Theorem mxidlprm 33569

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 20197	. . 3 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2	1	adantr 480	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑅 ∈ Ring)
3		eqid 2737	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
4	3	mxidlidl 33562	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))
5	1, 4	sylan 581	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))
6	3	mxidlnr 33563	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ≠ (Base‘𝑅))
7	1, 6	sylan 581	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ≠ (Base‘𝑅))
8		simpllr 776	. . . . . . . . . . . . 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 484	. . . . . . . . . . . . . 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 7386	. . . . . . . . . . . . 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 2772	. . . . . . . . . . . 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 7385	. . . . . . . . . . 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 733	. . . . . . . . . . . . 13 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → 𝑅 ∈ Ring)
14	13	ad5antr 735	. . . . . . . . . . . 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 792	. . . . . . . . . . . 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 2737	. . . . . . . . . . . . 13 ⊢ (.r‘𝑅) = (.r‘𝑅)
17		eqid 2737	. . . . . . . . . . . . 13 ⊢ (1r‘𝑅) = (1r‘𝑅)
18	3, 16, 17	ringlidm 20221	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ (Base‘𝑅)) → ((1r‘𝑅)(.r‘𝑅)𝑦) = 𝑦)
19	14, 15, 18	syl2anc 585	. . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . 17 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
21	3, 20	lidlss 21184	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ (LIdeal‘𝑅) → 𝑀 ⊆ (Base‘𝑅))
22	5, 21	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ⊆ (Base‘𝑅))
23	22	ad4antr 733	. . . . . . . . . . . . . 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 735	. . . . . . . . . . . . 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 786	. . . . . . . . . . . . 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 3936	. . . . . . . . . . . 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 769	. . . . . . . . . . . . 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 784	. . . . . . . . . . . . . 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 735	. . . . . . . . . . . . 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 20202	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑎(.r‘𝑅)𝑥) ∈ (Base‘𝑅))
31	14, 27, 29, 30	syl3anc 1374	. . . . . . . . . . . 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 2737	. . . . . . . . . . . . 13 ⊢ (+g‘𝑅) = (+g‘𝑅)
33	3, 32, 16	ringdir 20214	. . . . . . . . . . . 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 1375	. . . . . . . . . . 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 2780	. . . . . . . . . 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 786	. . . . . . . . . . . . 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 585	. . . . . . . . . . . 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 735	. . . . . . . . . . 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 795	. . . . . . . . . . . . 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 20203	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑢 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑅)𝑢) = (𝑢(.r‘𝑅)𝑦))
41	39, 15, 26, 40	syl3anc 1374	. . . . . . . . . . . 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 21197	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ (LIdeal‘𝑅)) ∧ (𝑦 ∈ (Base‘𝑅) ∧ 𝑢 ∈ 𝑀)) → (𝑦(.r‘𝑅)𝑢) ∈ 𝑀)
43	14, 38, 15, 25, 42	syl22anc 839	. . . . . . . . . . . 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 2838	. . . . . . . . . . 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 20205	. . . . . . . . . . . . 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 1375	. . . . . . . . . . . 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 790	. . . . . . . . . . . . 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 21197	. . . . . . . . . . . . 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 839	. . . . . . . . . . . 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 2837	. . . . . . . . . . 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 21193	. . . . . . . . . . 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 839	. . . . . . . . . 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 2837	. . . . . . . . 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 769	. . . . . . . . . 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 2737	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
56	55, 3	mgpbas 20097	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅))
57	55, 16	mgpplusg 20096	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (+g‘(mulGrp‘𝑅))
58		mxidlprm.1	. . . . . . . . . . . 12 ⊢ × = (LSSum‘(mulGrp‘𝑅))
59		fvexd 6859	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → (mulGrp‘𝑅) ∈ V)
60		ssidd 3959	. . . . . . . . . . . 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 33489	. . . . . . . . . . 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 731	. . . . . . . . . 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 232	. . . . . . . . 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 3146	. . . . . . . 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 20217	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
66	13, 65	syl 17	. . . . . . . . . 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 2737	. . . . . . . . . . . . 13 ⊢ (LSSum‘𝑅) = (LSSum‘𝑅)
68		eqid 2737	. . . . . . . . . . . . 13 ⊢ (RSpan‘𝑅) = (RSpan‘𝑅)
69		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (LPIdeal‘𝑅) = (LPIdeal‘𝑅)
70	69, 20	lpiss 21301	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ Ring → (LPIdeal‘𝑅) ⊆ (LIdeal‘𝑅))
71	13, 70	syl 17	. . . . . . . . . . . . . 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 33498	. . . . . . . . . . . . . 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 3936	. . . . . . . . . . . . 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 33500	. . . . . . . . . . . 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 21172	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod)
76	13, 75	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → (ringLMod‘𝑅) ∈ LMod)
77		rlmbas 21162	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝑅) = (Base‘(ringLMod‘𝑅))
78		rspval 21183	. . . . . . . . . . . . . . . 16 ⊢ (RSpan‘𝑅) = (LSpan‘(ringLMod‘𝑅))
79	77, 78	lspssid 20953	. . . . . . . . . . . . . . 15 ⊢ (((ringLMod‘𝑅) ∈ LMod ∧ 𝑀 ⊆ (Base‘𝑅)) → 𝑀 ⊆ ((RSpan‘𝑅)‘𝑀))
80	76, 23, 79	syl2anc 585	. . . . . . . . . . . . . 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 4767	. . . . . . . . . . . . . . . . 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 33494	. . . . . . . . . . . . . . . 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 4146	. . . . . . . . . . . . . . 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 4132	. . . . . . . . . . . . . . . 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 20952	. . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . 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 3946	. . . . . . . . . . . . 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 33499	. . . . . . . . . . . . 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 3973	. . . . . . . . . . . 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 33564	. . . . . . . . . . . 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 839	. . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . 17 ⊢ (0g‘𝑅) = (0g‘𝑅)
94	20, 93	lidl0cl 21192	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (LIdeal‘𝑅)) → (0g‘𝑅) ∈ 𝑀)
95	13, 37, 94	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → (0g‘𝑅) ∈ 𝑀)
96		oveq1 7377	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = (0g‘𝑅) → (𝑎(+g‘𝑅)𝑏) = ((0g‘𝑅)(+g‘𝑅)𝑏))
97	96	eqeq2d 2748	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = (0g‘𝑅) → (𝑥 = (𝑎(+g‘𝑅)𝑏) ↔ 𝑥 = ((0g‘𝑅)(+g‘𝑅)𝑏)))
98	97	rexbidv 3162	. . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . 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 7377	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = (1r‘𝑅) → (𝑎(.r‘𝑅)𝑥) = ((1r‘𝑅)(.r‘𝑅)𝑥))
101	100	eqeq2d 2748	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = (1r‘𝑅) → (𝑥 = (𝑎(.r‘𝑅)𝑥) ↔ 𝑥 = ((1r‘𝑅)(.r‘𝑅)𝑥)))
102	101	adantl 481	. . . . . . . . . . . . . . . . . 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 20221	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → ((1r‘𝑅)(.r‘𝑅)𝑥) = 𝑥)
104	13, 28, 103	syl2anc 585	. . . . . . . . . . . . . . . . . . 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 2743	. . . . . . . . . . . . . . . . . 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 3580	. . . . . . . . . . . . . . . . 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 33489	. . . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . . . 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 7378	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑥 → ((0g‘𝑅)(+g‘𝑅)𝑏) = ((0g‘𝑅)(+g‘𝑅)𝑥))
110	109	eqeq2d 2748	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑥 → (𝑥 = ((0g‘𝑅)(+g‘𝑅)𝑏) ↔ 𝑥 = ((0g‘𝑅)(+g‘𝑅)𝑥)))
111	110	adantl 481	. . . . . . . . . . . . . . . 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 20190	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
113	13, 112	syl 17	. . . . . . . . . . . . . . . . . 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 18914	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → ((0g‘𝑅)(+g‘𝑅)𝑥) = 𝑥)
115	113, 28, 114	syl2anc 585	. . . . . . . . . . . . . . . . 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 2743	. . . . . . . . . . . . . . . 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 3580	. . . . . . . . . . . . . . 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 3580	. . . . . . . . . . . . . 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 785	. . . . . . . . . . . . . . 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 19586	. . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . 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 257	. . . . . . . . . . . . 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 484	. . . . . . . . . . . . 13 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → ¬ 𝑥 ∈ 𝑀)
124		nelne1 3030	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) ∧ ¬ 𝑥 ∈ 𝑀) → (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) ≠ 𝑀)
125	122, 123, 124	syl2anc 585	. . . . . . . . . . . 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 2938	. . . . . . . . . . 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 879	. . . . . . . . . 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 2840	. . . . . . . . 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 19586	. . . . . . . . . 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 1374	. . . . . . . . 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 232	. . . . . . . 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 3198	. . . . . . 7 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → 𝑦 ∈ 𝑀)
133	132	ex 412	. . . . . 6 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) → (¬ 𝑥 ∈ 𝑀 → 𝑦 ∈ 𝑀))
134	133	orrd 864	. . . . 5 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) → (𝑥 ∈ 𝑀 ∨ 𝑦 ∈ 𝑀))
135	134	ex 412	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑥(.r‘𝑅)𝑦) ∈ 𝑀 → (𝑥 ∈ 𝑀 ∨ 𝑦 ∈ 𝑀)))
136	135	anasss 466	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((𝑥(.r‘𝑅)𝑦) ∈ 𝑀 → (𝑥 ∈ 𝑀 ∨ 𝑦 ∈ 𝑀)))
137	136	ralrimivva 3181	. 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 33540	. 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 839	1 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (PrmIdeal‘𝑅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  Vcvv 3442   ∪ cun 3901   ⊆ wss 3903  {csn 4582  ‘cfv 6502  (class class class)co 7370  Basecbs 17150  +_gcplusg 17191  ._rcmulr 17192  0_gc0g 17373  Grpcgrp 18880  LSSumclsm 19580  mulGrpcmgp 20092  1_rcur 20133  Ringcrg 20185  CRingccrg 20186  LModclmod 20828  ringLModcrglmod 21141  LIdealclidl 21178  RSpancrsp 21179  LPIdealclpidl 21292  PrmIdealcprmidl 33534  MaxIdealcmxidl 33558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-cnex 11096  ax-resscn 11097  ax-1cn 11098  ax-icn 11099  ax-addcl 11100  ax-addrcl 11101  ax-mulcl 11102  ax-mulrcl 11103  ax-mulcom 11104  ax-addass 11105  ax-mulass 11106  ax-distr 11107  ax-i2m1 11108  ax-1ne0 11109  ax-1rid 11110  ax-rnegex 11111  ax-rrecex 11112  ax-cnre 11113  ax-pre-lttri 11114  ax-pre-lttrn 11115  ax-pre-ltadd 11116  ax-pre-mulgt0 11117

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-om 7821  df-1st 7945  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-er 8647  df-en 8898  df-dom 8899  df-sdom 8900  df-pnf 11182  df-mnf 11183  df-xr 11184  df-ltxr 11185  df-le 11186  df-sub 11380  df-neg 11381  df-nn 12160  df-2 12222  df-3 12223  df-4 12224  df-5 12225  df-6 12226  df-7 12227  df-8 12228  df-sets 17105  df-slot 17123  df-ndx 17135  df-base 17151  df-ress 17172  df-plusg 17204  df-mulr 17205  df-sca 17207  df-vsca 17208  df-ip 17209  df-0g 17375  df-mgm 18579  df-sgrp 18658  df-mnd 18674  df-submnd 18723  df-grp 18883  df-minusg 18884  df-sbg 18885  df-subg 19070  df-cntz 19263  df-lsm 19582  df-cmn 19728  df-abl 19729  df-mgp 20093  df-rng 20105  df-ur 20134  df-ring 20187  df-cring 20188  df-subrg 20520  df-lmod 20830  df-lss 20900  df-lsp 20940  df-sra 21142  df-rgmod 21143  df-lidl 21180  df-rsp 21181  df-lpidl 21294  df-prmidl 33535  df-mxidl 33559

This theorem is referenced by:  rprmirredb  33631  1arithufdlem1  33643  zarcls1  34053  zarclssn  34057  zarmxt1  34064