Theorem mxidlprm 31640

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 19795	. . 3 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2	1	adantr 481	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑅 ∈ Ring)
3		eqid 2738	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
4	3	mxidlidl 31635	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))
5	1, 4	sylan 580	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))
6	3	mxidlnr 31636	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ≠ (Base‘𝑅))
7	1, 6	sylan 580	. 2 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ≠ (Base‘𝑅))
8		simpllr 773	. . . . . . . . . . . . 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 485	. . . . . . . . . . . . . 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 7291	. . . . . . . . . . . . 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 2778	. . . . . . . . . . . 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 7290	. . . . . . . . . . 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 729	. . . . . . . . . . . . 13 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → 𝑅 ∈ Ring)
14	13	ad5antr 731	. . . . . . . . . . . 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 789	. . . . . . . . . . . 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 2738	. . . . . . . . . . . . 13 ⊢ (.r‘𝑅) = (.r‘𝑅)
17		eqid 2738	. . . . . . . . . . . . 13 ⊢ (1r‘𝑅) = (1r‘𝑅)
18	3, 16, 17	ringlidm 19810	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ (Base‘𝑅)) → ((1r‘𝑅)(.r‘𝑅)𝑦) = 𝑦)
19	14, 15, 18	syl2anc 584	. . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . 17 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
21	3, 20	lidlss 20481	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ (LIdeal‘𝑅) → 𝑀 ⊆ (Base‘𝑅))
22	5, 21	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ⊆ (Base‘𝑅))
23	22	ad4antr 729	. . . . . . . . . . . . . 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 731	. . . . . . . . . . . . 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 783	. . . . . . . . . . . . 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 3922	. . . . . . . . . . . 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 766	. . . . . . . . . . . . 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 781	. . . . . . . . . . . . . 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 731	. . . . . . . . . . . . 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 19800	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑎(.r‘𝑅)𝑥) ∈ (Base‘𝑅))
31	14, 27, 29, 30	syl3anc 1370	. . . . . . . . . . . 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 2738	. . . . . . . . . . . . 13 ⊢ (+g‘𝑅) = (+g‘𝑅)
33	3, 32, 16	ringdir 19806	. . . . . . . . . . . 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 1371	. . . . . . . . . . 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 2786	. . . . . . . . . 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 783	. . . . . . . . . . . . 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 584	. . . . . . . . . . . 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 731	. . . . . . . . . . 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 792	. . . . . . . . . . . . 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 19801	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑢 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑅)𝑢) = (𝑢(.r‘𝑅)𝑦))
41	39, 15, 26, 40	syl3anc 1370	. . . . . . . . . . . 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 20488	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ (LIdeal‘𝑅)) ∧ (𝑦 ∈ (Base‘𝑅) ∧ 𝑢 ∈ 𝑀)) → (𝑦(.r‘𝑅)𝑢) ∈ 𝑀)
43	14, 38, 15, 25, 42	syl22anc 836	. . . . . . . . . . . 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 2840	. . . . . . . . . . 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 19803	. . . . . . . . . . . . 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 1371	. . . . . . . . . . . 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 787	. . . . . . . . . . . . 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 20488	. . . . . . . . . . . . 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 836	. . . . . . . . . . . 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 2839	. . . . . . . . . . 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 20484	. . . . . . . . . . 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 836	. . . . . . . . . 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 2839	. . . . . . . . 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 766	. . . . . . . . . 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 2738	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
56	55, 3	mgpbas 19726	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅))
57	55, 16	mgpplusg 19724	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (+g‘(mulGrp‘𝑅))
58		mxidlprm.1	. . . . . . . . . . . 12 ⊢ × = (LSSum‘(mulGrp‘𝑅))
59		fvexd 6789	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → (mulGrp‘𝑅) ∈ V)
60		ssidd 3944	. . . . . . . . . . . 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 31578	. . . . . . . . . . 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 727	. . . . . . . . . 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 231	. . . . . . . . 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 3218	. . . . . . . 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 19807	. . . . . . . . . . 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 2738	. . . . . . . . . . . . 13 ⊢ (LSSum‘𝑅) = (LSSum‘𝑅)
68		eqid 2738	. . . . . . . . . . . . 13 ⊢ (RSpan‘𝑅) = (RSpan‘𝑅)
69		eqid 2738	. . . . . . . . . . . . . . . 16 ⊢ (LPIdeal‘𝑅) = (LPIdeal‘𝑅)
70	69, 20	lpiss 20521	. . . . . . . . . . . . . . 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 31587	. . . . . . . . . . . . . 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 3922	. . . . . . . . . . . . 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 31589	. . . . . . . . . . . 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 20475	. . . . . . . . . . . . . . . 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 20465	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝑅) = (Base‘(ringLMod‘𝑅))
78		rspval 20463	. . . . . . . . . . . . . . . 16 ⊢ (RSpan‘𝑅) = (LSpan‘(ringLMod‘𝑅))
79	77, 78	lspssid 20247	. . . . . . . . . . . . . . 15 ⊢ (((ringLMod‘𝑅) ∈ LMod ∧ 𝑀 ⊆ (Base‘𝑅)) → 𝑀 ⊆ ((RSpan‘𝑅)‘𝑀))
80	76, 23, 79	syl2anc 584	. . . . . . . . . . . . . 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 4742	. . . . . . . . . . . . . . . . 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 31583	. . . . . . . . . . . . . . . 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 4120	. . . . . . . . . . . . . . 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 4106	. . . . . . . . . . . . . . . 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 20246	. . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . 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 3931	. . . . . . . . . . . . 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 31588	. . . . . . . . . . . . 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 3962	. . . . . . . . . . . 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 31637	. . . . . . . . . . . 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 836	. . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . 17 ⊢ (0g‘𝑅) = (0g‘𝑅)
94	20, 93	lidl0cl 20483	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (LIdeal‘𝑅)) → (0g‘𝑅) ∈ 𝑀)
95	13, 37, 94	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → (0g‘𝑅) ∈ 𝑀)
96		oveq1 7282	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = (0g‘𝑅) → (𝑎(+g‘𝑅)𝑏) = ((0g‘𝑅)(+g‘𝑅)𝑏))
97	96	eqeq2d 2749	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = (0g‘𝑅) → (𝑥 = (𝑎(+g‘𝑅)𝑏) ↔ 𝑥 = ((0g‘𝑅)(+g‘𝑅)𝑏)))
98	97	rexbidv 3226	. . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . 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 7282	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = (1r‘𝑅) → (𝑎(.r‘𝑅)𝑥) = ((1r‘𝑅)(.r‘𝑅)𝑥))
101	100	eqeq2d 2749	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = (1r‘𝑅) → (𝑥 = (𝑎(.r‘𝑅)𝑥) ↔ 𝑥 = ((1r‘𝑅)(.r‘𝑅)𝑥)))
102	101	adantl 482	. . . . . . . . . . . . . . . . . 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 19810	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → ((1r‘𝑅)(.r‘𝑅)𝑥) = 𝑥)
104	13, 28, 103	syl2anc 584	. . . . . . . . . . . . . . . . . . 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 2744	. . . . . . . . . . . . . . . . . 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 3563	. . . . . . . . . . . . . . . . 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 31578	. . . . . . . . . . . . . . . . 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 256	. . . . . . . . . . . . . . . 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 7283	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝑥 → ((0g‘𝑅)(+g‘𝑅)𝑏) = ((0g‘𝑅)(+g‘𝑅)𝑥))
110	109	eqeq2d 2749	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑥 → (𝑥 = ((0g‘𝑅)(+g‘𝑅)𝑏) ↔ 𝑥 = ((0g‘𝑅)(+g‘𝑅)𝑥)))
111	110	adantl 482	. . . . . . . . . . . . . . . 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 19788	. . . . . . . . . . . . . . . . . . 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 18609	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → ((0g‘𝑅)(+g‘𝑅)𝑥) = 𝑥)
115	113, 28, 114	syl2anc 584	. . . . . . . . . . . . . . . . 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 2744	. . . . . . . . . . . . . . . 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 3563	. . . . . . . . . . . . . . 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 3563	. . . . . . . . . . . . . 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 782	. . . . . . . . . . . . . . 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 19245	. . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . 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 256	. . . . . . . . . . . . 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 485	. . . . . . . . . . . . 13 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → ¬ 𝑥 ∈ 𝑀)
124		nelne1 3041	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) ∧ ¬ 𝑥 ∈ 𝑀) → (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) ≠ 𝑀)
125	122, 123, 124	syl2anc 584	. . . . . . . . . . . 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 2948	. . . . . . . . . . 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 876	. . . . . . . . . 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 2842	. . . . . . . . 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 19245	. . . . . . . . . 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 1370	. . . . . . . . 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 231	. . . . . . . 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 3266	. . . . . . 7 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥 ∈ 𝑀) → 𝑦 ∈ 𝑀)
133	132	ex 413	. . . . . 6 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) → (¬ 𝑥 ∈ 𝑀 → 𝑦 ∈ 𝑀))
134	133	orrd 860	. . . . 5 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r‘𝑅)𝑦) ∈ 𝑀) → (𝑥 ∈ 𝑀 ∨ 𝑦 ∈ 𝑀))
135	134	ex 413	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑥(.r‘𝑅)𝑦) ∈ 𝑀 → (𝑥 ∈ 𝑀 ∨ 𝑦 ∈ 𝑀)))
136	135	anasss 467	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((𝑥(.r‘𝑅)𝑦) ∈ 𝑀 → (𝑥 ∈ 𝑀 ∨ 𝑦 ∈ 𝑀)))
137	136	ralrimivva 3123	. 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 31616	. 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 836	1 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (PrmIdeal‘𝑅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ∪ cun 3885   ⊆ wss 3887  {csn 4561  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  +_gcplusg 16962  ._rcmulr 16963  0_gc0g 17150  Grpcgrp 18577  LSSumclsm 19239  mulGrpcmgp 19720  1_rcur 19737  Ringcrg 19783  CRingccrg 19784  LModclmod 20123  ringLModcrglmod 20431  LIdealclidl 20432  RSpancrsp 20433  LPIdealclpidl 20512  PrmIdealcprmidl 31610  MaxIdealcmxidl 31631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-mulr 16976  df-sca 16978  df-vsca 16979  df-ip 16980  df-0g 17152  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-submnd 18431  df-grp 18580  df-minusg 18581  df-sbg 18582  df-subg 18752  df-cntz 18923  df-lsm 19241  df-cmn 19388  df-abl 19389  df-mgp 19721  df-ur 19738  df-ring 19785  df-cring 19786  df-subrg 20022  df-lmod 20125  df-lss 20194  df-lsp 20234  df-sra 20434  df-rgmod 20435  df-lidl 20436  df-rsp 20437  df-lpidl 20514  df-prmidl 31611  df-mxidl 31632

This theorem is referenced by:  zarcls1  31819  zarclssn  31823  zarmxt1  31830