| Step |
Hyp |
Ref |
Expression |
| 1 |
|
zarclsx.1 |
⊢ 𝑉 = ( 𝑖 ∈ ( LIdeal ‘ 𝑅 ) ↦ { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖 ⊆ 𝑗 } ) |
| 2 |
|
zarclssn.1 |
⊢ 𝐵 = ( LIdeal ‘ 𝑅 ) |
| 3 |
|
crngring |
⊢ ( 𝑅 ∈ CRing → 𝑅 ∈ Ring ) |
| 4 |
3
|
ad2antrr |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) → 𝑅 ∈ Ring ) |
| 5 |
|
simplr |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) → 𝑀 ∈ 𝐵 ) |
| 6 |
5 2
|
eleqtrdi |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) → 𝑀 ∈ ( LIdeal ‘ 𝑅 ) ) |
| 7 |
|
simpr |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) → { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) |
| 8 |
|
snnzg |
⊢ ( 𝑀 ∈ 𝐵 → { 𝑀 } ≠ ∅ ) |
| 9 |
5 8
|
syl |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) → { 𝑀 } ≠ ∅ ) |
| 10 |
7 9
|
eqnetrrd |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) → ( 𝑉 ‘ 𝑀 ) ≠ ∅ ) |
| 11 |
|
simpll |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) → 𝑅 ∈ CRing ) |
| 12 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
| 13 |
1 12
|
zarcls1 |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( LIdeal ‘ 𝑅 ) ) → ( ( 𝑉 ‘ 𝑀 ) = ∅ ↔ 𝑀 = ( Base ‘ 𝑅 ) ) ) |
| 14 |
13
|
necon3bid |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( LIdeal ‘ 𝑅 ) ) → ( ( 𝑉 ‘ 𝑀 ) ≠ ∅ ↔ 𝑀 ≠ ( Base ‘ 𝑅 ) ) ) |
| 15 |
11 6 14
|
syl2anc |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) → ( ( 𝑉 ‘ 𝑀 ) ≠ ∅ ↔ 𝑀 ≠ ( Base ‘ 𝑅 ) ) ) |
| 16 |
10 15
|
mpbid |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) → 𝑀 ≠ ( Base ‘ 𝑅 ) ) |
| 17 |
|
simpr |
⊢ ( ( ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑀 ⊆ 𝑗 ) ∧ ¬ 𝑗 = ( Base ‘ 𝑅 ) ) ∧ 𝑚 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝑗 ⊆ 𝑚 ) → 𝑗 ⊆ 𝑚 ) |
| 18 |
11
|
ad5antr |
⊢ ( ( ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑀 ⊆ 𝑗 ) ∧ ¬ 𝑗 = ( Base ‘ 𝑅 ) ) ∧ 𝑚 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝑗 ⊆ 𝑚 ) → 𝑅 ∈ CRing ) |
| 19 |
|
simplr |
⊢ ( ( ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑀 ⊆ 𝑗 ) ∧ ¬ 𝑗 = ( Base ‘ 𝑅 ) ) ∧ 𝑚 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝑗 ⊆ 𝑚 ) → 𝑚 ∈ ( MaxIdeal ‘ 𝑅 ) ) |
| 20 |
|
eqid |
⊢ ( LSSum ‘ ( mulGrp ‘ 𝑅 ) ) = ( LSSum ‘ ( mulGrp ‘ 𝑅 ) ) |
| 21 |
20
|
mxidlprm |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑚 ∈ ( MaxIdeal ‘ 𝑅 ) ) → 𝑚 ∈ ( PrmIdeal ‘ 𝑅 ) ) |
| 22 |
18 19 21
|
syl2anc |
⊢ ( ( ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑀 ⊆ 𝑗 ) ∧ ¬ 𝑗 = ( Base ‘ 𝑅 ) ) ∧ 𝑚 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝑗 ⊆ 𝑚 ) → 𝑚 ∈ ( PrmIdeal ‘ 𝑅 ) ) |
| 23 |
|
simp-4r |
⊢ ( ( ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑀 ⊆ 𝑗 ) ∧ ¬ 𝑗 = ( Base ‘ 𝑅 ) ) ∧ 𝑚 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝑗 ⊆ 𝑚 ) → 𝑀 ⊆ 𝑗 ) |
| 24 |
23 17
|
sstrd |
⊢ ( ( ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑀 ⊆ 𝑗 ) ∧ ¬ 𝑗 = ( Base ‘ 𝑅 ) ) ∧ 𝑚 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝑗 ⊆ 𝑚 ) → 𝑀 ⊆ 𝑚 ) |
| 25 |
1
|
a1i |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) → 𝑉 = ( 𝑖 ∈ ( LIdeal ‘ 𝑅 ) ↦ { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖 ⊆ 𝑗 } ) ) |
| 26 |
|
sseq1 |
⊢ ( 𝑖 = 𝑀 → ( 𝑖 ⊆ 𝑗 ↔ 𝑀 ⊆ 𝑗 ) ) |
| 27 |
26
|
rabbidv |
⊢ ( 𝑖 = 𝑀 → { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖 ⊆ 𝑗 } = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑀 ⊆ 𝑗 } ) |
| 28 |
27
|
adantl |
⊢ ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) ∧ 𝑖 = 𝑀 ) → { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖 ⊆ 𝑗 } = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑀 ⊆ 𝑗 } ) |
| 29 |
|
fvex |
⊢ ( PrmIdeal ‘ 𝑅 ) ∈ V |
| 30 |
29
|
rabex |
⊢ { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑀 ⊆ 𝑗 } ∈ V |
| 31 |
30
|
a1i |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) → { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑀 ⊆ 𝑗 } ∈ V ) |
| 32 |
25 28 6 31
|
fvmptd |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) → ( 𝑉 ‘ 𝑀 ) = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑀 ⊆ 𝑗 } ) |
| 33 |
7 32
|
eqtr2d |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) → { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑀 ⊆ 𝑗 } = { 𝑀 } ) |
| 34 |
|
rabeqsn |
⊢ ( { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑀 ⊆ 𝑗 } = { 𝑀 } ↔ ∀ 𝑗 ( ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ↔ 𝑗 = 𝑀 ) ) |
| 35 |
33 34
|
sylib |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) → ∀ 𝑗 ( ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ↔ 𝑗 = 𝑀 ) ) |
| 36 |
35
|
ad5antr |
⊢ ( ( ( ( ( ( ( ( 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 ) ) |
| 37 |
|
vex |
⊢ 𝑚 ∈ V |
| 38 |
|
eleq1w |
⊢ ( 𝑗 = 𝑚 → ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ↔ 𝑚 ∈ ( PrmIdeal ‘ 𝑅 ) ) ) |
| 39 |
|
sseq2 |
⊢ ( 𝑗 = 𝑚 → ( 𝑀 ⊆ 𝑗 ↔ 𝑀 ⊆ 𝑚 ) ) |
| 40 |
38 39
|
anbi12d |
⊢ ( 𝑗 = 𝑚 → ( ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ↔ ( 𝑚 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑚 ) ) ) |
| 41 |
|
eqeq1 |
⊢ ( 𝑗 = 𝑚 → ( 𝑗 = 𝑀 ↔ 𝑚 = 𝑀 ) ) |
| 42 |
40 41
|
bibi12d |
⊢ ( 𝑗 = 𝑚 → ( ( ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ↔ 𝑗 = 𝑀 ) ↔ ( ( 𝑚 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑚 ) ↔ 𝑚 = 𝑀 ) ) ) |
| 43 |
37 42
|
spcv |
⊢ ( ∀ 𝑗 ( ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ↔ 𝑗 = 𝑀 ) → ( ( 𝑚 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑚 ) ↔ 𝑚 = 𝑀 ) ) |
| 44 |
36 43
|
syl |
⊢ ( ( ( ( ( ( ( ( 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 ) ) |
| 45 |
22 24 44
|
mpbi2and |
⊢ ( ( ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑀 ⊆ 𝑗 ) ∧ ¬ 𝑗 = ( Base ‘ 𝑅 ) ) ∧ 𝑚 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝑗 ⊆ 𝑚 ) → 𝑚 = 𝑀 ) |
| 46 |
17 45
|
sseqtrd |
⊢ ( ( ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑀 ⊆ 𝑗 ) ∧ ¬ 𝑗 = ( Base ‘ 𝑅 ) ) ∧ 𝑚 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝑗 ⊆ 𝑚 ) → 𝑗 ⊆ 𝑀 ) |
| 47 |
46 23
|
eqssd |
⊢ ( ( ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑀 ⊆ 𝑗 ) ∧ ¬ 𝑗 = ( Base ‘ 𝑅 ) ) ∧ 𝑚 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝑗 ⊆ 𝑚 ) → 𝑗 = 𝑀 ) |
| 48 |
3
|
ad5antr |
⊢ ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑀 ⊆ 𝑗 ) ∧ ¬ 𝑗 = ( Base ‘ 𝑅 ) ) → 𝑅 ∈ Ring ) |
| 49 |
|
simpllr |
⊢ ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑀 ⊆ 𝑗 ) ∧ ¬ 𝑗 = ( Base ‘ 𝑅 ) ) → 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) |
| 50 |
|
simpr |
⊢ ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑀 ⊆ 𝑗 ) ∧ ¬ 𝑗 = ( Base ‘ 𝑅 ) ) → ¬ 𝑗 = ( Base ‘ 𝑅 ) ) |
| 51 |
50
|
neqned |
⊢ ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑀 ⊆ 𝑗 ) ∧ ¬ 𝑗 = ( Base ‘ 𝑅 ) ) → 𝑗 ≠ ( Base ‘ 𝑅 ) ) |
| 52 |
12
|
ssmxidl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ∧ 𝑗 ≠ ( Base ‘ 𝑅 ) ) → ∃ 𝑚 ∈ ( MaxIdeal ‘ 𝑅 ) 𝑗 ⊆ 𝑚 ) |
| 53 |
48 49 51 52
|
syl3anc |
⊢ ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑀 ⊆ 𝑗 ) ∧ ¬ 𝑗 = ( Base ‘ 𝑅 ) ) → ∃ 𝑚 ∈ ( MaxIdeal ‘ 𝑅 ) 𝑗 ⊆ 𝑚 ) |
| 54 |
47 53
|
r19.29a |
⊢ ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑀 ⊆ 𝑗 ) ∧ ¬ 𝑗 = ( Base ‘ 𝑅 ) ) → 𝑗 = 𝑀 ) |
| 55 |
54
|
ex |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑀 ⊆ 𝑗 ) → ( ¬ 𝑗 = ( Base ‘ 𝑅 ) → 𝑗 = 𝑀 ) ) |
| 56 |
55
|
orrd |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑀 ⊆ 𝑗 ) → ( 𝑗 = ( Base ‘ 𝑅 ) ∨ 𝑗 = 𝑀 ) ) |
| 57 |
56
|
orcomd |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑀 ⊆ 𝑗 ) → ( 𝑗 = 𝑀 ∨ 𝑗 = ( Base ‘ 𝑅 ) ) ) |
| 58 |
57
|
ex |
⊢ ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) → ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = ( Base ‘ 𝑅 ) ) ) ) |
| 59 |
58
|
ralrimiva |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) → ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = ( Base ‘ 𝑅 ) ) ) ) |
| 60 |
6 16 59
|
3jca |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) → ( 𝑀 ∈ ( LIdeal ‘ 𝑅 ) ∧ 𝑀 ≠ ( Base ‘ 𝑅 ) ∧ ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = ( Base ‘ 𝑅 ) ) ) ) ) |
| 61 |
12
|
ismxidl |
⊢ ( 𝑅 ∈ Ring → ( 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ↔ ( 𝑀 ∈ ( LIdeal ‘ 𝑅 ) ∧ 𝑀 ≠ ( Base ‘ 𝑅 ) ∧ ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = ( Base ‘ 𝑅 ) ) ) ) ) ) |
| 62 |
61
|
biimpar |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑀 ∈ ( LIdeal ‘ 𝑅 ) ∧ 𝑀 ≠ ( Base ‘ 𝑅 ) ∧ ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = ( Base ‘ 𝑅 ) ) ) ) ) → 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) |
| 63 |
4 60 62
|
syl2anc |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) → 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) |
| 64 |
1
|
a1i |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) → 𝑉 = ( 𝑖 ∈ ( LIdeal ‘ 𝑅 ) ↦ { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖 ⊆ 𝑗 } ) ) |
| 65 |
27
|
adantl |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝑖 = 𝑀 ) → { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖 ⊆ 𝑗 } = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑀 ⊆ 𝑗 } ) |
| 66 |
12
|
mxidlidl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) → 𝑀 ∈ ( LIdeal ‘ 𝑅 ) ) |
| 67 |
3 66
|
sylan |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) → 𝑀 ∈ ( LIdeal ‘ 𝑅 ) ) |
| 68 |
30
|
a1i |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) → { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑀 ⊆ 𝑗 } ∈ V ) |
| 69 |
64 65 67 68
|
fvmptd |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) → ( 𝑉 ‘ 𝑀 ) = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑀 ⊆ 𝑗 } ) |
| 70 |
3
|
ad2antrr |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ) → 𝑅 ∈ Ring ) |
| 71 |
|
simplr |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ) → 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) |
| 72 |
|
simprl |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ) → 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) |
| 73 |
|
prmidlidl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) → 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) |
| 74 |
70 72 73
|
syl2anc |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ) → 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) |
| 75 |
|
simprr |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ) → 𝑀 ⊆ 𝑗 ) |
| 76 |
74 75
|
jca |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ) → ( 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ) |
| 77 |
12
|
mxidlmax |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ ( 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ) → ( 𝑗 = 𝑀 ∨ 𝑗 = ( Base ‘ 𝑅 ) ) ) |
| 78 |
70 71 76 77
|
syl21anc |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ) → ( 𝑗 = 𝑀 ∨ 𝑗 = ( Base ‘ 𝑅 ) ) ) |
| 79 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
| 80 |
12 79
|
prmidlnr |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) → 𝑗 ≠ ( Base ‘ 𝑅 ) ) |
| 81 |
70 72 80
|
syl2anc |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ) → 𝑗 ≠ ( Base ‘ 𝑅 ) ) |
| 82 |
81
|
neneqd |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ) → ¬ 𝑗 = ( Base ‘ 𝑅 ) ) |
| 83 |
78 82
|
olcnd |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ) → 𝑗 = 𝑀 ) |
| 84 |
|
simpr |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝑗 = 𝑀 ) → 𝑗 = 𝑀 ) |
| 85 |
20
|
mxidlprm |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) → 𝑀 ∈ ( PrmIdeal ‘ 𝑅 ) ) |
| 86 |
85
|
adantr |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝑗 = 𝑀 ) → 𝑀 ∈ ( PrmIdeal ‘ 𝑅 ) ) |
| 87 |
84 86
|
eqeltrd |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝑗 = 𝑀 ) → 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) |
| 88 |
|
ssidd |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝑗 = 𝑀 ) → 𝑗 ⊆ 𝑗 ) |
| 89 |
84 88
|
eqsstrrd |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝑗 = 𝑀 ) → 𝑀 ⊆ 𝑗 ) |
| 90 |
87 89
|
jca |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝑗 = 𝑀 ) → ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ) |
| 91 |
83 90
|
impbida |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) → ( ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ↔ 𝑗 = 𝑀 ) ) |
| 92 |
91
|
alrimiv |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) → ∀ 𝑗 ( ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ↔ 𝑗 = 𝑀 ) ) |
| 93 |
92 34
|
sylibr |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) → { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑀 ⊆ 𝑗 } = { 𝑀 } ) |
| 94 |
69 93
|
eqtr2d |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) → { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) |
| 95 |
94
|
adantlr |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) → { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) |
| 96 |
63 95
|
impbida |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ↔ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ) |