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Theorem zarclssn 32494
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 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗})
zarclssn.1 𝐡 = (LIdealβ€˜π‘…)
Assertion
Ref Expression
zarclssn ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ ({𝑀} = (π‘‰β€˜π‘€) ↔ 𝑀 ∈ (MaxIdealβ€˜π‘…)))
Distinct variable groups:   𝑅,𝑖,𝑗   𝑖,𝑉   𝐡,𝑖,𝑗   𝑖,𝑀,𝑗   𝑗,𝑉

Proof of Theorem zarclssn
Dummy variable π‘š is distinct from all other variables.
StepHypRef Expression
1 crngring 19983 . . . 4 (𝑅 ∈ CRing β†’ 𝑅 ∈ Ring)
21ad2antrr 725 . . 3 (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) β†’ 𝑅 ∈ Ring)
3 simplr 768 . . . . 5 (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) β†’ 𝑀 ∈ 𝐡)
4 zarclssn.1 . . . . 5 𝐡 = (LIdealβ€˜π‘…)
53, 4eleqtrdi 2848 . . . 4 (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) β†’ 𝑀 ∈ (LIdealβ€˜π‘…))
6 simpr 486 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) β†’ {𝑀} = (π‘‰β€˜π‘€))
73snn0d 4741 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) β†’ {𝑀} β‰  βˆ…)
86, 7eqnetrrd 3013 . . . . 5 (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) β†’ (π‘‰β€˜π‘€) β‰  βˆ…)
9 simpll 766 . . . . . 6 (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) β†’ 𝑅 ∈ CRing)
10 zarclsx.1 . . . . . . . 8 𝑉 = (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗})
11 eqid 2737 . . . . . . . 8 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
1210, 11zarcls1 32490 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (LIdealβ€˜π‘…)) β†’ ((π‘‰β€˜π‘€) = βˆ… ↔ 𝑀 = (Baseβ€˜π‘…)))
1312necon3bid 2989 . . . . . 6 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (LIdealβ€˜π‘…)) β†’ ((π‘‰β€˜π‘€) β‰  βˆ… ↔ 𝑀 β‰  (Baseβ€˜π‘…)))
149, 5, 13syl2anc 585 . . . . 5 (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) β†’ ((π‘‰β€˜π‘€) β‰  βˆ… ↔ 𝑀 β‰  (Baseβ€˜π‘…)))
158, 14mpbid 231 . . . 4 (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) β†’ 𝑀 β‰  (Baseβ€˜π‘…))
16 simpr 486 . . . . . . . . . . . 12 ((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) ∧ 𝑗 ∈ (LIdealβ€˜π‘…)) ∧ 𝑀 βŠ† 𝑗) ∧ Β¬ 𝑗 = (Baseβ€˜π‘…)) ∧ π‘š ∈ (MaxIdealβ€˜π‘…)) ∧ 𝑗 βŠ† π‘š) β†’ 𝑗 βŠ† π‘š)
179ad5antr 733 . . . . . . . . . . . . . 14 ((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) ∧ 𝑗 ∈ (LIdealβ€˜π‘…)) ∧ 𝑀 βŠ† 𝑗) ∧ Β¬ 𝑗 = (Baseβ€˜π‘…)) ∧ π‘š ∈ (MaxIdealβ€˜π‘…)) ∧ 𝑗 βŠ† π‘š) β†’ 𝑅 ∈ CRing)
18 simplr 768 . . . . . . . . . . . . . 14 ((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) ∧ 𝑗 ∈ (LIdealβ€˜π‘…)) ∧ 𝑀 βŠ† 𝑗) ∧ Β¬ 𝑗 = (Baseβ€˜π‘…)) ∧ π‘š ∈ (MaxIdealβ€˜π‘…)) ∧ 𝑗 βŠ† π‘š) β†’ π‘š ∈ (MaxIdealβ€˜π‘…))
19 eqid 2737 . . . . . . . . . . . . . . 15 (LSSumβ€˜(mulGrpβ€˜π‘…)) = (LSSumβ€˜(mulGrpβ€˜π‘…))
2019mxidlprm 32277 . . . . . . . . . . . . . 14 ((𝑅 ∈ CRing ∧ π‘š ∈ (MaxIdealβ€˜π‘…)) β†’ π‘š ∈ (PrmIdealβ€˜π‘…))
2117, 18, 20syl2anc 585 . . . . . . . . . . . . 13 ((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) ∧ 𝑗 ∈ (LIdealβ€˜π‘…)) ∧ 𝑀 βŠ† 𝑗) ∧ Β¬ 𝑗 = (Baseβ€˜π‘…)) ∧ π‘š ∈ (MaxIdealβ€˜π‘…)) ∧ 𝑗 βŠ† π‘š) β†’ π‘š ∈ (PrmIdealβ€˜π‘…))
22 simp-4r 783 . . . . . . . . . . . . . 14 ((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) ∧ 𝑗 ∈ (LIdealβ€˜π‘…)) ∧ 𝑀 βŠ† 𝑗) ∧ Β¬ 𝑗 = (Baseβ€˜π‘…)) ∧ π‘š ∈ (MaxIdealβ€˜π‘…)) ∧ 𝑗 βŠ† π‘š) β†’ 𝑀 βŠ† 𝑗)
2322, 16sstrd 3959 . . . . . . . . . . . . 13 ((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) ∧ 𝑗 ∈ (LIdealβ€˜π‘…)) ∧ 𝑀 βŠ† 𝑗) ∧ Β¬ 𝑗 = (Baseβ€˜π‘…)) ∧ π‘š ∈ (MaxIdealβ€˜π‘…)) ∧ 𝑗 βŠ† π‘š) β†’ 𝑀 βŠ† π‘š)
2410a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) β†’ 𝑉 = (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗}))
25 sseq1 3974 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 𝑀 β†’ (𝑖 βŠ† 𝑗 ↔ 𝑀 βŠ† 𝑗))
2625rabbidv 3418 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 𝑀 β†’ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗} = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑀 βŠ† 𝑗})
2726adantl 483 . . . . . . . . . . . . . . . . . 18 ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) ∧ 𝑖 = 𝑀) β†’ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗} = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑀 βŠ† 𝑗})
28 fvex 6860 . . . . . . . . . . . . . . . . . . . 20 (PrmIdealβ€˜π‘…) ∈ V
2928rabex 5294 . . . . . . . . . . . . . . . . . . 19 {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑀 βŠ† 𝑗} ∈ V
3029a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) β†’ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑀 βŠ† 𝑗} ∈ V)
3124, 27, 5, 30fvmptd 6960 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) β†’ (π‘‰β€˜π‘€) = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑀 βŠ† 𝑗})
326, 31eqtr2d 2778 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) β†’ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑀 βŠ† 𝑗} = {𝑀})
33 rabeqsn 4632 . . . . . . . . . . . . . . . 16 ({𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑀 βŠ† 𝑗} = {𝑀} ↔ βˆ€π‘—((𝑗 ∈ (PrmIdealβ€˜π‘…) ∧ 𝑀 βŠ† 𝑗) ↔ 𝑗 = 𝑀))
3432, 33sylib 217 . . . . . . . . . . . . . . 15 (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) β†’ βˆ€π‘—((𝑗 ∈ (PrmIdealβ€˜π‘…) ∧ 𝑀 βŠ† 𝑗) ↔ 𝑗 = 𝑀))
3534ad5antr 733 . . . . . . . . . . . . . 14 ((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) ∧ 𝑗 ∈ (LIdealβ€˜π‘…)) ∧ 𝑀 βŠ† 𝑗) ∧ Β¬ 𝑗 = (Baseβ€˜π‘…)) ∧ π‘š ∈ (MaxIdealβ€˜π‘…)) ∧ 𝑗 βŠ† π‘š) β†’ βˆ€π‘—((𝑗 ∈ (PrmIdealβ€˜π‘…) ∧ 𝑀 βŠ† 𝑗) ↔ 𝑗 = 𝑀))
36 vex 3452 . . . . . . . . . . . . . . 15 π‘š ∈ V
37 eleq1w 2821 . . . . . . . . . . . . . . . . 17 (𝑗 = π‘š β†’ (𝑗 ∈ (PrmIdealβ€˜π‘…) ↔ π‘š ∈ (PrmIdealβ€˜π‘…)))
38 sseq2 3975 . . . . . . . . . . . . . . . . 17 (𝑗 = π‘š β†’ (𝑀 βŠ† 𝑗 ↔ 𝑀 βŠ† π‘š))
3937, 38anbi12d 632 . . . . . . . . . . . . . . . 16 (𝑗 = π‘š β†’ ((𝑗 ∈ (PrmIdealβ€˜π‘…) ∧ 𝑀 βŠ† 𝑗) ↔ (π‘š ∈ (PrmIdealβ€˜π‘…) ∧ 𝑀 βŠ† π‘š)))
40 eqeq1 2741 . . . . . . . . . . . . . . . 16 (𝑗 = π‘š β†’ (𝑗 = 𝑀 ↔ π‘š = 𝑀))
4139, 40bibi12d 346 . . . . . . . . . . . . . . 15 (𝑗 = π‘š β†’ (((𝑗 ∈ (PrmIdealβ€˜π‘…) ∧ 𝑀 βŠ† 𝑗) ↔ 𝑗 = 𝑀) ↔ ((π‘š ∈ (PrmIdealβ€˜π‘…) ∧ 𝑀 βŠ† π‘š) ↔ π‘š = 𝑀)))
4236, 41spcv 3567 . . . . . . . . . . . . . 14 (βˆ€π‘—((𝑗 ∈ (PrmIdealβ€˜π‘…) ∧ 𝑀 βŠ† 𝑗) ↔ 𝑗 = 𝑀) β†’ ((π‘š ∈ (PrmIdealβ€˜π‘…) ∧ 𝑀 βŠ† π‘š) ↔ π‘š = 𝑀))
4335, 42syl 17 . . . . . . . . . . . . 13 ((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) ∧ 𝑗 ∈ (LIdealβ€˜π‘…)) ∧ 𝑀 βŠ† 𝑗) ∧ Β¬ 𝑗 = (Baseβ€˜π‘…)) ∧ π‘š ∈ (MaxIdealβ€˜π‘…)) ∧ 𝑗 βŠ† π‘š) β†’ ((π‘š ∈ (PrmIdealβ€˜π‘…) ∧ 𝑀 βŠ† π‘š) ↔ π‘š = 𝑀))
4421, 23, 43mpbi2and 711 . . . . . . . . . . . 12 ((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) ∧ 𝑗 ∈ (LIdealβ€˜π‘…)) ∧ 𝑀 βŠ† 𝑗) ∧ Β¬ 𝑗 = (Baseβ€˜π‘…)) ∧ π‘š ∈ (MaxIdealβ€˜π‘…)) ∧ 𝑗 βŠ† π‘š) β†’ π‘š = 𝑀)
4516, 44sseqtrd 3989 . . . . . . . . . . 11 ((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) ∧ 𝑗 ∈ (LIdealβ€˜π‘…)) ∧ 𝑀 βŠ† 𝑗) ∧ Β¬ 𝑗 = (Baseβ€˜π‘…)) ∧ π‘š ∈ (MaxIdealβ€˜π‘…)) ∧ 𝑗 βŠ† π‘š) β†’ 𝑗 βŠ† 𝑀)
4645, 22eqssd 3966 . . . . . . . . . 10 ((((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) ∧ 𝑗 ∈ (LIdealβ€˜π‘…)) ∧ 𝑀 βŠ† 𝑗) ∧ Β¬ 𝑗 = (Baseβ€˜π‘…)) ∧ π‘š ∈ (MaxIdealβ€˜π‘…)) ∧ 𝑗 βŠ† π‘š) β†’ 𝑗 = 𝑀)
471ad5antr 733 . . . . . . . . . . 11 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) ∧ 𝑗 ∈ (LIdealβ€˜π‘…)) ∧ 𝑀 βŠ† 𝑗) ∧ Β¬ 𝑗 = (Baseβ€˜π‘…)) β†’ 𝑅 ∈ Ring)
48 simpllr 775 . . . . . . . . . . 11 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) ∧ 𝑗 ∈ (LIdealβ€˜π‘…)) ∧ 𝑀 βŠ† 𝑗) ∧ Β¬ 𝑗 = (Baseβ€˜π‘…)) β†’ 𝑗 ∈ (LIdealβ€˜π‘…))
49 simpr 486 . . . . . . . . . . . 12 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) ∧ 𝑗 ∈ (LIdealβ€˜π‘…)) ∧ 𝑀 βŠ† 𝑗) ∧ Β¬ 𝑗 = (Baseβ€˜π‘…)) β†’ Β¬ 𝑗 = (Baseβ€˜π‘…))
5049neqned 2951 . . . . . . . . . . 11 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) ∧ 𝑗 ∈ (LIdealβ€˜π‘…)) ∧ 𝑀 βŠ† 𝑗) ∧ Β¬ 𝑗 = (Baseβ€˜π‘…)) β†’ 𝑗 β‰  (Baseβ€˜π‘…))
5111ssmxidl 32279 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝑗 ∈ (LIdealβ€˜π‘…) ∧ 𝑗 β‰  (Baseβ€˜π‘…)) β†’ βˆƒπ‘š ∈ (MaxIdealβ€˜π‘…)𝑗 βŠ† π‘š)
5247, 48, 50, 51syl3anc 1372 . . . . . . . . . 10 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) ∧ 𝑗 ∈ (LIdealβ€˜π‘…)) ∧ 𝑀 βŠ† 𝑗) ∧ Β¬ 𝑗 = (Baseβ€˜π‘…)) β†’ βˆƒπ‘š ∈ (MaxIdealβ€˜π‘…)𝑗 βŠ† π‘š)
5346, 52r19.29a 3160 . . . . . . . . 9 ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) ∧ 𝑗 ∈ (LIdealβ€˜π‘…)) ∧ 𝑀 βŠ† 𝑗) ∧ Β¬ 𝑗 = (Baseβ€˜π‘…)) β†’ 𝑗 = 𝑀)
5453ex 414 . . . . . . . 8 (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) ∧ 𝑗 ∈ (LIdealβ€˜π‘…)) ∧ 𝑀 βŠ† 𝑗) β†’ (Β¬ 𝑗 = (Baseβ€˜π‘…) β†’ 𝑗 = 𝑀))
5554orrd 862 . . . . . . 7 (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) ∧ 𝑗 ∈ (LIdealβ€˜π‘…)) ∧ 𝑀 βŠ† 𝑗) β†’ (𝑗 = (Baseβ€˜π‘…) ∨ 𝑗 = 𝑀))
5655orcomd 870 . . . . . 6 (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) ∧ 𝑗 ∈ (LIdealβ€˜π‘…)) ∧ 𝑀 βŠ† 𝑗) β†’ (𝑗 = 𝑀 ∨ 𝑗 = (Baseβ€˜π‘…)))
5756ex 414 . . . . 5 ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) ∧ 𝑗 ∈ (LIdealβ€˜π‘…)) β†’ (𝑀 βŠ† 𝑗 β†’ (𝑗 = 𝑀 ∨ 𝑗 = (Baseβ€˜π‘…))))
5857ralrimiva 3144 . . . 4 (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) β†’ βˆ€π‘— ∈ (LIdealβ€˜π‘…)(𝑀 βŠ† 𝑗 β†’ (𝑗 = 𝑀 ∨ 𝑗 = (Baseβ€˜π‘…))))
595, 15, 583jca 1129 . . 3 (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) β†’ (𝑀 ∈ (LIdealβ€˜π‘…) ∧ 𝑀 β‰  (Baseβ€˜π‘…) ∧ βˆ€π‘— ∈ (LIdealβ€˜π‘…)(𝑀 βŠ† 𝑗 β†’ (𝑗 = 𝑀 ∨ 𝑗 = (Baseβ€˜π‘…)))))
6011ismxidl 32271 . . . 4 (𝑅 ∈ Ring β†’ (𝑀 ∈ (MaxIdealβ€˜π‘…) ↔ (𝑀 ∈ (LIdealβ€˜π‘…) ∧ 𝑀 β‰  (Baseβ€˜π‘…) ∧ βˆ€π‘— ∈ (LIdealβ€˜π‘…)(𝑀 βŠ† 𝑗 β†’ (𝑗 = 𝑀 ∨ 𝑗 = (Baseβ€˜π‘…))))))
6160biimpar 479 . . 3 ((𝑅 ∈ Ring ∧ (𝑀 ∈ (LIdealβ€˜π‘…) ∧ 𝑀 β‰  (Baseβ€˜π‘…) ∧ βˆ€π‘— ∈ (LIdealβ€˜π‘…)(𝑀 βŠ† 𝑗 β†’ (𝑗 = 𝑀 ∨ 𝑗 = (Baseβ€˜π‘…))))) β†’ 𝑀 ∈ (MaxIdealβ€˜π‘…))
622, 59, 61syl2anc 585 . 2 (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ {𝑀} = (π‘‰β€˜π‘€)) β†’ 𝑀 ∈ (MaxIdealβ€˜π‘…))
6310a1i 11 . . . . 5 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) β†’ 𝑉 = (𝑖 ∈ (LIdealβ€˜π‘…) ↦ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗}))
6426adantl 483 . . . . 5 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) ∧ 𝑖 = 𝑀) β†’ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑖 βŠ† 𝑗} = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑀 βŠ† 𝑗})
6511mxidlidl 32272 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) β†’ 𝑀 ∈ (LIdealβ€˜π‘…))
661, 65sylan 581 . . . . 5 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) β†’ 𝑀 ∈ (LIdealβ€˜π‘…))
6729a1i 11 . . . . 5 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) β†’ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑀 βŠ† 𝑗} ∈ V)
6863, 64, 66, 67fvmptd 6960 . . . 4 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) β†’ (π‘‰β€˜π‘€) = {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑀 βŠ† 𝑗})
691ad2antrr 725 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) ∧ (𝑗 ∈ (PrmIdealβ€˜π‘…) ∧ 𝑀 βŠ† 𝑗)) β†’ 𝑅 ∈ Ring)
70 simplr 768 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) ∧ (𝑗 ∈ (PrmIdealβ€˜π‘…) ∧ 𝑀 βŠ† 𝑗)) β†’ 𝑀 ∈ (MaxIdealβ€˜π‘…))
71 simprl 770 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) ∧ (𝑗 ∈ (PrmIdealβ€˜π‘…) ∧ 𝑀 βŠ† 𝑗)) β†’ 𝑗 ∈ (PrmIdealβ€˜π‘…))
72 prmidlidl 32256 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝑗 ∈ (PrmIdealβ€˜π‘…)) β†’ 𝑗 ∈ (LIdealβ€˜π‘…))
7369, 71, 72syl2anc 585 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) ∧ (𝑗 ∈ (PrmIdealβ€˜π‘…) ∧ 𝑀 βŠ† 𝑗)) β†’ 𝑗 ∈ (LIdealβ€˜π‘…))
74 simprr 772 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) ∧ (𝑗 ∈ (PrmIdealβ€˜π‘…) ∧ 𝑀 βŠ† 𝑗)) β†’ 𝑀 βŠ† 𝑗)
7573, 74jca 513 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) ∧ (𝑗 ∈ (PrmIdealβ€˜π‘…) ∧ 𝑀 βŠ† 𝑗)) β†’ (𝑗 ∈ (LIdealβ€˜π‘…) ∧ 𝑀 βŠ† 𝑗))
7611mxidlmax 32274 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) ∧ (𝑗 ∈ (LIdealβ€˜π‘…) ∧ 𝑀 βŠ† 𝑗)) β†’ (𝑗 = 𝑀 ∨ 𝑗 = (Baseβ€˜π‘…)))
7769, 70, 75, 76syl21anc 837 . . . . . . . 8 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) ∧ (𝑗 ∈ (PrmIdealβ€˜π‘…) ∧ 𝑀 βŠ† 𝑗)) β†’ (𝑗 = 𝑀 ∨ 𝑗 = (Baseβ€˜π‘…)))
78 eqid 2737 . . . . . . . . . . 11 (.rβ€˜π‘…) = (.rβ€˜π‘…)
7911, 78prmidlnr 32251 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑗 ∈ (PrmIdealβ€˜π‘…)) β†’ 𝑗 β‰  (Baseβ€˜π‘…))
8069, 71, 79syl2anc 585 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) ∧ (𝑗 ∈ (PrmIdealβ€˜π‘…) ∧ 𝑀 βŠ† 𝑗)) β†’ 𝑗 β‰  (Baseβ€˜π‘…))
8180neneqd 2949 . . . . . . . 8 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) ∧ (𝑗 ∈ (PrmIdealβ€˜π‘…) ∧ 𝑀 βŠ† 𝑗)) β†’ Β¬ 𝑗 = (Baseβ€˜π‘…))
8277, 81olcnd 876 . . . . . . 7 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) ∧ (𝑗 ∈ (PrmIdealβ€˜π‘…) ∧ 𝑀 βŠ† 𝑗)) β†’ 𝑗 = 𝑀)
83 simpr 486 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) ∧ 𝑗 = 𝑀) β†’ 𝑗 = 𝑀)
8419mxidlprm 32277 . . . . . . . . . 10 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) β†’ 𝑀 ∈ (PrmIdealβ€˜π‘…))
8584adantr 482 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) ∧ 𝑗 = 𝑀) β†’ 𝑀 ∈ (PrmIdealβ€˜π‘…))
8683, 85eqeltrd 2838 . . . . . . . 8 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) ∧ 𝑗 = 𝑀) β†’ 𝑗 ∈ (PrmIdealβ€˜π‘…))
87 ssidd 3972 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) ∧ 𝑗 = 𝑀) β†’ 𝑗 βŠ† 𝑗)
8883, 87eqsstrrd 3988 . . . . . . . 8 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) ∧ 𝑗 = 𝑀) β†’ 𝑀 βŠ† 𝑗)
8986, 88jca 513 . . . . . . 7 (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) ∧ 𝑗 = 𝑀) β†’ (𝑗 ∈ (PrmIdealβ€˜π‘…) ∧ 𝑀 βŠ† 𝑗))
9082, 89impbida 800 . . . . . 6 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) β†’ ((𝑗 ∈ (PrmIdealβ€˜π‘…) ∧ 𝑀 βŠ† 𝑗) ↔ 𝑗 = 𝑀))
9190alrimiv 1931 . . . . 5 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) β†’ βˆ€π‘—((𝑗 ∈ (PrmIdealβ€˜π‘…) ∧ 𝑀 βŠ† 𝑗) ↔ 𝑗 = 𝑀))
9291, 33sylibr 233 . . . 4 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) β†’ {𝑗 ∈ (PrmIdealβ€˜π‘…) ∣ 𝑀 βŠ† 𝑗} = {𝑀})
9368, 92eqtr2d 2778 . . 3 ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) β†’ {𝑀} = (π‘‰β€˜π‘€))
9493adantlr 714 . 2 (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) β†’ {𝑀} = (π‘‰β€˜π‘€))
9562, 94impbida 800 1 ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐡) β†’ ({𝑀} = (π‘‰β€˜π‘€) ↔ 𝑀 ∈ (MaxIdealβ€˜π‘…)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∧ w3a 1088  βˆ€wal 1540   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆ€wral 3065  βˆƒwrex 3074  {crab 3410  Vcvv 3448   βŠ† wss 3915  βˆ…c0 4287  {csn 4591   ↦ cmpt 5193  β€˜cfv 6501  Basecbs 17090  .rcmulr 17141  LSSumclsm 19423  mulGrpcmgp 19903  Ringcrg 19971  CRingccrg 19972  LIdealclidl 20647  PrmIdealcprmidl 32247  MaxIdealcmxidl 32268
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-ac2 10406  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-rpss 7665  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-oadd 8421  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-dju 9844  df-card 9882  df-ac 10059  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-2 12223  df-3 12224  df-4 12225  df-5 12226  df-6 12227  df-7 12228  df-8 12229  df-sets 17043  df-slot 17061  df-ndx 17073  df-base 17091  df-ress 17120  df-plusg 17153  df-mulr 17154  df-sca 17156  df-vsca 17157  df-ip 17158  df-0g 17330  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-submnd 18609  df-grp 18758  df-minusg 18759  df-sbg 18760  df-subg 18932  df-cntz 19104  df-lsm 19425  df-cmn 19571  df-abl 19572  df-mgp 19904  df-ur 19921  df-ring 19973  df-cring 19974  df-subrg 20236  df-lmod 20340  df-lss 20409  df-lsp 20449  df-sra 20649  df-rgmod 20650  df-lidl 20651  df-rsp 20652  df-lpidl 20729  df-prmidl 32248  df-mxidl 32269
This theorem is referenced by:  zarmxt1  32501
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