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Theorem mxidlprm 32586
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.
StepHypRef Expression
1 crngring 20068 . . 3 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
21adantr 482 . 2 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑅 ∈ Ring)
3 eqid 2733 . . . 4 (Base‘𝑅) = (Base‘𝑅)
43mxidlidl 32579 . . 3 ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))
51, 4sylan 581 . 2 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))
63mxidlnr 32580 . . 3 ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ≠ (Base‘𝑅))
71, 6sylan 581 . 2 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ≠ (Base‘𝑅))
8 simpllr 775 . . . . . . . . . . . . 13 (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) ∧ 𝑢𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r𝑅) = (𝑢(+g𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r𝑅)𝑥)) → (1r𝑅) = (𝑢(+g𝑅)𝑘))
9 simpr 486 . . . . . . . . . . . . . 14 (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) ∧ 𝑢𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r𝑅) = (𝑢(+g𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r𝑅)𝑥)) → 𝑘 = (𝑎(.r𝑅)𝑥))
109oveq2d 7425 . . . . . . . . . . . . 13 (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) ∧ 𝑢𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r𝑅) = (𝑢(+g𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r𝑅)𝑥)) → (𝑢(+g𝑅)𝑘) = (𝑢(+g𝑅)(𝑎(.r𝑅)𝑥)))
118, 10eqtrd 2773 . . . . . . . . . . . 12 (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) ∧ 𝑢𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r𝑅) = (𝑢(+g𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r𝑅)𝑥)) → (1r𝑅) = (𝑢(+g𝑅)(𝑎(.r𝑅)𝑥)))
1211oveq1d 7424 . . . . . . . . . . 11 (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) ∧ 𝑢𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r𝑅) = (𝑢(+g𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r𝑅)𝑥)) → ((1r𝑅)(.r𝑅)𝑦) = ((𝑢(+g𝑅)(𝑎(.r𝑅)𝑥))(.r𝑅)𝑦))
132ad4antr 731 . . . . . . . . . . . . 13 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → 𝑅 ∈ Ring)
1413ad5antr 733 . . . . . . . . . . . 12 (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) ∧ 𝑢𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r𝑅) = (𝑢(+g𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r𝑅)𝑥)) → 𝑅 ∈ Ring)
15 simp-8r 791 . . . . . . . . . . . 12 (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) ∧ 𝑢𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r𝑅) = (𝑢(+g𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r𝑅)𝑥)) → 𝑦 ∈ (Base‘𝑅))
16 eqid 2733 . . . . . . . . . . . . 13 (.r𝑅) = (.r𝑅)
17 eqid 2733 . . . . . . . . . . . . 13 (1r𝑅) = (1r𝑅)
183, 16, 17ringlidm 20086 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝑦 ∈ (Base‘𝑅)) → ((1r𝑅)(.r𝑅)𝑦) = 𝑦)
1914, 15, 18syl2anc 585 . . . . . . . . . . 11 (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) ∧ 𝑢𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r𝑅) = (𝑢(+g𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r𝑅)𝑥)) → ((1r𝑅)(.r𝑅)𝑦) = 𝑦)
20 eqid 2733 . . . . . . . . . . . . . . . . 17 (LIdeal‘𝑅) = (LIdeal‘𝑅)
213, 20lidlss 20833 . . . . . . . . . . . . . . . 16 (𝑀 ∈ (LIdeal‘𝑅) → 𝑀 ⊆ (Base‘𝑅))
225, 21syl 17 . . . . . . . . . . . . . . 15 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ⊆ (Base‘𝑅))
2322ad4antr 731 . . . . . . . . . . . . . 14 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → 𝑀 ⊆ (Base‘𝑅))
2423ad5antr 733 . . . . . . . . . . . . 13 (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) ∧ 𝑢𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r𝑅) = (𝑢(+g𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r𝑅)𝑥)) → 𝑀 ⊆ (Base‘𝑅))
25 simp-5r 785 . . . . . . . . . . . . 13 (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) ∧ 𝑢𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r𝑅) = (𝑢(+g𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r𝑅)𝑥)) → 𝑢𝑀)
2624, 25sseldd 3984 . . . . . . . . . . . 12 (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) ∧ 𝑢𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r𝑅) = (𝑢(+g𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r𝑅)𝑥)) → 𝑢 ∈ (Base‘𝑅))
27 simplr 768 . . . . . . . . . . . . 13 (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) ∧ 𝑢𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r𝑅) = (𝑢(+g𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r𝑅)𝑥)) → 𝑎 ∈ (Base‘𝑅))
28 simp-4r 783 . . . . . . . . . . . . . 14 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → 𝑥 ∈ (Base‘𝑅))
2928ad5antr 733 . . . . . . . . . . . . 13 (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) ∧ 𝑢𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r𝑅) = (𝑢(+g𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r𝑅)𝑥)) → 𝑥 ∈ (Base‘𝑅))
303, 16ringcl 20073 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑎(.r𝑅)𝑥) ∈ (Base‘𝑅))
3114, 27, 29, 30syl3anc 1372 . . . . . . . . . . . 12 (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) ∧ 𝑢𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r𝑅) = (𝑢(+g𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r𝑅)𝑥)) → (𝑎(.r𝑅)𝑥) ∈ (Base‘𝑅))
32 eqid 2733 . . . . . . . . . . . . 13 (+g𝑅) = (+g𝑅)
333, 32, 16ringdir 20082 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ (𝑢 ∈ (Base‘𝑅) ∧ (𝑎(.r𝑅)𝑥) ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((𝑢(+g𝑅)(𝑎(.r𝑅)𝑥))(.r𝑅)𝑦) = ((𝑢(.r𝑅)𝑦)(+g𝑅)((𝑎(.r𝑅)𝑥)(.r𝑅)𝑦)))
3414, 26, 31, 15, 33syl13anc 1373 . . . . . . . . . . 11 (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) ∧ 𝑢𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r𝑅) = (𝑢(+g𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r𝑅)𝑥)) → ((𝑢(+g𝑅)(𝑎(.r𝑅)𝑥))(.r𝑅)𝑦) = ((𝑢(.r𝑅)𝑦)(+g𝑅)((𝑎(.r𝑅)𝑥)(.r𝑅)𝑦)))
3512, 19, 343eqtr3d 2781 . . . . . . . . . 10 (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) ∧ 𝑢𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r𝑅) = (𝑢(+g𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r𝑅)𝑥)) → 𝑦 = ((𝑢(.r𝑅)𝑦)(+g𝑅)((𝑎(.r𝑅)𝑥)(.r𝑅)𝑦)))
36 simp-5r 785 . . . . . . . . . . . . 13 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → 𝑀 ∈ (MaxIdeal‘𝑅))
3713, 36, 4syl2anc 585 . . . . . . . . . . . 12 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → 𝑀 ∈ (LIdeal‘𝑅))
3837ad5antr 733 . . . . . . . . . . 11 (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) ∧ 𝑢𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r𝑅) = (𝑢(+g𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r𝑅)𝑥)) → 𝑀 ∈ (LIdeal‘𝑅))
39 simp-10l 794 . . . . . . . . . . . . 13 (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) ∧ 𝑢𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r𝑅) = (𝑢(+g𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r𝑅)𝑥)) → 𝑅 ∈ CRing)
403, 16crngcom 20074 . . . . . . . . . . . . 13 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑢 ∈ (Base‘𝑅)) → (𝑦(.r𝑅)𝑢) = (𝑢(.r𝑅)𝑦))
4139, 15, 26, 40syl3anc 1372 . . . . . . . . . . . 12 (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) ∧ 𝑢𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r𝑅) = (𝑢(+g𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r𝑅)𝑥)) → (𝑦(.r𝑅)𝑢) = (𝑢(.r𝑅)𝑦))
4220, 3, 16lidlmcl 20840 . . . . . . . . . . . . 13 (((𝑅 ∈ Ring ∧ 𝑀 ∈ (LIdeal‘𝑅)) ∧ (𝑦 ∈ (Base‘𝑅) ∧ 𝑢𝑀)) → (𝑦(.r𝑅)𝑢) ∈ 𝑀)
4314, 38, 15, 25, 42syl22anc 838 . . . . . . . . . . . 12 (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) ∧ 𝑢𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r𝑅) = (𝑢(+g𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r𝑅)𝑥)) → (𝑦(.r𝑅)𝑢) ∈ 𝑀)
4441, 43eqeltrrd 2835 . . . . . . . . . . 11 (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) ∧ 𝑢𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r𝑅) = (𝑢(+g𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r𝑅)𝑥)) → (𝑢(.r𝑅)𝑦) ∈ 𝑀)
453, 16ringass 20076 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ (𝑎 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((𝑎(.r𝑅)𝑥)(.r𝑅)𝑦) = (𝑎(.r𝑅)(𝑥(.r𝑅)𝑦)))
4614, 27, 29, 15, 45syl13anc 1373 . . . . . . . . . . . 12 (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) ∧ 𝑢𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r𝑅) = (𝑢(+g𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r𝑅)𝑥)) → ((𝑎(.r𝑅)𝑥)(.r𝑅)𝑦) = (𝑎(.r𝑅)(𝑥(.r𝑅)𝑦)))
47 simp-7r 789 . . . . . . . . . . . . 13 (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) ∧ 𝑢𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r𝑅) = (𝑢(+g𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r𝑅)𝑥)) → (𝑥(.r𝑅)𝑦) ∈ 𝑀)
4820, 3, 16lidlmcl 20840 . . . . . . . . . . . . 13 (((𝑅 ∈ Ring ∧ 𝑀 ∈ (LIdeal‘𝑅)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀)) → (𝑎(.r𝑅)(𝑥(.r𝑅)𝑦)) ∈ 𝑀)
4914, 38, 27, 47, 48syl22anc 838 . . . . . . . . . . . 12 (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) ∧ 𝑢𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r𝑅) = (𝑢(+g𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r𝑅)𝑥)) → (𝑎(.r𝑅)(𝑥(.r𝑅)𝑦)) ∈ 𝑀)
5046, 49eqeltrd 2834 . . . . . . . . . . 11 (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) ∧ 𝑢𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r𝑅) = (𝑢(+g𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r𝑅)𝑥)) → ((𝑎(.r𝑅)𝑥)(.r𝑅)𝑦) ∈ 𝑀)
5120, 32lidlacl 20836 . . . . . . . . . . 11 (((𝑅 ∈ Ring ∧ 𝑀 ∈ (LIdeal‘𝑅)) ∧ ((𝑢(.r𝑅)𝑦) ∈ 𝑀 ∧ ((𝑎(.r𝑅)𝑥)(.r𝑅)𝑦) ∈ 𝑀)) → ((𝑢(.r𝑅)𝑦)(+g𝑅)((𝑎(.r𝑅)𝑥)(.r𝑅)𝑦)) ∈ 𝑀)
5214, 38, 44, 50, 51syl22anc 838 . . . . . . . . . 10 (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) ∧ 𝑢𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r𝑅) = (𝑢(+g𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r𝑅)𝑥)) → ((𝑢(.r𝑅)𝑦)(+g𝑅)((𝑎(.r𝑅)𝑥)(.r𝑅)𝑦)) ∈ 𝑀)
5335, 52eqeltrd 2834 . . . . . . . . 9 (((((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) ∧ 𝑢𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r𝑅) = (𝑢(+g𝑅)𝑘)) ∧ 𝑎 ∈ (Base‘𝑅)) ∧ 𝑘 = (𝑎(.r𝑅)𝑥)) → 𝑦𝑀)
54 simplr 768 . . . . . . . . . 10 (((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) ∧ 𝑢𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r𝑅) = (𝑢(+g𝑅)𝑘)) → 𝑘 ∈ ((Base‘𝑅) × {𝑥}))
55 eqid 2733 . . . . . . . . . . . . 13 (mulGrp‘𝑅) = (mulGrp‘𝑅)
5655, 3mgpbas 19993 . . . . . . . . . . . 12 (Base‘𝑅) = (Base‘(mulGrp‘𝑅))
5755, 16mgpplusg 19991 . . . . . . . . . . . 12 (.r𝑅) = (+g‘(mulGrp‘𝑅))
58 mxidlprm.1 . . . . . . . . . . . 12 × = (LSSum‘(mulGrp‘𝑅))
59 fvexd 6907 . . . . . . . . . . . 12 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → (mulGrp‘𝑅) ∈ V)
60 ssidd 4006 . . . . . . . . . . . 12 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → (Base‘𝑅) ⊆ (Base‘𝑅))
6156, 57, 58, 59, 60, 28elgrplsmsn 32500 . . . . . . . . . . 11 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → (𝑘 ∈ ((Base‘𝑅) × {𝑥}) ↔ ∃𝑎 ∈ (Base‘𝑅)𝑘 = (𝑎(.r𝑅)𝑥)))
6261ad3antrrr 729 . . . . . . . . . 10 (((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) ∧ 𝑢𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r𝑅) = (𝑢(+g𝑅)𝑘)) → (𝑘 ∈ ((Base‘𝑅) × {𝑥}) ↔ ∃𝑎 ∈ (Base‘𝑅)𝑘 = (𝑎(.r𝑅)𝑥)))
6354, 62mpbid 231 . . . . . . . . 9 (((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) ∧ 𝑢𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r𝑅) = (𝑢(+g𝑅)𝑘)) → ∃𝑎 ∈ (Base‘𝑅)𝑘 = (𝑎(.r𝑅)𝑥))
6453, 63r19.29a 3163 . . . . . . . 8 (((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) ∧ 𝑢𝑀) ∧ 𝑘 ∈ ((Base‘𝑅) × {𝑥})) ∧ (1r𝑅) = (𝑢(+g𝑅)𝑘)) → 𝑦𝑀)
653, 17ringidcl 20083 . . . . . . . . . . 11 (𝑅 ∈ Ring → (1r𝑅) ∈ (Base‘𝑅))
6613, 65syl 17 . . . . . . . . . 10 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → (1r𝑅) ∈ (Base‘𝑅))
67 eqid 2733 . . . . . . . . . . . . 13 (LSSum‘𝑅) = (LSSum‘𝑅)
68 eqid 2733 . . . . . . . . . . . . 13 (RSpan‘𝑅) = (RSpan‘𝑅)
69 eqid 2733 . . . . . . . . . . . . . . . 16 (LPIdeal‘𝑅) = (LPIdeal‘𝑅)
7069, 20lpiss 20888 . . . . . . . . . . . . . . 15 (𝑅 ∈ Ring → (LPIdeal‘𝑅) ⊆ (LIdeal‘𝑅))
7113, 70syl 17 . . . . . . . . . . . . . 14 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → (LPIdeal‘𝑅) ⊆ (LIdeal‘𝑅))
723, 55, 58, 68, 13, 28lsmsnidl 32509 . . . . . . . . . . . . . 14 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → ((Base‘𝑅) × {𝑥}) ∈ (LPIdeal‘𝑅))
7371, 72sseldd 3984 . . . . . . . . . . . . 13 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → ((Base‘𝑅) × {𝑥}) ∈ (LIdeal‘𝑅))
743, 67, 68, 13, 37, 73lsmidl 32511 . . . . . . . . . . . 12 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) ∈ (LIdeal‘𝑅))
75 rlmlmod 20827 . . . . . . . . . . . . . . . 16 (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod)
7613, 75syl 17 . . . . . . . . . . . . . . 15 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → (ringLMod‘𝑅) ∈ LMod)
77 rlmbas 20817 . . . . . . . . . . . . . . . 16 (Base‘𝑅) = (Base‘(ringLMod‘𝑅))
78 rspval 20815 . . . . . . . . . . . . . . . 16 (RSpan‘𝑅) = (LSpan‘(ringLMod‘𝑅))
7977, 78lspssid 20596 . . . . . . . . . . . . . . 15 (((ringLMod‘𝑅) ∈ LMod ∧ 𝑀 ⊆ (Base‘𝑅)) → 𝑀 ⊆ ((RSpan‘𝑅)‘𝑀))
8076, 23, 79syl2anc 585 . . . . . . . . . . . . . 14 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → 𝑀 ⊆ ((RSpan‘𝑅)‘𝑀))
8128snssd 4813 . . . . . . . . . . . . . . . . 17 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → {𝑥} ⊆ (Base‘𝑅))
823, 55, 58, 13, 60, 81ringlsmss 32505 . . . . . . . . . . . . . . . 16 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → ((Base‘𝑅) × {𝑥}) ⊆ (Base‘𝑅))
8323, 82unssd 4187 . . . . . . . . . . . . . . 15 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → (𝑀 ∪ ((Base‘𝑅) × {𝑥})) ⊆ (Base‘𝑅))
84 ssun1 4173 . . . . . . . . . . . . . . . 16 𝑀 ⊆ (𝑀 ∪ ((Base‘𝑅) × {𝑥}))
8584a1i 11 . . . . . . . . . . . . . . 15 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → 𝑀 ⊆ (𝑀 ∪ ((Base‘𝑅) × {𝑥})))
8677, 78lspss 20595 . . . . . . . . . . . . . . 15 (((ringLMod‘𝑅) ∈ LMod ∧ (𝑀 ∪ ((Base‘𝑅) × {𝑥})) ⊆ (Base‘𝑅) ∧ 𝑀 ⊆ (𝑀 ∪ ((Base‘𝑅) × {𝑥}))) → ((RSpan‘𝑅)‘𝑀) ⊆ ((RSpan‘𝑅)‘(𝑀 ∪ ((Base‘𝑅) × {𝑥}))))
8776, 83, 85, 86syl3anc 1372 . . . . . . . . . . . . . 14 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → ((RSpan‘𝑅)‘𝑀) ⊆ ((RSpan‘𝑅)‘(𝑀 ∪ ((Base‘𝑅) × {𝑥}))))
8880, 87sstrd 3993 . . . . . . . . . . . . 13 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → 𝑀 ⊆ ((RSpan‘𝑅)‘(𝑀 ∪ ((Base‘𝑅) × {𝑥}))))
893, 67, 68, 13, 37, 73lsmidllsp 32510 . . . . . . . . . . . . 13 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) = ((RSpan‘𝑅)‘(𝑀 ∪ ((Base‘𝑅) × {𝑥}))))
9088, 89sseqtrrd 4024 . . . . . . . . . . . 12 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → 𝑀 ⊆ (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})))
913mxidlmax 32581 . . . . . . . . . . . 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)))
9213, 36, 74, 90, 91syl22anc 838 . . . . . . . . . . 11 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → ((𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) = 𝑀 ∨ (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) = (Base‘𝑅)))
93 eqid 2733 . . . . . . . . . . . . . . . . 17 (0g𝑅) = (0g𝑅)
9420, 93lidl0cl 20835 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Ring ∧ 𝑀 ∈ (LIdeal‘𝑅)) → (0g𝑅) ∈ 𝑀)
9513, 37, 94syl2anc 585 . . . . . . . . . . . . . . 15 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → (0g𝑅) ∈ 𝑀)
96 oveq1 7416 . . . . . . . . . . . . . . . . . 18 (𝑎 = (0g𝑅) → (𝑎(+g𝑅)𝑏) = ((0g𝑅)(+g𝑅)𝑏))
9796eqeq2d 2744 . . . . . . . . . . . . . . . . 17 (𝑎 = (0g𝑅) → (𝑥 = (𝑎(+g𝑅)𝑏) ↔ 𝑥 = ((0g𝑅)(+g𝑅)𝑏)))
9897rexbidv 3179 . . . . . . . . . . . . . . . 16 (𝑎 = (0g𝑅) → (∃𝑏 ∈ ((Base‘𝑅) × {𝑥})𝑥 = (𝑎(+g𝑅)𝑏) ↔ ∃𝑏 ∈ ((Base‘𝑅) × {𝑥})𝑥 = ((0g𝑅)(+g𝑅)𝑏)))
9998adantl 483 . . . . . . . . . . . . . . 15 (((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) ∧ 𝑎 = (0g𝑅)) → (∃𝑏 ∈ ((Base‘𝑅) × {𝑥})𝑥 = (𝑎(+g𝑅)𝑏) ↔ ∃𝑏 ∈ ((Base‘𝑅) × {𝑥})𝑥 = ((0g𝑅)(+g𝑅)𝑏)))
100 oveq1 7416 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = (1r𝑅) → (𝑎(.r𝑅)𝑥) = ((1r𝑅)(.r𝑅)𝑥))
101100eqeq2d 2744 . . . . . . . . . . . . . . . . . . 19 (𝑎 = (1r𝑅) → (𝑥 = (𝑎(.r𝑅)𝑥) ↔ 𝑥 = ((1r𝑅)(.r𝑅)𝑥)))
102101adantl 483 . . . . . . . . . . . . . . . . . 18 (((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) ∧ 𝑎 = (1r𝑅)) → (𝑥 = (𝑎(.r𝑅)𝑥) ↔ 𝑥 = ((1r𝑅)(.r𝑅)𝑥)))
1033, 16, 17ringlidm 20086 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → ((1r𝑅)(.r𝑅)𝑥) = 𝑥)
10413, 28, 103syl2anc 585 . . . . . . . . . . . . . . . . . . 19 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → ((1r𝑅)(.r𝑅)𝑥) = 𝑥)
105104eqcomd 2739 . . . . . . . . . . . . . . . . . 18 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → 𝑥 = ((1r𝑅)(.r𝑅)𝑥))
10666, 102, 105rspcedvd 3615 . . . . . . . . . . . . . . . . 17 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → ∃𝑎 ∈ (Base‘𝑅)𝑥 = (𝑎(.r𝑅)𝑥))
10756, 57, 58, 59, 60, 28elgrplsmsn 32500 . . . . . . . . . . . . . . . . 17 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → (𝑥 ∈ ((Base‘𝑅) × {𝑥}) ↔ ∃𝑎 ∈ (Base‘𝑅)𝑥 = (𝑎(.r𝑅)𝑥)))
108106, 107mpbird 257 . . . . . . . . . . . . . . . 16 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → 𝑥 ∈ ((Base‘𝑅) × {𝑥}))
109 oveq2 7417 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑥 → ((0g𝑅)(+g𝑅)𝑏) = ((0g𝑅)(+g𝑅)𝑥))
110109eqeq2d 2744 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑥 → (𝑥 = ((0g𝑅)(+g𝑅)𝑏) ↔ 𝑥 = ((0g𝑅)(+g𝑅)𝑥)))
111110adantl 483 . . . . . . . . . . . . . . . 16 (((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) ∧ 𝑏 = 𝑥) → (𝑥 = ((0g𝑅)(+g𝑅)𝑏) ↔ 𝑥 = ((0g𝑅)(+g𝑅)𝑥)))
112 ringgrp 20061 . . . . . . . . . . . . . . . . . . 19 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
11313, 112syl 17 . . . . . . . . . . . . . . . . . 18 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → 𝑅 ∈ Grp)
1143, 32, 93grplid 18852 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → ((0g𝑅)(+g𝑅)𝑥) = 𝑥)
115113, 28, 114syl2anc 585 . . . . . . . . . . . . . . . . 17 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → ((0g𝑅)(+g𝑅)𝑥) = 𝑥)
116115eqcomd 2739 . . . . . . . . . . . . . . . 16 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → 𝑥 = ((0g𝑅)(+g𝑅)𝑥))
117108, 111, 116rspcedvd 3615 . . . . . . . . . . . . . . 15 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → ∃𝑏 ∈ ((Base‘𝑅) × {𝑥})𝑥 = ((0g𝑅)(+g𝑅)𝑏))
11895, 99, 117rspcedvd 3615 . . . . . . . . . . . . . 14 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → ∃𝑎𝑀𝑏 ∈ ((Base‘𝑅) × {𝑥})𝑥 = (𝑎(+g𝑅)𝑏))
119 simp-5l 784 . . . . . . . . . . . . . . 15 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → 𝑅 ∈ CRing)
1203, 32, 67lsmelvalx 19508 . . . . . . . . . . . . . . 15 ((𝑅 ∈ CRing ∧ 𝑀 ⊆ (Base‘𝑅) ∧ ((Base‘𝑅) × {𝑥}) ⊆ (Base‘𝑅)) → (𝑥 ∈ (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) ↔ ∃𝑎𝑀𝑏 ∈ ((Base‘𝑅) × {𝑥})𝑥 = (𝑎(+g𝑅)𝑏)))
121119, 23, 82, 120syl3anc 1372 . . . . . . . . . . . . . 14 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → (𝑥 ∈ (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) ↔ ∃𝑎𝑀𝑏 ∈ ((Base‘𝑅) × {𝑥})𝑥 = (𝑎(+g𝑅)𝑏)))
122118, 121mpbird 257 . . . . . . . . . . . . 13 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → 𝑥 ∈ (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})))
123 simpr 486 . . . . . . . . . . . . 13 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → ¬ 𝑥𝑀)
124 nelne1 3040 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) ∧ ¬ 𝑥𝑀) → (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) ≠ 𝑀)
125122, 123, 124syl2anc 585 . . . . . . . . . . . 12 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) ≠ 𝑀)
126125neneqd 2946 . . . . . . . . . . 11 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → ¬ (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) = 𝑀)
12792, 126orcnd 878 . . . . . . . . . 10 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) = (Base‘𝑅))
12866, 127eleqtrrd 2837 . . . . . . . . 9 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → (1r𝑅) ∈ (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})))
1293, 32, 67lsmelvalx 19508 . . . . . . . . . 10 ((𝑅 ∈ CRing ∧ 𝑀 ⊆ (Base‘𝑅) ∧ ((Base‘𝑅) × {𝑥}) ⊆ (Base‘𝑅)) → ((1r𝑅) ∈ (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) ↔ ∃𝑢𝑀𝑘 ∈ ((Base‘𝑅) × {𝑥})(1r𝑅) = (𝑢(+g𝑅)𝑘)))
130119, 23, 82, 129syl3anc 1372 . . . . . . . . 9 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → ((1r𝑅) ∈ (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) ↔ ∃𝑢𝑀𝑘 ∈ ((Base‘𝑅) × {𝑥})(1r𝑅) = (𝑢(+g𝑅)𝑘)))
131128, 130mpbid 231 . . . . . . . 8 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → ∃𝑢𝑀𝑘 ∈ ((Base‘𝑅) × {𝑥})(1r𝑅) = (𝑢(+g𝑅)𝑘))
13264, 131r19.29vva 3214 . . . . . . 7 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → 𝑦𝑀)
133132ex 414 . . . . . 6 (((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) → (¬ 𝑥𝑀𝑦𝑀))
134133orrd 862 . . . . 5 (((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) → (𝑥𝑀𝑦𝑀))
135134ex 414 . . . 4 ((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑥(.r𝑅)𝑦) ∈ 𝑀 → (𝑥𝑀𝑦𝑀)))
136135anasss 468 . . 3 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((𝑥(.r𝑅)𝑦) ∈ 𝑀 → (𝑥𝑀𝑦𝑀)))
137136ralrimivva 3201 . 2 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) ∈ 𝑀 → (𝑥𝑀𝑦𝑀)))
1383, 16prmidl2 32559 . 2 (((𝑅 ∈ Ring ∧ 𝑀 ∈ (LIdeal‘𝑅)) ∧ (𝑀 ≠ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) ∈ 𝑀 → (𝑥𝑀𝑦𝑀)))) → 𝑀 ∈ (PrmIdeal‘𝑅))
1392, 5, 7, 137, 138syl22anc 838 1 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (PrmIdeal‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846   = wceq 1542  wcel 2107  wne 2941  wral 3062  wrex 3071  Vcvv 3475  cun 3947  wss 3949  {csn 4629  cfv 6544  (class class class)co 7409  Basecbs 17144  +gcplusg 17197  .rcmulr 17198  0gc0g 17385  Grpcgrp 18819  LSSumclsm 19502  mulGrpcmgp 19987  1rcur 20004  Ringcrg 20056  CRingccrg 20057  LModclmod 20471  ringLModcrglmod 20782  LIdealclidl 20783  RSpancrsp 20784  LPIdealclpidl 20879  PrmIdealcprmidl 32553  MaxIdealcmxidl 32575
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-mulr 17211  df-sca 17213  df-vsca 17214  df-ip 17215  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-submnd 18672  df-grp 18822  df-minusg 18823  df-sbg 18824  df-subg 19003  df-cntz 19181  df-lsm 19504  df-cmn 19650  df-abl 19651  df-mgp 19988  df-ur 20005  df-ring 20058  df-cring 20059  df-subrg 20317  df-lmod 20473  df-lss 20543  df-lsp 20583  df-sra 20785  df-rgmod 20786  df-lidl 20787  df-rsp 20788  df-lpidl 20881  df-prmidl 32554  df-mxidl 32576
This theorem is referenced by:  zarcls1  32849  zarclssn  32853  zarmxt1  32860
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