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Theorem mxidlprm 32860
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 20139 . . 3 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
21adantr 479 . 2 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑅 ∈ Ring)
3 eqid 2730 . . . 4 (Base‘𝑅) = (Base‘𝑅)
43mxidlidl 32853 . . 3 ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))
51, 4sylan 578 . 2 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))
63mxidlnr 32854 . . 3 ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ≠ (Base‘𝑅))
71, 6sylan 578 . 2 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ≠ (Base‘𝑅))
8 simpllr 772 . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . 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 7427 . . . . . . . . . . . . 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 2770 . . . . . . . . . . . 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 7426 . . . . . . . . . . 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 728 . . . . . . . . . . . . 13 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → 𝑅 ∈ Ring)
1413ad5antr 730 . . . . . . . . . . . 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 788 . . . . . . . . . . . 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 2730 . . . . . . . . . . . . 13 (.r𝑅) = (.r𝑅)
17 eqid 2730 . . . . . . . . . . . . 13 (1r𝑅) = (1r𝑅)
183, 16, 17ringlidm 20157 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝑦 ∈ (Base‘𝑅)) → ((1r𝑅)(.r𝑅)𝑦) = 𝑦)
1914, 15, 18syl2anc 582 . . . . . . . . . . 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 2730 . . . . . . . . . . . . . . . . 17 (LIdeal‘𝑅) = (LIdeal‘𝑅)
213, 20lidlss 20978 . . . . . . . . . . . . . . . 16 (𝑀 ∈ (LIdeal‘𝑅) → 𝑀 ⊆ (Base‘𝑅))
225, 21syl 17 . . . . . . . . . . . . . . 15 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ⊆ (Base‘𝑅))
2322ad4antr 728 . . . . . . . . . . . . . 14 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → 𝑀 ⊆ (Base‘𝑅))
2423ad5antr 730 . . . . . . . . . . . . 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 782 . . . . . . . . . . . . 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 3982 . . . . . . . . . . . 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 765 . . . . . . . . . . . . 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 780 . . . . . . . . . . . . . 14 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → 𝑥 ∈ (Base‘𝑅))
2928ad5antr 730 . . . . . . . . . . . . 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 20144 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑎 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑎(.r𝑅)𝑥) ∈ (Base‘𝑅))
3114, 27, 29, 30syl3anc 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) ∈ (Base‘𝑅))
32 eqid 2730 . . . . . . . . . . . . 13 (+g𝑅) = (+g𝑅)
333, 32, 16ringdir 20153 . . . . . . . . . . . 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 1370 . . . . . . . . . . 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 2778 . . . . . . . . . 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 782 . . . . . . . . . . . . 13 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → 𝑀 ∈ (MaxIdeal‘𝑅))
3713, 36, 4syl2anc 582 . . . . . . . . . . . 12 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → 𝑀 ∈ (LIdeal‘𝑅))
3837ad5antr 730 . . . . . . . . . . 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 791 . . . . . . . . . . . . 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 20145 . . . . . . . . . . . . 13 ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑢 ∈ (Base‘𝑅)) → (𝑦(.r𝑅)𝑢) = (𝑢(.r𝑅)𝑦))
4139, 15, 26, 40syl3anc 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)) → (𝑦(.r𝑅)𝑢) = (𝑢(.r𝑅)𝑦))
4220, 3, 16lidlmcl 20989 . . . . . . . . . . . . 13 (((𝑅 ∈ Ring ∧ 𝑀 ∈ (LIdeal‘𝑅)) ∧ (𝑦 ∈ (Base‘𝑅) ∧ 𝑢𝑀)) → (𝑦(.r𝑅)𝑢) ∈ 𝑀)
4314, 38, 15, 25, 42syl22anc 835 . . . . . . . . . . . 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 2832 . . . . . . . . . . 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 20147 . . . . . . . . . . . . 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 1370 . . . . . . . . . . . 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 786 . . . . . . . . . . . . 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 20989 . . . . . . . . . . . . 13 (((𝑅 ∈ Ring ∧ 𝑀 ∈ (LIdeal‘𝑅)) ∧ (𝑎 ∈ (Base‘𝑅) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀)) → (𝑎(.r𝑅)(𝑥(.r𝑅)𝑦)) ∈ 𝑀)
4914, 38, 27, 47, 48syl22anc 835 . . . . . . . . . . . 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 2831 . . . . . . . . . . 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 20985 . . . . . . . . . . 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 835 . . . . . . . . . 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 2831 . . . . . . . . 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 765 . . . . . . . . . 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 2730 . . . . . . . . . . . . 13 (mulGrp‘𝑅) = (mulGrp‘𝑅)
5655, 3mgpbas 20034 . . . . . . . . . . . 12 (Base‘𝑅) = (Base‘(mulGrp‘𝑅))
5755, 16mgpplusg 20032 . . . . . . . . . . . 12 (.r𝑅) = (+g‘(mulGrp‘𝑅))
58 mxidlprm.1 . . . . . . . . . . . 12 × = (LSSum‘(mulGrp‘𝑅))
59 fvexd 6905 . . . . . . . . . . . 12 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → (mulGrp‘𝑅) ∈ V)
60 ssidd 4004 . . . . . . . . . . . 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 32774 . . . . . . . . . . 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 726 . . . . . . . . . 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 3160 . . . . . . . 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 20154 . . . . . . . . . . 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 2730 . . . . . . . . . . . . 13 (LSSum‘𝑅) = (LSSum‘𝑅)
68 eqid 2730 . . . . . . . . . . . . 13 (RSpan‘𝑅) = (RSpan‘𝑅)
69 eqid 2730 . . . . . . . . . . . . . . . 16 (LPIdeal‘𝑅) = (LPIdeal‘𝑅)
7069, 20lpiss 21088 . . . . . . . . . . . . . . 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 32783 . . . . . . . . . . . . . 14 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → ((Base‘𝑅) × {𝑥}) ∈ (LPIdeal‘𝑅))
7371, 72sseldd 3982 . . . . . . . . . . . . 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 32785 . . . . . . . . . . . 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 20972 . . . . . . . . . . . . . . . 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 20962 . . . . . . . . . . . . . . . 16 (Base‘𝑅) = (Base‘(ringLMod‘𝑅))
78 rspval 20960 . . . . . . . . . . . . . . . 16 (RSpan‘𝑅) = (LSpan‘(ringLMod‘𝑅))
7977, 78lspssid 20740 . . . . . . . . . . . . . . 15 (((ringLMod‘𝑅) ∈ LMod ∧ 𝑀 ⊆ (Base‘𝑅)) → 𝑀 ⊆ ((RSpan‘𝑅)‘𝑀))
8076, 23, 79syl2anc 582 . . . . . . . . . . . . . 14 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → 𝑀 ⊆ ((RSpan‘𝑅)‘𝑀))
8128snssd 4811 . . . . . . . . . . . . . . . . 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 32779 . . . . . . . . . . . . . . . 16 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → ((Base‘𝑅) × {𝑥}) ⊆ (Base‘𝑅))
8323, 82unssd 4185 . . . . . . . . . . . . . . 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 4171 . . . . . . . . . . . . . . . 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 20739 . . . . . . . . . . . . . . 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 1369 . . . . . . . . . . . . . 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 3991 . . . . . . . . . . . . 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 32784 . . . . . . . . . . . . 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 4022 . . . . . . . . . . . 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 32855 . . . . . . . . . . . 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 835 . . . . . . . . . . 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 2730 . . . . . . . . . . . . . . . . 17 (0g𝑅) = (0g𝑅)
9420, 93lidl0cl 20984 . . . . . . . . . . . . . . . 16 ((𝑅 ∈ Ring ∧ 𝑀 ∈ (LIdeal‘𝑅)) → (0g𝑅) ∈ 𝑀)
9513, 37, 94syl2anc 582 . . . . . . . . . . . . . . 15 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → (0g𝑅) ∈ 𝑀)
96 oveq1 7418 . . . . . . . . . . . . . . . . . 18 (𝑎 = (0g𝑅) → (𝑎(+g𝑅)𝑏) = ((0g𝑅)(+g𝑅)𝑏))
9796eqeq2d 2741 . . . . . . . . . . . . . . . . 17 (𝑎 = (0g𝑅) → (𝑥 = (𝑎(+g𝑅)𝑏) ↔ 𝑥 = ((0g𝑅)(+g𝑅)𝑏)))
9897rexbidv 3176 . . . . . . . . . . . . . . . 16 (𝑎 = (0g𝑅) → (∃𝑏 ∈ ((Base‘𝑅) × {𝑥})𝑥 = (𝑎(+g𝑅)𝑏) ↔ ∃𝑏 ∈ ((Base‘𝑅) × {𝑥})𝑥 = ((0g𝑅)(+g𝑅)𝑏)))
9998adantl 480 . . . . . . . . . . . . . . 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 7418 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = (1r𝑅) → (𝑎(.r𝑅)𝑥) = ((1r𝑅)(.r𝑅)𝑥))
101100eqeq2d 2741 . . . . . . . . . . . . . . . . . . 19 (𝑎 = (1r𝑅) → (𝑥 = (𝑎(.r𝑅)𝑥) ↔ 𝑥 = ((1r𝑅)(.r𝑅)𝑥)))
102101adantl 480 . . . . . . . . . . . . . . . . . 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 20157 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → ((1r𝑅)(.r𝑅)𝑥) = 𝑥)
10413, 28, 103syl2anc 582 . . . . . . . . . . . . . . . . . . 19 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → ((1r𝑅)(.r𝑅)𝑥) = 𝑥)
105104eqcomd 2736 . . . . . . . . . . . . . . . . . 18 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → 𝑥 = ((1r𝑅)(.r𝑅)𝑥))
10666, 102, 105rspcedvd 3613 . . . . . . . . . . . . . . . . 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 32774 . . . . . . . . . . . . . . . . 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 256 . . . . . . . . . . . . . . . 16 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → 𝑥 ∈ ((Base‘𝑅) × {𝑥}))
109 oveq2 7419 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑥 → ((0g𝑅)(+g𝑅)𝑏) = ((0g𝑅)(+g𝑅)𝑥))
110109eqeq2d 2741 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑥 → (𝑥 = ((0g𝑅)(+g𝑅)𝑏) ↔ 𝑥 = ((0g𝑅)(+g𝑅)𝑥)))
111110adantl 480 . . . . . . . . . . . . . . . 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 20132 . . . . . . . . . . . . . . . . . . 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 18888 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → ((0g𝑅)(+g𝑅)𝑥) = 𝑥)
115113, 28, 114syl2anc 582 . . . . . . . . . . . . . . . . 17 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → ((0g𝑅)(+g𝑅)𝑥) = 𝑥)
116115eqcomd 2736 . . . . . . . . . . . . . . . 16 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → 𝑥 = ((0g𝑅)(+g𝑅)𝑥))
117108, 111, 116rspcedvd 3613 . . . . . . . . . . . . . . 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 3613 . . . . . . . . . . . . . 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 781 . . . . . . . . . . . . . . 15 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → 𝑅 ∈ CRing)
1203, 32, 67lsmelvalx 19549 . . . . . . . . . . . . . . 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 1369 . . . . . . . . . . . . . 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 256 . . . . . . . . . . . . 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 483 . . . . . . . . . . . . 13 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → ¬ 𝑥𝑀)
124 nelne1 3037 . . . . . . . . . . . . 13 ((𝑥 ∈ (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) ∧ ¬ 𝑥𝑀) → (𝑀(LSSum‘𝑅)((Base‘𝑅) × {𝑥})) ≠ 𝑀)
125122, 123, 124syl2anc 582 . . . . . . . . . . . 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 2943 . . . . . . . . . . 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 875 . . . . . . . . . 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 2834 . . . . . . . . 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 19549 . . . . . . . . . 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 1369 . . . . . . . . 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 3211 . . . . . . 7 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) ∧ ¬ 𝑥𝑀) → 𝑦𝑀)
133132ex 411 . . . . . 6 (((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) → (¬ 𝑥𝑀𝑦𝑀))
134133orrd 859 . . . . 5 (((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝑀) → (𝑥𝑀𝑦𝑀))
135134ex 411 . . . 4 ((((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑥(.r𝑅)𝑦) ∈ 𝑀 → (𝑥𝑀𝑦𝑀)))
136135anasss 465 . . 3 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((𝑥(.r𝑅)𝑦) ∈ 𝑀 → (𝑥𝑀𝑦𝑀)))
137136ralrimivva 3198 . 2 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) ∈ 𝑀 → (𝑥𝑀𝑦𝑀)))
1383, 16prmidl2 32833 . 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 835 1 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (PrmIdeal‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wo 843   = wceq 1539  wcel 2104  wne 2938  wral 3059  wrex 3068  Vcvv 3472  cun 3945  wss 3947  {csn 4627  cfv 6542  (class class class)co 7411  Basecbs 17148  +gcplusg 17201  .rcmulr 17202  0gc0g 17389  Grpcgrp 18855  LSSumclsm 19543  mulGrpcmgp 20028  1rcur 20075  Ringcrg 20127  CRingccrg 20128  LModclmod 20614  ringLModcrglmod 20927  LIdealclidl 20928  RSpancrsp 20929  LPIdealclpidl 21079  PrmIdealcprmidl 32827  MaxIdealcmxidl 32849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-plusg 17214  df-mulr 17215  df-sca 17217  df-vsca 17218  df-ip 17219  df-0g 17391  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-submnd 18706  df-grp 18858  df-minusg 18859  df-sbg 18860  df-subg 19039  df-cntz 19222  df-lsm 19545  df-cmn 19691  df-abl 19692  df-mgp 20029  df-rng 20047  df-ur 20076  df-ring 20129  df-cring 20130  df-subrg 20459  df-lmod 20616  df-lss 20687  df-lsp 20727  df-sra 20930  df-rgmod 20931  df-lidl 20932  df-rsp 20933  df-lpidl 21081  df-prmidl 32828  df-mxidl 32850
This theorem is referenced by:  zarcls1  33147  zarclssn  33151  zarmxt1  33158
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