| Step | Hyp | Ref
| Expression |
| 1 | | crngring 20242 |
. . . 4
⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) |
| 2 | 1 | ad2antrr 726 |
. . 3
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → 𝑅 ∈ Ring) |
| 3 | | simplr 769 |
. . . . 5
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → 𝑀 ∈ 𝐵) |
| 4 | | zarclssn.1 |
. . . . 5
⊢ 𝐵 = (LIdeal‘𝑅) |
| 5 | 3, 4 | eleqtrdi 2851 |
. . . 4
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → 𝑀 ∈ (LIdeal‘𝑅)) |
| 6 | | simpr 484 |
. . . . . 6
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → {𝑀} = (𝑉‘𝑀)) |
| 7 | 3 | snn0d 4775 |
. . . . . 6
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → {𝑀} ≠ ∅) |
| 8 | 6, 7 | eqnetrrd 3009 |
. . . . 5
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → (𝑉‘𝑀) ≠ ∅) |
| 9 | | simpll 767 |
. . . . . 6
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → 𝑅 ∈ CRing) |
| 10 | | zarclsx.1 |
. . . . . . . 8
⊢ 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗}) |
| 11 | | eqid 2737 |
. . . . . . . 8
⊢
(Base‘𝑅) =
(Base‘𝑅) |
| 12 | 10, 11 | zarcls1 33868 |
. . . . . . 7
⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (LIdeal‘𝑅)) → ((𝑉‘𝑀) = ∅ ↔ 𝑀 = (Base‘𝑅))) |
| 13 | 12 | necon3bid 2985 |
. . . . . 6
⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (LIdeal‘𝑅)) → ((𝑉‘𝑀) ≠ ∅ ↔ 𝑀 ≠ (Base‘𝑅))) |
| 14 | 9, 5, 13 | syl2anc 584 |
. . . . 5
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → ((𝑉‘𝑀) ≠ ∅ ↔ 𝑀 ≠ (Base‘𝑅))) |
| 15 | 8, 14 | mpbid 232 |
. . . 4
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → 𝑀 ≠ (Base‘𝑅)) |
| 16 | | simpr 484 |
. . . . . . . . . . . 12
⊢
((((((((𝑅 ∈
CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 ⊆ 𝑚) → 𝑗 ⊆ 𝑚) |
| 17 | 9 | ad5antr 734 |
. . . . . . . . . . . . . 14
⊢
((((((((𝑅 ∈
CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 ⊆ 𝑚) → 𝑅 ∈ CRing) |
| 18 | | simplr 769 |
. . . . . . . . . . . . . 14
⊢
((((((((𝑅 ∈
CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 ⊆ 𝑚) → 𝑚 ∈ (MaxIdeal‘𝑅)) |
| 19 | | eqid 2737 |
. . . . . . . . . . . . . . 15
⊢
(LSSum‘(mulGrp‘𝑅)) = (LSSum‘(mulGrp‘𝑅)) |
| 20 | 19 | mxidlprm 33498 |
. . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ CRing ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) → 𝑚 ∈ (PrmIdeal‘𝑅)) |
| 21 | 17, 18, 20 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢
((((((((𝑅 ∈
CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 ⊆ 𝑚) → 𝑚 ∈ (PrmIdeal‘𝑅)) |
| 22 | | simp-4r 784 |
. . . . . . . . . . . . . 14
⊢
((((((((𝑅 ∈
CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 ⊆ 𝑚) → 𝑀 ⊆ 𝑗) |
| 23 | 22, 16 | sstrd 3994 |
. . . . . . . . . . . . 13
⊢
((((((((𝑅 ∈
CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 ⊆ 𝑚) → 𝑀 ⊆ 𝑚) |
| 24 | 10 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})) |
| 25 | | sseq1 4009 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 = 𝑀 → (𝑖 ⊆ 𝑗 ↔ 𝑀 ⊆ 𝑗)) |
| 26 | 25 | rabbidv 3444 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 = 𝑀 → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀 ⊆ 𝑗}) |
| 27 | 26 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑖 = 𝑀) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀 ⊆ 𝑗}) |
| 28 | | fvex 6919 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(PrmIdeal‘𝑅)
∈ V |
| 29 | 28 | rabex 5339 |
. . . . . . . . . . . . . . . . . . 19
⊢ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀 ⊆ 𝑗} ∈ V |
| 30 | 29 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀 ⊆ 𝑗} ∈ V) |
| 31 | 24, 27, 5, 30 | fvmptd 7023 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → (𝑉‘𝑀) = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀 ⊆ 𝑗}) |
| 32 | 6, 31 | eqtr2d 2778 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀 ⊆ 𝑗} = {𝑀}) |
| 33 | | rabeqsn 4667 |
. . . . . . . . . . . . . . . 16
⊢ ({𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀 ⊆ 𝑗} = {𝑀} ↔ ∀𝑗((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗) ↔ 𝑗 = 𝑀)) |
| 34 | 32, 33 | sylib 218 |
. . . . . . . . . . . . . . 15
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → ∀𝑗((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗) ↔ 𝑗 = 𝑀)) |
| 35 | 34 | ad5antr 734 |
. . . . . . . . . . . . . 14
⊢
((((((((𝑅 ∈
CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 ⊆ 𝑚) → ∀𝑗((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗) ↔ 𝑗 = 𝑀)) |
| 36 | | vex 3484 |
. . . . . . . . . . . . . . 15
⊢ 𝑚 ∈ V |
| 37 | | eleq1w 2824 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑚 → (𝑗 ∈ (PrmIdeal‘𝑅) ↔ 𝑚 ∈ (PrmIdeal‘𝑅))) |
| 38 | | sseq2 4010 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑚 → (𝑀 ⊆ 𝑗 ↔ 𝑀 ⊆ 𝑚)) |
| 39 | 37, 38 | anbi12d 632 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑚 → ((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗) ↔ (𝑚 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑚))) |
| 40 | | eqeq1 2741 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑚 → (𝑗 = 𝑀 ↔ 𝑚 = 𝑀)) |
| 41 | 39, 40 | bibi12d 345 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 𝑚 → (((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗) ↔ 𝑗 = 𝑀) ↔ ((𝑚 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑚) ↔ 𝑚 = 𝑀))) |
| 42 | 36, 41 | spcv 3605 |
. . . . . . . . . . . . . 14
⊢
(∀𝑗((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗) ↔ 𝑗 = 𝑀) → ((𝑚 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑚) ↔ 𝑚 = 𝑀)) |
| 43 | 35, 42 | syl 17 |
. . . . . . . . . . . . 13
⊢
((((((((𝑅 ∈
CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 ⊆ 𝑚) → ((𝑚 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑚) ↔ 𝑚 = 𝑀)) |
| 44 | 21, 23, 43 | mpbi2and 712 |
. . . . . . . . . . . 12
⊢
((((((((𝑅 ∈
CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 ⊆ 𝑚) → 𝑚 = 𝑀) |
| 45 | 16, 44 | sseqtrd 4020 |
. . . . . . . . . . 11
⊢
((((((((𝑅 ∈
CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 ⊆ 𝑚) → 𝑗 ⊆ 𝑀) |
| 46 | 45, 22 | eqssd 4001 |
. . . . . . . . . 10
⊢
((((((((𝑅 ∈
CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) ∧ 𝑚 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 ⊆ 𝑚) → 𝑗 = 𝑀) |
| 47 | 1 | ad5antr 734 |
. . . . . . . . . . 11
⊢
((((((𝑅 ∈ CRing
∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) → 𝑅 ∈ Ring) |
| 48 | | simpllr 776 |
. . . . . . . . . . 11
⊢
((((((𝑅 ∈ CRing
∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) → 𝑗 ∈ (LIdeal‘𝑅)) |
| 49 | | simpr 484 |
. . . . . . . . . . . 12
⊢
((((((𝑅 ∈ CRing
∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) → ¬ 𝑗 = (Base‘𝑅)) |
| 50 | 49 | neqned 2947 |
. . . . . . . . . . 11
⊢
((((((𝑅 ∈ CRing
∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) → 𝑗 ≠ (Base‘𝑅)) |
| 51 | 11 | ssmxidl 33502 |
. . . . . . . . . . 11
⊢ ((𝑅 ∈ Ring ∧ 𝑗 ∈ (LIdeal‘𝑅) ∧ 𝑗 ≠ (Base‘𝑅)) → ∃𝑚 ∈ (MaxIdeal‘𝑅)𝑗 ⊆ 𝑚) |
| 52 | 47, 48, 50, 51 | syl3anc 1373 |
. . . . . . . . . 10
⊢
((((((𝑅 ∈ CRing
∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) → ∃𝑚 ∈ (MaxIdeal‘𝑅)𝑗 ⊆ 𝑚) |
| 53 | 46, 52 | r19.29a 3162 |
. . . . . . . . 9
⊢
((((((𝑅 ∈ CRing
∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) ∧ ¬ 𝑗 = (Base‘𝑅)) → 𝑗 = 𝑀) |
| 54 | 53 | ex 412 |
. . . . . . . 8
⊢
(((((𝑅 ∈ CRing
∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) → (¬ 𝑗 = (Base‘𝑅) → 𝑗 = 𝑀)) |
| 55 | 54 | orrd 864 |
. . . . . . 7
⊢
(((((𝑅 ∈ CRing
∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) → (𝑗 = (Base‘𝑅) ∨ 𝑗 = 𝑀)) |
| 56 | 55 | orcomd 872 |
. . . . . 6
⊢
(((((𝑅 ∈ CRing
∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) ∧ 𝑀 ⊆ 𝑗) → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))) |
| 57 | 56 | ex 412 |
. . . . 5
⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) ∧ 𝑗 ∈ (LIdeal‘𝑅)) → (𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅)))) |
| 58 | 57 | ralrimiva 3146 |
. . . 4
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅)))) |
| 59 | 5, 15, 58 | 3jca 1129 |
. . 3
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ (Base‘𝑅) ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))))) |
| 60 | 11 | ismxidl 33490 |
. . . 4
⊢ (𝑅 ∈ Ring → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ (Base‘𝑅) ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅)))))) |
| 61 | 60 | biimpar 477 |
. . 3
⊢ ((𝑅 ∈ Ring ∧ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ (Base‘𝑅) ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))))) → 𝑀 ∈ (MaxIdeal‘𝑅)) |
| 62 | 2, 59, 61 | syl2anc 584 |
. 2
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ {𝑀} = (𝑉‘𝑀)) → 𝑀 ∈ (MaxIdeal‘𝑅)) |
| 63 | 10 | a1i 11 |
. . . . 5
⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑉 = (𝑖 ∈ (LIdeal‘𝑅) ↦ {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗})) |
| 64 | 26 | adantl 481 |
. . . . 5
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑖 = 𝑀) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑖 ⊆ 𝑗} = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀 ⊆ 𝑗}) |
| 65 | 11 | mxidlidl 33491 |
. . . . . 6
⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅)) |
| 66 | 1, 65 | sylan 580 |
. . . . 5
⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅)) |
| 67 | 29 | a1i 11 |
. . . . 5
⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀 ⊆ 𝑗} ∈ V) |
| 68 | 63, 64, 66, 67 | fvmptd 7023 |
. . . 4
⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → (𝑉‘𝑀) = {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀 ⊆ 𝑗}) |
| 69 | 1 | ad2antrr 726 |
. . . . . . . . 9
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → 𝑅 ∈ Ring) |
| 70 | | simplr 769 |
. . . . . . . . 9
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → 𝑀 ∈ (MaxIdeal‘𝑅)) |
| 71 | | simprl 771 |
. . . . . . . . . . 11
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → 𝑗 ∈ (PrmIdeal‘𝑅)) |
| 72 | | prmidlidl 33472 |
. . . . . . . . . . 11
⊢ ((𝑅 ∈ Ring ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) → 𝑗 ∈ (LIdeal‘𝑅)) |
| 73 | 69, 71, 72 | syl2anc 584 |
. . . . . . . . . 10
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → 𝑗 ∈ (LIdeal‘𝑅)) |
| 74 | | simprr 773 |
. . . . . . . . . 10
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → 𝑀 ⊆ 𝑗) |
| 75 | 73, 74 | jca 511 |
. . . . . . . . 9
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → (𝑗 ∈ (LIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) |
| 76 | 11 | mxidlmax 33493 |
. . . . . . . . 9
⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (LIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))) |
| 77 | 69, 70, 75, 76 | syl21anc 838 |
. . . . . . . 8
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → (𝑗 = 𝑀 ∨ 𝑗 = (Base‘𝑅))) |
| 78 | | eqid 2737 |
. . . . . . . . . . 11
⊢
(.r‘𝑅) = (.r‘𝑅) |
| 79 | 11, 78 | prmidlnr 33467 |
. . . . . . . . . 10
⊢ ((𝑅 ∈ Ring ∧ 𝑗 ∈ (PrmIdeal‘𝑅)) → 𝑗 ≠ (Base‘𝑅)) |
| 80 | 69, 71, 79 | syl2anc 584 |
. . . . . . . . 9
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → 𝑗 ≠ (Base‘𝑅)) |
| 81 | 80 | neneqd 2945 |
. . . . . . . 8
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → ¬ 𝑗 = (Base‘𝑅)) |
| 82 | 77, 81 | olcnd 878 |
. . . . . . 7
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) → 𝑗 = 𝑀) |
| 83 | | simpr 484 |
. . . . . . . . 9
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 = 𝑀) → 𝑗 = 𝑀) |
| 84 | 19 | mxidlprm 33498 |
. . . . . . . . . 10
⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (PrmIdeal‘𝑅)) |
| 85 | 84 | adantr 480 |
. . . . . . . . 9
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 = 𝑀) → 𝑀 ∈ (PrmIdeal‘𝑅)) |
| 86 | 83, 85 | eqeltrd 2841 |
. . . . . . . 8
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 = 𝑀) → 𝑗 ∈ (PrmIdeal‘𝑅)) |
| 87 | | ssidd 4007 |
. . . . . . . . 9
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 = 𝑀) → 𝑗 ⊆ 𝑗) |
| 88 | 83, 87 | eqsstrrd 4019 |
. . . . . . . 8
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 = 𝑀) → 𝑀 ⊆ 𝑗) |
| 89 | 86, 88 | jca 511 |
. . . . . . 7
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ 𝑗 = 𝑀) → (𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗)) |
| 90 | 82, 89 | impbida 801 |
. . . . . 6
⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → ((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗) ↔ 𝑗 = 𝑀)) |
| 91 | 90 | alrimiv 1927 |
. . . . 5
⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → ∀𝑗((𝑗 ∈ (PrmIdeal‘𝑅) ∧ 𝑀 ⊆ 𝑗) ↔ 𝑗 = 𝑀)) |
| 92 | 91, 33 | sylibr 234 |
. . . 4
⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → {𝑗 ∈ (PrmIdeal‘𝑅) ∣ 𝑀 ⊆ 𝑗} = {𝑀}) |
| 93 | 68, 92 | eqtr2d 2778 |
. . 3
⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → {𝑀} = (𝑉‘𝑀)) |
| 94 | 93 | adantlr 715 |
. 2
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → {𝑀} = (𝑉‘𝑀)) |
| 95 | 62, 94 | impbida 801 |
1
⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ({𝑀} = (𝑉‘𝑀) ↔ 𝑀 ∈ (MaxIdeal‘𝑅))) |