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Theorem mxidlprm 31000
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 19299 . . 3 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
21adantr 484 . 2 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑅 ∈ Ring)
3 eqid 2824 . . . 4 (Base‘𝑅) = (Base‘𝑅)
43mxidlidl 30995 . . 3 ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))
51, 4sylan 583 . 2 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))
63mxidlnr 30996 . . 3 ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ≠ (Base‘𝑅))
71, 6sylan 583 . 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 488 . . . . . . . . . . . . . 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 7156 . . . . . . . . . . . . 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 2859 . . . . . . . . . . . 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 7155 . . . . . . . . . . 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 2824 . . . . . . . . . . . . 13 (.r𝑅) = (.r𝑅)
17 eqid 2824 . . . . . . . . . . . . 13 (1r𝑅) = (1r𝑅)
183, 16, 17ringlidm 19312 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝑦 ∈ (Base‘𝑅)) → ((1r𝑅)(.r𝑅)𝑦) = 𝑦)
1914, 15, 18syl2anc 587 . . . . . . . . . . 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 2824 . . . . . . . . . . . . . . . . 17 (LIdeal‘𝑅) = (LIdeal‘𝑅)
213, 20lidlss 19971 . . . . . . . . . . . . . . . 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 3952 . . . . . . . . . . . 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 19302 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑎(.r𝑅)𝑥) ∈ (Base‘𝑅))
3114, 27, 29, 30syl3anc 1368 . . . . . . . . . . . 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 2824 . . . . . . . . . . . . 13 (+g𝑅) = (+g𝑅)
333, 32, 16ringdir 19308 . . . . . . . . . . . 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 1369 . . . . . . . . . . 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 2867 . . . . . . . . . 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 587 . . . . . . . . . . . 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 19303 . . . . . . . . . . . . 13 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑢 ∈ (Base‘𝑅)) → (𝑦(.r𝑅)𝑢) = (𝑢(.r𝑅)𝑦))
4139, 15, 26, 40syl3anc 1368 . . . . . . . . . . . 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 19978 . . . . . . . . . . . . 13 (((𝑅 ∈ Ring ∧ 𝑀 ∈ (LIdeal‘𝑅)) ∧ (𝑦 ∈ (Base‘𝑅) ∧ 𝑢𝑀)) → (𝑦(.r𝑅)𝑢) ∈ 𝑀)
4314, 38, 15, 25, 42syl22anc 837 . . . . . . . . . . . 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 2917 . . . . . . . . . . 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 19305 . . . . . . . . . . . . 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 1369 . . . . . . . . . . . 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 19978 . . . . . . . . . . . . 13 (((𝑅 ∈ Ring ∧ 𝑀 ∈ (LIdeal‘𝑅)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀)) → (𝑎(.r𝑅)(𝑥(.r𝑅)𝑦)) ∈ 𝑀)
4914, 38, 27, 47, 48syl22anc 837 . . . . . . . . . . . 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 2916 . . . . . . . . . . 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 19974 . . . . . . . . . . 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 837 . . . . . . . . . 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 2916 . . . . . . . . 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 2824 . . . . . . . . . . . . 13 (mulGrp‘𝑅) = (mulGrp‘𝑅)
5655, 3mgpbas 19236 . . . . . . . . . . . 12 (Base‘𝑅) = (Base‘(mulGrp‘𝑅))
5755, 16mgpplusg 19234 . . . . . . . . . . . 12 (.r𝑅) = (+g‘(mulGrp‘𝑅))
58 mxidlprm.1 . . . . . . . . . . . 12 × = (LSSum‘(mulGrp‘𝑅))
59 fvexd 6668 . . . . . . . . . . . 12 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → (mulGrp‘𝑅) ∈ V)
60 ssidd 3974 . . . . . . . . . . . 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 30965 . . . . . . . . . . 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 235 . . . . . . . . 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 3281 . . . . . . . 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 19309 . . . . . . . . . . 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 2824 . . . . . . . . . . . . 13 (LSSum‘𝑅) = (LSSum‘𝑅)
68 eqid 2824 . . . . . . . . . . . . 13 (RSpan‘𝑅) = (RSpan‘𝑅)
69 eqid 2824 . . . . . . . . . . . . . . . 16 (LPIdeal‘𝑅) = (LPIdeal‘𝑅)
7069, 20lpiss 20011 . . . . . . . . . . . . . . 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 30971 . . . . . . . . . . . . . 14 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → ((Base‘𝑅) × {𝑥}) ∈ (LPIdeal‘𝑅))
7371, 72sseldd 3952 . . . . . . . . . . . . 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 30973 . . . . . . . . . . . 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 19965 . . . . . . . . . . . . . . . 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 19955 . . . . . . . . . . . . . . . 16 (Base‘𝑅) = (Base‘(ringLMod‘𝑅))
78 rspval 19953 . . . . . . . . . . . . . . . 16 (RSpan‘𝑅) = (LSpan‘(ringLMod‘𝑅))
7977, 78lspssid 19745 . . . . . . . . . . . . . . 15 (((ringLMod‘𝑅) ∈ LMod ∧ 𝑀 ⊆ (Base‘𝑅)) → 𝑀 ⊆ ((RSpan‘𝑅)‘𝑀))
8076, 23, 79syl2anc 587 . . . . . . . . . . . . . 14 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → 𝑀 ⊆ ((RSpan‘𝑅)‘𝑀))
8128snssd 4725 . . . . . . . . . . . . . . . . 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 30969 . . . . . . . . . . . . . . . 16 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → ((Base‘𝑅) × {𝑥}) ⊆ (Base‘𝑅))
8323, 82unssd 4146 . . . . . . . . . . . . . . 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 4132 . . . . . . . . . . . . . . . 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 19744 . . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . . 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 3961 . . . . . . . . . . . . 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 30972 . . . . . . . . . . . . 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 3992 . . . . . . . . . . . 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 30997 . . . . . . . . . . . 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 837 . . . . . . . . . . 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 2824 . . . . . . . . . . . . . . . . 17 (0g𝑅) = (0g𝑅)
9420, 93lidl0cl 19973 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Ring ∧ 𝑀 ∈ (LIdeal‘𝑅)) → (0g𝑅) ∈ 𝑀)
9513, 37, 94syl2anc 587 . . . . . . . . . . . . . . 15 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → (0g𝑅) ∈ 𝑀)
96 oveq1 7147 . . . . . . . . . . . . . . . . . 18 (𝑎 = (0g𝑅) → (𝑎(+g𝑅)𝑏) = ((0g𝑅)(+g𝑅)𝑏))
9796eqeq2d 2835 . . . . . . . . . . . . . . . . 17 (𝑎 = (0g𝑅) → (𝑥 = (𝑎(+g𝑅)𝑏) ↔ 𝑥 = ((0g𝑅)(+g𝑅)𝑏)))
9897rexbidv 3289 . . . . . . . . . . . . . . . 16 (𝑎 = (0g𝑅) → (∃𝑏 ∈ ((Base‘𝑅) × {𝑥})𝑥 = (𝑎(+g𝑅)𝑏) ↔ ∃𝑏 ∈ ((Base‘𝑅) × {𝑥})𝑥 = ((0g𝑅)(+g𝑅)𝑏)))
9998adantl 485 . . . . . . . . . . . . . . 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 7147 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = (1r𝑅) → (𝑎(.r𝑅)𝑥) = ((1r𝑅)(.r𝑅)𝑥))
101100eqeq2d 2835 . . . . . . . . . . . . . . . . . . 19 (𝑎 = (1r𝑅) → (𝑥 = (𝑎(.r𝑅)𝑥) ↔ 𝑥 = ((1r𝑅)(.r𝑅)𝑥)))
102101adantl 485 . . . . . . . . . . . . . . . . . 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 19312 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → ((1r𝑅)(.r𝑅)𝑥) = 𝑥)
10413, 28, 103syl2anc 587 . . . . . . . . . . . . . . . . . . 19 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → ((1r𝑅)(.r𝑅)𝑥) = 𝑥)
105104eqcomd 2830 . . . . . . . . . . . . . . . . . 18 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → 𝑥 = ((1r𝑅)(.r𝑅)𝑥))
10666, 102, 105rspcedvd 3611 . . . . . . . . . . . . . . . . 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 30965 . . . . . . . . . . . . . . . . 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 260 . . . . . . . . . . . . . . . 16 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → 𝑥 ∈ ((Base‘𝑅) × {𝑥}))
109 oveq2 7148 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑥 → ((0g𝑅)(+g𝑅)𝑏) = ((0g𝑅)(+g𝑅)𝑥))
110109eqeq2d 2835 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑥 → (𝑥 = ((0g𝑅)(+g𝑅)𝑏) ↔ 𝑥 = ((0g𝑅)(+g𝑅)𝑥)))
111110adantl 485 . . . . . . . . . . . . . . . 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 19293 . . . . . . . . . . . . . . . . . . 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 18124 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → ((0g𝑅)(+g𝑅)𝑥) = 𝑥)
115113, 28, 114syl2anc 587 . . . . . . . . . . . . . . . . 17 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → ((0g𝑅)(+g𝑅)𝑥) = 𝑥)
116115eqcomd 2830 . . . . . . . . . . . . . . . 16 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → 𝑥 = ((0g𝑅)(+g𝑅)𝑥))
117108, 111, 116rspcedvd 3611 . . . . . . . . . . . . . . 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 3611 . . . . . . . . . . . . . 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 18756 . . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . . 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 260 . . . . . . . . . . . . 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 488 . . . . . . . . . . . . 13 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → ¬ 𝑥𝑀)
124 nelne1 3109 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) ∧ ¬ 𝑥𝑀) → (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) ≠ 𝑀)
125122, 123, 124syl2anc 587 . . . . . . . . . . . 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 3018 . . . . . . . . . . 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 876 . . . . . . . . . 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 2919 . . . . . . . . 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 18756 . . . . . . . . . 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 1368 . . . . . . . . 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 235 . . . . . . . 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 3327 . . . . . . 7 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → 𝑦𝑀)
133132ex 416 . . . . . 6 (((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) → (¬ 𝑥𝑀𝑦𝑀))
134133orrd 860 . . . . 5 (((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) → (𝑥𝑀𝑦𝑀))
135134ex 416 . . . 4 ((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑥(.r𝑅)𝑦) ∈ 𝑀 → (𝑥𝑀𝑦𝑀)))
136135anasss 470 . . 3 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((𝑥(.r𝑅)𝑦) ∈ 𝑀 → (𝑥𝑀𝑦𝑀)))
137136ralrimivva 3185 . 2 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) ∈ 𝑀 → (𝑥𝑀𝑦𝑀)))
1383, 16prmidl2 30980 . 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 837 1 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (PrmIdeal‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844   = wceq 1538  wcel 2115  wne 3013  wral 3132  wrex 3133  Vcvv 3479  cun 3916  wss 3918  {csn 4548  cfv 6338  (class class class)co 7140  Basecbs 16474  +gcplusg 16556  .rcmulr 16557  0gc0g 16704  Grpcgrp 18094  LSSumclsm 18750  mulGrpcmgp 19230  1rcur 19242  Ringcrg 19288  CRingccrg 19289  LModclmod 19622  ringLModcrglmod 19929  LIdealclidl 19930  RSpancrsp 19931  LPIdealclpidl 20002  PrmIdealcprmidl 30974  MaxIdealcmxidl 30991
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5173  ax-sep 5186  ax-nul 5193  ax-pow 5249  ax-pr 5313  ax-un 7446  ax-cnex 10580  ax-resscn 10581  ax-1cn 10582  ax-icn 10583  ax-addcl 10584  ax-addrcl 10585  ax-mulcl 10586  ax-mulrcl 10587  ax-mulcom 10588  ax-addass 10589  ax-mulass 10590  ax-distr 10591  ax-i2m1 10592  ax-1ne0 10593  ax-1rid 10594  ax-rnegex 10595  ax-rrecex 10596  ax-cnre 10597  ax-pre-lttri 10598  ax-pre-lttrn 10599  ax-pre-ltadd 10600  ax-pre-mulgt0 10601
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-nel 3118  df-ral 3137  df-rex 3138  df-reu 3139  df-rmo 3140  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4822  df-int 4860  df-iun 4904  df-br 5050  df-opab 5112  df-mpt 5130  df-tr 5156  df-id 5443  df-eprel 5448  df-po 5457  df-so 5458  df-fr 5497  df-we 5499  df-xp 5544  df-rel 5545  df-cnv 5546  df-co 5547  df-dm 5548  df-rn 5549  df-res 5550  df-ima 5551  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6297  df-fun 6340  df-fn 6341  df-f 6342  df-f1 6343  df-fo 6344  df-f1o 6345  df-fv 6346  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7674  df-2nd 7675  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-er 8274  df-en 8495  df-dom 8496  df-sdom 8497  df-pnf 10664  df-mnf 10665  df-xr 10666  df-ltxr 10667  df-le 10668  df-sub 10859  df-neg 10860  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-ndx 16477  df-slot 16478  df-base 16480  df-sets 16481  df-ress 16482  df-plusg 16569  df-mulr 16570  df-sca 16572  df-vsca 16573  df-ip 16574  df-0g 16706  df-mgm 17843  df-sgrp 17892  df-mnd 17903  df-submnd 17948  df-grp 18097  df-minusg 18098  df-sbg 18099  df-subg 18267  df-cntz 18438  df-lsm 18752  df-cmn 18899  df-abl 18900  df-mgp 19231  df-ur 19243  df-ring 19290  df-cring 19291  df-subrg 19521  df-lmod 19624  df-lss 19692  df-lsp 19732  df-sra 19932  df-rgmod 19933  df-lidl 19934  df-rsp 19935  df-lpidl 20004  df-prmidl 30975  df-mxidl 30992
This theorem is referenced by: (None)
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