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Theorem zarclssn 31226
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 19305 . . . 4 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
21ad2antrr 725 . . 3 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) → 𝑅 ∈ Ring)
3 simplr 768 . . . . 5 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) → 𝑀𝐵)
4 zarclssn.1 . . . . 5 𝐵 = (LIdeal‘𝑅)
53, 4eleqtrdi 2903 . . . 4 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) → 𝑀 ∈ (LIdeal‘𝑅))
6 simpr 488 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) → {𝑀} = (𝑉𝑀))
7 snnzg 4673 . . . . . . 7 (𝑀𝐵 → {𝑀} ≠ ∅)
83, 7syl 17 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) → {𝑀} ≠ ∅)
96, 8eqnetrrd 3058 . . . . 5 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) → (𝑉𝑀) ≠ ∅)
10 simpll 766 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) → 𝑅 ∈ CRing)
11 zarclsx.1 . . . . . . . 8 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗})
12 eqid 2801 . . . . . . . 8 (Base‘𝑅) = (Base‘𝑅)
1311, 12zarcls1 31222 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (LIdeal‘𝑅)) → ((𝑉𝑀) = ∅ ↔ 𝑀 = (Base‘𝑅)))
1413necon3bid 3034 . . . . . 6 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (LIdeal‘𝑅)) → ((𝑉𝑀) ≠ ∅ ↔ 𝑀 ≠ (Base‘𝑅)))
1510, 5, 14syl2anc 587 . . . . 5 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) → ((𝑉𝑀) ≠ ∅ ↔ 𝑀 ≠ (Base‘𝑅)))
169, 15mpbid 235 . . . 4 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) → 𝑀 ≠ (Base‘𝑅))
17 simpr 488 . . . . . . . . . . . 12 ((((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗𝑚) → 𝑗𝑚)
1810ad5antr 733 . . . . . . . . . . . . . 14 ((((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗𝑚) → 𝑅 ∈ CRing)
19 simplr 768 . . . . . . . . . . . . . 14 ((((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗𝑚) → 𝑚 ∈ (MaxIdeal‘𝑅))
20 eqid 2801 . . . . . . . . . . . . . . 15 (LSSum‘(mulGrp‘𝑅)) = (LSSum‘(mulGrp‘𝑅))
2120mxidlprm 31048 . . . . . . . . . . . . . 14 ((𝑅 ∈ CRing ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ∈ (PrmIdeal‘𝑅))
2218, 19, 21syl2anc 587 . . . . . . . . . . . . 13 ((((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗𝑚) → 𝑚 ∈ (PrmIdeal‘𝑅))
23 simp-4r 783 . . . . . . . . . . . . . 14 ((((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗𝑚) → 𝑀𝑗)
2423, 17sstrd 3928 . . . . . . . . . . . . 13 ((((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗𝑚) → 𝑀𝑚)
2511a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) → 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗}))
26 sseq1 3943 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 𝑀 → (𝑖𝑗𝑀𝑗))
2726rabbidv 3430 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 𝑀 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀𝑗})
2827adantl 485 . . . . . . . . . . . . . . . . . 18 ((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑖 = 𝑀) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀𝑗})
29 fvex 6662 . . . . . . . . . . . . . . . . . . . 20 (PrmIdeal‘𝑅) ∈ V
3029rabex 5202 . . . . . . . . . . . . . . . . . . 19 {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀𝑗} ∈ V
3130a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀𝑗} ∈ V)
3225, 28, 5, 31fvmptd 6756 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) → (𝑉𝑀) = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀𝑗})
336, 32eqtr2d 2837 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀𝑗} = {𝑀})
34 rabeqsn 4569 . . . . . . . . . . . . . . . 16 ({𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀𝑗} = {𝑀} ↔ ∀𝑗((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗) ↔ 𝑗 = 𝑀))
3533, 34sylib 221 . . . . . . . . . . . . . . 15 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) → ∀𝑗((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗) ↔ 𝑗 = 𝑀))
3635ad5antr 733 . . . . . . . . . . . . . 14 ((((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗𝑚) → ∀𝑗((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗) ↔ 𝑗 = 𝑀))
37 vex 3447 . . . . . . . . . . . . . . 15 𝑚 ∈ V
38 eleq1w 2875 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑚 → (𝑗 ∈ (PrmIdeal‘𝑅) ↔ 𝑚 ∈ (PrmIdeal‘𝑅)))
39 sseq2 3944 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑚 → (𝑀𝑗𝑀𝑚))
4038, 39anbi12d 633 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑚 → ((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗) ↔ (𝑚 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑚)))
41 eqeq1 2805 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑚 → (𝑗 = 𝑀𝑚 = 𝑀))
4240, 41bibi12d 349 . . . . . . . . . . . . . . 15 (𝑗 = 𝑚 → (((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗) ↔ 𝑗 = 𝑀) ↔ ((𝑚 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑚) ↔ 𝑚 = 𝑀)))
4337, 42spcv 3557 . . . . . . . . . . . . . 14 (∀𝑗((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗) ↔ 𝑗 = 𝑀) → ((𝑚 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑚) ↔ 𝑚 = 𝑀))
4436, 43syl 17 . . . . . . . . . . . . 13 ((((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗𝑚) → ((𝑚 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑚) ↔ 𝑚 = 𝑀))
4522, 24, 44mpbi2and 711 . . . . . . . . . . . 12 ((((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗𝑚) → 𝑚 = 𝑀)
4617, 45sseqtrd 3958 . . . . . . . . . . 11 ((((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗𝑚) → 𝑗𝑀)
4746, 23eqssd 3935 . . . . . . . . . 10 ((((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗𝑚) → 𝑗 = 𝑀)
481ad5antr 733 . . . . . . . . . . 11 ((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) → 𝑅 ∈ Ring)
49 simpllr 775 . . . . . . . . . . 11 ((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) → 𝑗 ∈ (LIdeal‘𝑅))
50 simpr 488 . . . . . . . . . . . 12 ((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) → ¬ 𝑗 = (Base‘𝑅))
5150neqned 2997 . . . . . . . . . . 11 ((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) → 𝑗 ≠ (Base‘𝑅))
5212ssmxidl 31050 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝑗 ∈ (LIdeal‘𝑅) ∧ 𝑗 ≠ (Base‘𝑅)) → ∃𝑚 ∈ (MaxIdeal‘𝑅)𝑗𝑚)
5348, 49, 51, 52syl3anc 1368 . . . . . . . . . 10 ((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) → ∃𝑚 ∈ (MaxIdeal‘𝑅)𝑗𝑚)
5447, 53r19.29a 3251 . . . . . . . . 9 ((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) → 𝑗 = 𝑀)
5554ex 416 . . . . . . . 8 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) → (¬ 𝑗 = (Base‘𝑅) → 𝑗 = 𝑀))
5655orrd 860 . . . . . . 7 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) → (𝑗 = (Base‘𝑅) ∨ 𝑗 = 𝑀))
5756orcomd 868 . . . . . 6 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀𝑗) → (𝑗 = 𝑀𝑗 = (Base‘𝑅)))
5857ex 416 . . . . 5 ((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) → (𝑀𝑗 → (𝑗 = 𝑀𝑗 = (Base‘𝑅))))
5958ralrimiva 3152 . . . 4 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) → ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = (Base‘𝑅))))
605, 16, 593jca 1125 . . 3 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) → (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ (Base‘𝑅) ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = (Base‘𝑅)))))
6112ismxidl 31042 . . . 4 (𝑅 ∈ Ring → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ (Base‘𝑅) ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = (Base‘𝑅))))))
6261biimpar 481 . . 3 ((𝑅 ∈ Ring ∧ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ (Base‘𝑅) ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = (Base‘𝑅))))) → 𝑀 ∈ (MaxIdeal‘𝑅))
632, 60, 62syl2anc 587 . 2 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ {𝑀} = (𝑉𝑀)) → 𝑀 ∈ (MaxIdeal‘𝑅))
6411a1i 11 . . . . 5 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗}))
6527adantl 485 . . . . 5 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑖 = 𝑀) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀𝑗})
6612mxidlidl 31043 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))
671, 66sylan 583 . . . . 5 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))
6830a1i 11 . . . . 5 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀𝑗} ∈ V)
6964, 65, 67, 68fvmptd 6756 . . . 4 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → (𝑉𝑀) = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀𝑗})
701ad2antrr 725 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗)) → 𝑅 ∈ Ring)
71 simplr 768 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗)) → 𝑀 ∈ (MaxIdeal‘𝑅))
72 simprl 770 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗)) → 𝑗 ∈ (PrmIdeal‘𝑅))
73 prmidlidl 31027 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) → 𝑗 ∈ (LIdeal‘𝑅))
7470, 72, 73syl2anc 587 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗)) → 𝑗 ∈ (LIdeal‘𝑅))
75 simprr 772 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗)) → 𝑀𝑗)
7674, 75jca 515 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗)) → (𝑗 ∈ (LIdeal‘𝑅) ∧ 𝑀𝑗))
7712mxidlmax 31045 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (LIdeal‘𝑅) ∧ 𝑀𝑗)) → (𝑗 = 𝑀𝑗 = (Base‘𝑅)))
7870, 71, 76, 77syl21anc 836 . . . . . . . 8 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗)) → (𝑗 = 𝑀𝑗 = (Base‘𝑅)))
79 eqid 2801 . . . . . . . . . . 11 (.r𝑅) = (.r𝑅)
8012, 79prmidlnr 31022 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) → 𝑗 ≠ (Base‘𝑅))
8170, 72, 80syl2anc 587 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗)) → 𝑗 ≠ (Base‘𝑅))
8281neneqd 2995 . . . . . . . 8 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗)) → ¬ 𝑗 = (Base‘𝑅))
8378, 82olcnd 874 . . . . . . 7 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗)) → 𝑗 = 𝑀)
84 simpr 488 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 = 𝑀) → 𝑗 = 𝑀)
8520mxidlprm 31048 . . . . . . . . . 10 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (PrmIdeal‘𝑅))
8685adantr 484 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 = 𝑀) → 𝑀 ∈ (PrmIdeal‘𝑅))
8784, 86eqeltrd 2893 . . . . . . . 8 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 = 𝑀) → 𝑗 ∈ (PrmIdeal‘𝑅))
88 ssidd 3941 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 = 𝑀) → 𝑗𝑗)
8984, 88eqsstrrd 3957 . . . . . . . 8 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 = 𝑀) → 𝑀𝑗)
9087, 89jca 515 . . . . . . 7 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 = 𝑀) → (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗))
9183, 90impbida 800 . . . . . 6 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → ((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗) ↔ 𝑗 = 𝑀))
9291alrimiv 1928 . . . . 5 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → ∀𝑗((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀𝑗) ↔ 𝑗 = 𝑀))
9392, 34sylibr 237 . . . 4 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀𝑗} = {𝑀})
9469, 93eqtr2d 2837 . . 3 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → {𝑀} = (𝑉𝑀))
9594adantlr 714 . 2 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → {𝑀} = (𝑉𝑀))
9663, 95impbida 800 1 ((𝑅 ∈ CRing ∧ 𝑀𝐵) → ({𝑀} = (𝑉𝑀) ↔ 𝑀 ∈ (MaxIdeal‘𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844  w3a 1084  wal 1536   = wceq 1538  wcel 2112  wne 2990  wral 3109  wrex 3110  {crab 3113  Vcvv 3444  wss 3884  c0 4246  {csn 4528  cmpt 5113  cfv 6328  Basecbs 16478  .rcmulr 16561  LSSumclsm 18754  mulGrpcmgp 19235  Ringcrg 19293  CRingccrg 19294  LIdealclidl 19938  PrmIdealcprmidl 31018  MaxIdealcmxidl 31039
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-ac2 9878  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
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 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-rpss 7433  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-dju 9318  df-card 9356  df-ac 9531  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-ndx 16481  df-slot 16482  df-base 16484  df-sets 16485  df-ress 16486  df-plusg 16573  df-mulr 16574  df-sca 16576  df-vsca 16577  df-ip 16578  df-0g 16710  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-submnd 17952  df-grp 18101  df-minusg 18102  df-sbg 18103  df-subg 18271  df-cntz 18442  df-lsm 18756  df-cmn 18903  df-abl 18904  df-mgp 19236  df-ur 19248  df-ring 19295  df-cring 19296  df-subrg 19529  df-lmod 19632  df-lss 19700  df-lsp 19740  df-sra 19940  df-rgmod 19941  df-lidl 19942  df-rsp 19943  df-lpidl 20012  df-prmidl 31019  df-mxidl 31040
This theorem is referenced by:  zarmxt1  31233
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