑎(.r‘𝑅)𝑥)))
108	106, 107	mpbird 260	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → 𝑥 ∈ ((Base‘𝑅) × {𝑥}))
109		oveq2 7420	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑥 → ((0g‘𝑅)(+g‘𝑅)𝑏) = ((0g‘𝑅)(+g‘𝑅)𝑥))
110	109	eqeq2d 2772	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑥 → (𝑥 = ((0g‘𝑅)(+g‘𝑅)𝑏) ↔ 𝑥 = ((0g‘𝑅)(+g‘𝑅)𝑥)))
111	110	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) ∧ 𝑏 = 𝑥) → (𝑥 = ((0g‘𝑅)(+g‘𝑅)𝑏) ↔ 𝑥 = ((0g‘𝑅)(+g‘𝑅)𝑥)))
112		ringgrp 20444	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
113	13, 112	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → 𝑅 ∈ Grp)
114	3, 32, 93	grplid 19158	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → ((0g‘𝑅)(+g‘𝑅)𝑥) = 𝑥)
115	113, 28, 114	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → ((0g‘𝑅)(+g‘𝑅)𝑥) = 𝑥)
116	115	eqcomd 2767	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → 𝑥 = ((0g‘𝑅)(+g‘𝑅)𝑥))
117	108, 111, 116	rspcedvd 3579	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → ∃𝑏 ∈ ((Base‘𝑅) × {𝑥})𝑥 = ((0g‘𝑅)(+g‘𝑅)𝑏))
118	95, 99, 117	rspcedvd 3579	. . . . . . . . . . . . . 14 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → ∃𝑎 ∈ 𝑀 ∃𝑏 ∈ ((Base‘𝑅) × {𝑥})𝑥 = (𝑎(+g‘𝑅)𝑏))
119		simp-5l 797	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → 𝑅 ∈ CRing)
120	3, 32, 67	lsmelvalx 19834	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ⊆ (Base‘𝑅) ∧ ((Base‘𝑅) × {𝑥}) ⊆ (Base‘𝑅)) → (𝑥 ∈ (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) ↔ ∃𝑎 ∈ 𝑀 ∃𝑏 ∈ ((Base‘𝑅) × {𝑥})𝑥 = (𝑎(+g‘𝑅)𝑏)))
121	119, 23, 82, 120	syl3anc 1398	. . . . . . . . . . . . . 14 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → (𝑥 ∈ (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) ↔ ∃𝑎 ∈ 𝑀 ∃𝑏 ∈ ((Base‘𝑅) × {𝑥})𝑥 = (𝑎(+g‘𝑅)𝑏)))
122	118, 121	mpbird 260	. . . . . . . . . . . . 13 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → 𝑥 ∈ (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})))
123		simpr 490	. . . . . . . . . . . . 13 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → ¬ 𝑥 ∈ 𝑀)
124		nelne1 3053	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) ∧ ¬ 𝑥 ∈ 𝑀) → (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) ≠ 𝑀)
125	122, 123, 124	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) ≠ 𝑀)
126	125	neneqd 2961	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → ¬ (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) = 𝑀)
127	92, 126	orcnd 892	. . . . . . . . . 10 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) = (Base‘𝑅))
128	66, 127	eleqtrrd 2864	. . . . . . . . 9 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → (1r‘𝑅) ∈ (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})))
129	3, 32, 67	lsmelvalx 19834	. . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ⊆ (Base‘𝑅) ∧ ((Base‘𝑅) × {𝑥}) ⊆ (Base‘𝑅)) → ((1r‘𝑅) ∈ (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) ↔ ∃𝑢 ∈ 𝑀 ∃𝑘 ∈ ((Base‘𝑅) × {𝑥})(1r‘𝑅) = (𝑢(+g‘𝑅)𝑘)))
130	119, 23, 82, 129	syl3anc 1398	. . . . . . . . 9 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → ((1r‘𝑅) ∈ (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) ↔ ∃𝑢 ∈ 𝑀 ∃𝑘 ∈ ((Base‘𝑅) × {𝑥})(1r‘𝑅) = (𝑢(+g‘𝑅)𝑘)))
131	128, 130	mpbid 235	. . . . . . . 8 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → ∃𝑢 ∈ 𝑀 ∃𝑘 ∈ ((Base‘𝑅) × {𝑥})(1r‘𝑅) = (𝑢(+g‘𝑅)𝑘))
132	64, 131	r19.29vva 3223	. . . . . . 7 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → 𝑦 ∈ 𝑀)
133	132	ex 418	. . . . . 6 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) → (¬ 𝑥 ∈ 𝑀 → 𝑦 ∈ 𝑀))
134	133	orrd 877	. . . . 5 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) → (𝑥 ∈ 𝑀 ∨ 𝑦 ∈ 𝑀))
135	134	ex 418	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑥(.r‘𝑅)𝑦) ∈ 𝑀 → (𝑥 ∈ 𝑀 ∨ 𝑦 ∈ 𝑀)))
136	135	anasss 472	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((𝑥(.r‘𝑅)𝑦) ∈ 𝑀 → (𝑥 ∈ 𝑀 ∨ 𝑦 ∈ 𝑀)))
137	136	ralrimivva 3206	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) ∈ 𝑀 → (𝑥 ∈ 𝑀 ∨ 𝑦 ∈ 𝑀)))
138	3, 16	prmidl2 21602	. 2 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ (LIdeal‘𝑅)) ∧ (𝑀 ≠ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) ∈ 𝑀 → (𝑥 ∈ 𝑀 ∨ 𝑦 ∈ 𝑀)))) → 𝑀 ∈ (PrmIdeal‘𝑅))
139	2, 5, 7, 137, 138	syl22anc 852	1 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (PrmIdeal‘𝑅))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  ∃wrex 3087  Vcvv 3451   ∪ cun 3897   ⊆ wss 3899  {csn 4584  ‘cfv 6531  (class class class)co 7412  Basecbs 17367  +_gcplusg 17408  ._rcmulr 17409  0_gc0g 17590  Grpcgrp 19124  LSSumclsm 19828  mulGrpcmgp 20340  1_rcur 20387  Ringcrg 20439  CRingccrg 20440  LModclmod 21115  ringLModcrglmod 21427  LIdealclidl 21464  RSpancrsp 21465  PrmIdealcprmidl 21596  LPIdealclpidl 21624  MaxIdealcmxidl 33966

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-er 8701  df-en 8958  df-dom 8959  df-sdom 8960  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-nn 12317  df-2 12386  df-3 12387  df-4 12388  df-5 12389  df-6 12390  df-7 12391  df-8 12392  df-sets 17322  df-slot 17340  df-ndx 17352  df-base 17368  df-ress 17389  df-plusg 17421  df-mulr 17422  df-sca 17424  df-vsca 17425  df-ip 17426  df-0g 17592  df-mgm 18796  df-sgrp 18888  df-mnd 18904  df-submnd 18959  df-grp 19127  df-minusg 19128  df-sbg 19129  df-subg 19313  df-cntz 19511  df-lsm 19830  df-cmn 19976  df-abl 19977  df-mgp 20341  df-rng 20355  df-ur 20388  df-ring 20441  df-cring 20442  df-subrg 20802  df-lmod 21117  df-lss 21187  df-lsp 21227  df-sra 21428  df-rgmod 21429  df-lidl 21466  df-rsp 21467  df-prmidl 21597  df-lpidl 21626  df-mxidl 33967

This theorem is used by:  rprmirredb  34046  1arithufdlem1  34058  zarcls1  34483  zarclssn  34487  zarmxt1  34494