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Theorem zarclssn 33834
Description: The closed points of Zariski topology are the maximal ideals. (Contributed by Thierry Arnoux, 16-Jun-2024.)
Hypotheses
Ref Expression
zarclsx.1 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗})
zarclssn.1 𝐵 = (LIdeal‘𝑅)
Assertion
Ref Expression
zarclssn ((𝑅 ∈ CRing ∧ 𝑀𝐵) → ({𝑀} = (𝑉𝑀) ↔ 𝑀 ∈ (MaxIdeal‘𝑅)))
Distinct variable groups:   𝑅,𝑖,𝑗   𝑖,𝑉   𝐵,𝑖,𝑗   𝑖,𝑀,𝑗   𝑗,𝑉

Proof of Theorem zarclssn
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 crngring 20263 . . . 4 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
21ad2antrr 726 . . 3 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) → 𝑅 ∈ Ring)
3 simplr 769 . . . . 5 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) → 𝑀𝐵)
4 zarclssn.1 . . . . 5 𝐵 = (LIdeal‘𝑅)
53, 4eleqtrdi 2849 . . . 4 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) → 𝑀 ∈ (LIdeal‘𝑅))
6 simpr 484 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) → {𝑀} = (𝑉𝑀))
73snn0d 4780 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) → {𝑀} ≠ ∅)
86, 7eqnetrrd 3007 . . . . 5 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) → (𝑉𝑀) ≠ ∅)
9 simpll 767 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) → 𝑅 ∈ CRing)
10 zarclsx.1 . . . . . . . 8 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗})
11 eqid 2735 . . . . . . . 8 (Base‘𝑅) = (Base‘𝑅)
1210, 11zarcls1 33830 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (LIdeal‘𝑅)) → ((𝑉𝑀) = ∅ ↔ 𝑀 = (Base‘𝑅)))
1312necon3bid 2983 . . . . . 6 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (LIdeal‘𝑅)) → ((𝑉𝑀) ≠ ∅ ↔ 𝑀 ≠ (Base‘𝑅)))
149, 5, 13syl2anc 584 . . . . 5 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) → ((𝑉𝑀) ≠ ∅ ↔ 𝑀 ≠ (Base‘𝑅)))
158, 14mpbid 232 . . . 4 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) → 𝑀 ≠ (Base‘𝑅))
16 simpr 484 . . . . . . . . . . . 12 ((((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗𝑚) → 𝑗𝑚)
179ad5antr 734 . . . . . . . . . . . . . 14 ((((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗𝑚) → 𝑅 ∈ CRing)
18 simplr 769 . . . . . . . . . . . . . 14 ((((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗𝑚) → 𝑚 ∈ (MaxIdeal‘𝑅))
19 eqid 2735 . . . . . . . . . . . . . . 15 (LSSum‘(mulGrp‘𝑅)) = (LSSum‘(mulGrp‘𝑅))
2019mxidlprm 33478 . . . . . . . . . . . . . 14 ((𝑅 ∈ CRing ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ∈ (PrmIdeal‘𝑅))
2117, 18, 20syl2anc 584 . . . . . . . . . . . . 13 ((((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗𝑚) → 𝑚 ∈ (PrmIdeal‘𝑅))
22 simp-4r 784 . . . . . . . . . . . . . 14 ((((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗𝑚) → 𝑀𝑗)
2322, 16sstrd 4006 . . . . . . . . . . . . 13 ((((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗𝑚) → 𝑀𝑚)
2410a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) → 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗}))
25 sseq1 4021 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 𝑀 → (𝑖𝑗𝑀𝑗))
2625rabbidv 3441 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 𝑀 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀𝑗})
2726adantl 481 . . . . . . . . . . . . . . . . . 18 ((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑖 = 𝑀) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀𝑗})
28 fvex 6920 . . . . . . . . . . . . . . . . . . . 20 (PrmIdeal‘𝑅) ∈ V
2928rabex 5345 . . . . . . . . . . . . . . . . . . 19 {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀𝑗} ∈ V
3029a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀𝑗} ∈ V)
3124, 27, 5, 30fvmptd 7023 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) → (𝑉𝑀) = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀𝑗})
326, 31eqtr2d 2776 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀𝑗} = {𝑀})
33 rabeqsn 4672 . . . . . . . . . . . . . . . 16 ({𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀𝑗} = {𝑀} ↔ ∀𝑗((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗) ↔ 𝑗 = 𝑀))
3432, 33sylib 218 . . . . . . . . . . . . . . 15 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) → ∀𝑗((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗) ↔ 𝑗 = 𝑀))
3534ad5antr 734 . . . . . . . . . . . . . 14 ((((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗𝑚) → ∀𝑗((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗) ↔ 𝑗 = 𝑀))
36 vex 3482 . . . . . . . . . . . . . . 15 𝑚 ∈ V
37 eleq1w 2822 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑚 → (𝑗 ∈ (PrmIdeal‘𝑅) ↔ 𝑚 ∈ (PrmIdeal‘𝑅)))
38 sseq2 4022 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑚 → (𝑀𝑗𝑀𝑚))
3937, 38anbi12d 632 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑚 → ((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗) ↔ (𝑚 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑚)))
40 eqeq1 2739 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑚 → (𝑗 = 𝑀𝑚 = 𝑀))
4139, 40bibi12d 345 . . . . . . . . . . . . . . 15 (𝑗 = 𝑚 → (((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗) ↔ 𝑗 = 𝑀) ↔ ((𝑚 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑚) ↔ 𝑚 = 𝑀)))
4236, 41spcv 3605 . . . . . . . . . . . . . 14 (∀𝑗((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗) ↔ 𝑗 = 𝑀) → ((𝑚 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑚) ↔ 𝑚 = 𝑀))
4335, 42syl 17 . . . . . . . . . . . . 13 ((((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗𝑚) → ((𝑚 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑚) ↔ 𝑚 = 𝑀))
4421, 23, 43mpbi2and 712 . . . . . . . . . . . 12 ((((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗𝑚) → 𝑚 = 𝑀)
4516, 44sseqtrd 4036 . . . . . . . . . . 11 ((((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗𝑚) → 𝑗𝑀)
4645, 22eqssd 4013 . . . . . . . . . 10 ((((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗𝑚) → 𝑗 = 𝑀)
471ad5antr 734 . . . . . . . . . . 11 ((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) → 𝑅 ∈ Ring)
48 simpllr 776 . . . . . . . . . . 11 ((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) → 𝑗 ∈ (LIdeal‘𝑅))
49 simpr 484 . . . . . . . . . . . 12 ((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) → ¬ 𝑗 = (Base‘𝑅))
5049neqned 2945 . . . . . . . . . . 11 ((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) → 𝑗 ≠ (Base‘𝑅))
5111ssmxidl 33482 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝑗 ∈ (LIdeal‘𝑅) ∧ 𝑗 ≠ (Base‘𝑅)) → ∃𝑚 ∈ (MaxIdeal‘𝑅)𝑗𝑚)
5247, 48, 50, 51syl3anc 1370 . . . . . . . . . 10 ((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) → ∃𝑚 ∈ (MaxIdeal‘𝑅)𝑗𝑚)
5346, 52r19.29a 3160 . . . . . . . . 9 ((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) → 𝑗 = 𝑀)
5453ex 412 . . . . . . . 8 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) → (¬ 𝑗 = (Base‘𝑅) → 𝑗 = 𝑀))
5554orrd 863 . . . . . . 7 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) → (𝑗 = (Base‘𝑅) ∨ 𝑗 = 𝑀))
5655orcomd 871 . . . . . 6 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) → (𝑗 = 𝑀𝑗 = (Base‘𝑅)))
5756ex 412 . . . . 5 ((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) → (𝑀𝑗 → (𝑗 = 𝑀𝑗 = (Base‘𝑅))))
5857ralrimiva 3144 . . . 4 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) → ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = (Base‘𝑅))))
595, 15, 583jca 1127 . . 3 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) → (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ (Base‘𝑅) ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = (Base‘𝑅)))))
6011ismxidl 33470 . . . 4 (𝑅 ∈ Ring → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ (Base‘𝑅) ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = (Base‘𝑅))))))
6160biimpar 477 . . 3 ((𝑅 ∈ Ring ∧ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ (Base‘𝑅) ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = (Base‘𝑅))))) → 𝑀 ∈ (MaxIdeal‘𝑅))
622, 59, 61syl2anc 584 . 2 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) → 𝑀 ∈ (MaxIdeal‘𝑅))
6310a1i 11 . . . . 5 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗}))
6426adantl 481 . . . . 5 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑖 = 𝑀) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀𝑗})
6511mxidlidl 33471 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))
661, 65sylan 580 . . . . 5 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))
6729a1i 11 . . . . 5 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀𝑗} ∈ V)
6863, 64, 66, 67fvmptd 7023 . . . 4 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → (𝑉𝑀) = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀𝑗})
691ad2antrr 726 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗)) → 𝑅 ∈ Ring)
70 simplr 769 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗)) → 𝑀 ∈ (MaxIdeal‘𝑅))
71 simprl 771 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗)) → 𝑗 ∈ (PrmIdeal‘𝑅))
72 prmidlidl 33452 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) → 𝑗 ∈ (LIdeal‘𝑅))
7369, 71, 72syl2anc 584 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗)) → 𝑗 ∈ (LIdeal‘𝑅))
74 simprr 773 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗)) → 𝑀𝑗)
7573, 74jca 511 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗)) → (𝑗 ∈ (LIdeal‘𝑅) ∧ 𝑀𝑗))
7611mxidlmax 33473 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (LIdeal‘𝑅) ∧ 𝑀𝑗)) → (𝑗 = 𝑀𝑗 = (Base‘𝑅)))
7769, 70, 75, 76syl21anc 838 . . . . . . . 8 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗)) → (𝑗 = 𝑀𝑗 = (Base‘𝑅)))
78 eqid 2735 . . . . . . . . . . 11 (.r𝑅) = (.r𝑅)
7911, 78prmidlnr 33447 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) → 𝑗 ≠ (Base‘𝑅))
8069, 71, 79syl2anc 584 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗)) → 𝑗 ≠ (Base‘𝑅))
8180neneqd 2943 . . . . . . . 8 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗)) → ¬ 𝑗 = (Base‘𝑅))
8277, 81olcnd 877 . . . . . . 7 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗)) → 𝑗 = 𝑀)
83 simpr 484 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 = 𝑀) → 𝑗 = 𝑀)
8419mxidlprm 33478 . . . . . . . . . 10 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (PrmIdeal‘𝑅))
8584adantr 480 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 = 𝑀) → 𝑀 ∈ (PrmIdeal‘𝑅))
8683, 85eqeltrd 2839 . . . . . . . 8 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 = 𝑀) → 𝑗 ∈ (PrmIdeal‘𝑅))
87 ssidd 4019 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 = 𝑀) → 𝑗𝑗)
8883, 87eqsstrrd 4035 . . . . . . . 8 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 = 𝑀) → 𝑀𝑗)
8986, 88jca 511 . . . . . . 7 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 = 𝑀) → (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗))
9082, 89impbida 801 . . . . . 6 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → ((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗) ↔ 𝑗 = 𝑀))
9190alrimiv 1925 . . . . 5 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → ∀𝑗((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗) ↔ 𝑗 = 𝑀))
9291, 33sylibr 234 . . . 4 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀𝑗} = {𝑀})
9368, 92eqtr2d 2776 . . 3 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → {𝑀} = (𝑉𝑀))
9493adantlr 715 . 2 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → {𝑀} = (𝑉𝑀))
9562, 94impbida 801 1 ((𝑅 ∈ CRing ∧ 𝑀𝐵) → ({𝑀} = (𝑉𝑀) ↔ 𝑀 ∈ (MaxIdeal‘𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847  w3a 1086  wal 1535   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  wss 3963  c0 4339  {csn 4631  cmpt 5231  cfv 6563  Basecbs 17245  .rcmulr 17299  LSSumclsm 19667  mulGrpcmgp 20152  Ringcrg 20251  CRingccrg 20252  LIdealclidl 21234  PrmIdealcprmidl 33443  MaxIdealcmxidl 33467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-ac2 10501  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-rpss 7742  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-dju 9939  df-card 9977  df-ac 10154  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-submnd 18810  df-grp 18967  df-minusg 18968  df-sbg 18969  df-subg 19154  df-cntz 19348  df-lsm 19669  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-cring 20254  df-subrg 20587  df-lmod 20877  df-lss 20948  df-lsp 20988  df-sra 21190  df-rgmod 21191  df-lidl 21236  df-rsp 21237  df-lpidl 21350  df-prmidl 33444  df-mxidl 33468
This theorem is referenced by:  zarmxt1  33841
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