| 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 |
5
|
snn0d |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) → { 𝑀 } ≠ ∅ ) |
| 9 |
7 8
|
eqnetrrd |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) → ( 𝑉 ‘ 𝑀 ) ≠ ∅ ) |
| 10 |
|
simpll |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) → 𝑅 ∈ CRing ) |
| 11 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
| 12 |
1 11
|
zarcls1 |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( LIdeal ‘ 𝑅 ) ) → ( ( 𝑉 ‘ 𝑀 ) = ∅ ↔ 𝑀 = ( Base ‘ 𝑅 ) ) ) |
| 13 |
12
|
necon3bid |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( LIdeal ‘ 𝑅 ) ) → ( ( 𝑉 ‘ 𝑀 ) ≠ ∅ ↔ 𝑀 ≠ ( Base ‘ 𝑅 ) ) ) |
| 14 |
10 6 13
|
syl2anc |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) → ( ( 𝑉 ‘ 𝑀 ) ≠ ∅ ↔ 𝑀 ≠ ( Base ‘ 𝑅 ) ) ) |
| 15 |
9 14
|
mpbid |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) → 𝑀 ≠ ( Base ‘ 𝑅 ) ) |
| 16 |
|
simpr |
⊢ ( ( ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑀 ⊆ 𝑗 ) ∧ ¬ 𝑗 = ( Base ‘ 𝑅 ) ) ∧ 𝑚 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝑗 ⊆ 𝑚 ) → 𝑗 ⊆ 𝑚 ) |
| 17 |
10
|
ad5antr |
⊢ ( ( ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑀 ⊆ 𝑗 ) ∧ ¬ 𝑗 = ( Base ‘ 𝑅 ) ) ∧ 𝑚 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝑗 ⊆ 𝑚 ) → 𝑅 ∈ CRing ) |
| 18 |
|
simplr |
⊢ ( ( ( ( ( ( ( ( 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 |
⊢ ( LSSum ‘ ( mulGrp ‘ 𝑅 ) ) = ( LSSum ‘ ( mulGrp ‘ 𝑅 ) ) |
| 20 |
19
|
mxidlprm |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑚 ∈ ( MaxIdeal ‘ 𝑅 ) ) → 𝑚 ∈ ( PrmIdeal ‘ 𝑅 ) ) |
| 21 |
17 18 20
|
syl2anc |
⊢ ( ( ( ( ( ( ( ( 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 |
⊢ ( ( ( ( ( ( ( ( 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 |
⊢ ( ( ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑀 ⊆ 𝑗 ) ∧ ¬ 𝑗 = ( Base ‘ 𝑅 ) ) ∧ 𝑚 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝑗 ⊆ 𝑚 ) → 𝑀 ⊆ 𝑚 ) |
| 24 |
1
|
a1i |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) → 𝑉 = ( 𝑖 ∈ ( LIdeal ‘ 𝑅 ) ↦ { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖 ⊆ 𝑗 } ) ) |
| 25 |
|
sseq1 |
⊢ ( 𝑖 = 𝑀 → ( 𝑖 ⊆ 𝑗 ↔ 𝑀 ⊆ 𝑗 ) ) |
| 26 |
25
|
rabbidv |
⊢ ( 𝑖 = 𝑀 → { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖 ⊆ 𝑗 } = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑀 ⊆ 𝑗 } ) |
| 27 |
26
|
adantl |
⊢ ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) ∧ 𝑖 = 𝑀 ) → { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖 ⊆ 𝑗 } = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑀 ⊆ 𝑗 } ) |
| 28 |
|
fvex |
⊢ ( PrmIdeal ‘ 𝑅 ) ∈ V |
| 29 |
28
|
rabex |
⊢ { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑀 ⊆ 𝑗 } ∈ V |
| 30 |
29
|
a1i |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) → { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑀 ⊆ 𝑗 } ∈ V ) |
| 31 |
24 27 6 30
|
fvmptd |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) → ( 𝑉 ‘ 𝑀 ) = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑀 ⊆ 𝑗 } ) |
| 32 |
7 31
|
eqtr2d |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) → { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑀 ⊆ 𝑗 } = { 𝑀 } ) |
| 33 |
|
rabeqsn |
⊢ ( { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑀 ⊆ 𝑗 } = { 𝑀 } ↔ ∀ 𝑗 ( ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ↔ 𝑗 = 𝑀 ) ) |
| 34 |
32 33
|
sylib |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) → ∀ 𝑗 ( ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ↔ 𝑗 = 𝑀 ) ) |
| 35 |
34
|
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 ) ) |
| 36 |
|
vex |
⊢ 𝑚 ∈ V |
| 37 |
|
eleq1w |
⊢ ( 𝑗 = 𝑚 → ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ↔ 𝑚 ∈ ( PrmIdeal ‘ 𝑅 ) ) ) |
| 38 |
|
sseq2 |
⊢ ( 𝑗 = 𝑚 → ( 𝑀 ⊆ 𝑗 ↔ 𝑀 ⊆ 𝑚 ) ) |
| 39 |
37 38
|
anbi12d |
⊢ ( 𝑗 = 𝑚 → ( ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ↔ ( 𝑚 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑚 ) ) ) |
| 40 |
|
eqeq1 |
⊢ ( 𝑗 = 𝑚 → ( 𝑗 = 𝑀 ↔ 𝑚 = 𝑀 ) ) |
| 41 |
39 40
|
bibi12d |
⊢ ( 𝑗 = 𝑚 → ( ( ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ↔ 𝑗 = 𝑀 ) ↔ ( ( 𝑚 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑚 ) ↔ 𝑚 = 𝑀 ) ) ) |
| 42 |
36 41
|
spcv |
⊢ ( ∀ 𝑗 ( ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ↔ 𝑗 = 𝑀 ) → ( ( 𝑚 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑚 ) ↔ 𝑚 = 𝑀 ) ) |
| 43 |
35 42
|
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 ) ) |
| 44 |
21 23 43
|
mpbi2and |
⊢ ( ( ( ( ( ( ( ( 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 |
⊢ ( ( ( ( ( ( ( ( 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 |
⊢ ( ( ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑀 ⊆ 𝑗 ) ∧ ¬ 𝑗 = ( Base ‘ 𝑅 ) ) ∧ 𝑚 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝑗 ⊆ 𝑚 ) → 𝑗 = 𝑀 ) |
| 47 |
3
|
ad5antr |
⊢ ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑀 ⊆ 𝑗 ) ∧ ¬ 𝑗 = ( Base ‘ 𝑅 ) ) → 𝑅 ∈ Ring ) |
| 48 |
|
simpllr |
⊢ ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑀 ⊆ 𝑗 ) ∧ ¬ 𝑗 = ( Base ‘ 𝑅 ) ) → 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) |
| 49 |
|
simpr |
⊢ ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑀 ⊆ 𝑗 ) ∧ ¬ 𝑗 = ( Base ‘ 𝑅 ) ) → ¬ 𝑗 = ( Base ‘ 𝑅 ) ) |
| 50 |
49
|
neqned |
⊢ ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑀 ⊆ 𝑗 ) ∧ ¬ 𝑗 = ( Base ‘ 𝑅 ) ) → 𝑗 ≠ ( Base ‘ 𝑅 ) ) |
| 51 |
11
|
ssmxidl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ∧ 𝑗 ≠ ( Base ‘ 𝑅 ) ) → ∃ 𝑚 ∈ ( MaxIdeal ‘ 𝑅 ) 𝑗 ⊆ 𝑚 ) |
| 52 |
47 48 50 51
|
syl3anc |
⊢ ( ( ( ( ( ( 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 |
⊢ ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑀 ⊆ 𝑗 ) ∧ ¬ 𝑗 = ( Base ‘ 𝑅 ) ) → 𝑗 = 𝑀 ) |
| 54 |
53
|
ex |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑀 ⊆ 𝑗 ) → ( ¬ 𝑗 = ( Base ‘ 𝑅 ) → 𝑗 = 𝑀 ) ) |
| 55 |
54
|
orrd |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑀 ⊆ 𝑗 ) → ( 𝑗 = ( Base ‘ 𝑅 ) ∨ 𝑗 = 𝑀 ) ) |
| 56 |
55
|
orcomd |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝑀 ⊆ 𝑗 ) → ( 𝑗 = 𝑀 ∨ 𝑗 = ( Base ‘ 𝑅 ) ) ) |
| 57 |
56
|
ex |
⊢ ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) → ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = ( Base ‘ 𝑅 ) ) ) ) |
| 58 |
57
|
ralrimiva |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) → ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = ( Base ‘ 𝑅 ) ) ) ) |
| 59 |
6 15 58
|
3jca |
⊢ ( ( ( 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 |
⊢ ( 𝑅 ∈ Ring → ( 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ↔ ( 𝑀 ∈ ( LIdeal ‘ 𝑅 ) ∧ 𝑀 ≠ ( Base ‘ 𝑅 ) ∧ ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = ( Base ‘ 𝑅 ) ) ) ) ) ) |
| 61 |
60
|
biimpar |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑀 ∈ ( LIdeal ‘ 𝑅 ) ∧ 𝑀 ≠ ( Base ‘ 𝑅 ) ∧ ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = ( Base ‘ 𝑅 ) ) ) ) ) → 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) |
| 62 |
4 59 61
|
syl2anc |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) → 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) |
| 63 |
1
|
a1i |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) → 𝑉 = ( 𝑖 ∈ ( LIdeal ‘ 𝑅 ) ↦ { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖 ⊆ 𝑗 } ) ) |
| 64 |
26
|
adantl |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝑖 = 𝑀 ) → { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖 ⊆ 𝑗 } = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑀 ⊆ 𝑗 } ) |
| 65 |
11
|
mxidlidl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) → 𝑀 ∈ ( LIdeal ‘ 𝑅 ) ) |
| 66 |
3 65
|
sylan |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) → 𝑀 ∈ ( LIdeal ‘ 𝑅 ) ) |
| 67 |
29
|
a1i |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) → { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑀 ⊆ 𝑗 } ∈ V ) |
| 68 |
63 64 66 67
|
fvmptd |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) → ( 𝑉 ‘ 𝑀 ) = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑀 ⊆ 𝑗 } ) |
| 69 |
3
|
ad2antrr |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ) → 𝑅 ∈ Ring ) |
| 70 |
|
simplr |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ) → 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) |
| 71 |
|
simprl |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ) → 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) |
| 72 |
|
prmidlidl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) → 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) |
| 73 |
69 71 72
|
syl2anc |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ) → 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) |
| 74 |
|
simprr |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ) → 𝑀 ⊆ 𝑗 ) |
| 75 |
73 74
|
jca |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ) → ( 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ) |
| 76 |
11
|
mxidlmax |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ ( 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ) → ( 𝑗 = 𝑀 ∨ 𝑗 = ( Base ‘ 𝑅 ) ) ) |
| 77 |
69 70 75 76
|
syl21anc |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ) → ( 𝑗 = 𝑀 ∨ 𝑗 = ( Base ‘ 𝑅 ) ) ) |
| 78 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
| 79 |
11 78
|
prmidlnr |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) → 𝑗 ≠ ( Base ‘ 𝑅 ) ) |
| 80 |
69 71 79
|
syl2anc |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ) → 𝑗 ≠ ( Base ‘ 𝑅 ) ) |
| 81 |
80
|
neneqd |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ) → ¬ 𝑗 = ( Base ‘ 𝑅 ) ) |
| 82 |
77 81
|
olcnd |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ) → 𝑗 = 𝑀 ) |
| 83 |
|
simpr |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝑗 = 𝑀 ) → 𝑗 = 𝑀 ) |
| 84 |
19
|
mxidlprm |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) → 𝑀 ∈ ( PrmIdeal ‘ 𝑅 ) ) |
| 85 |
84
|
adantr |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝑗 = 𝑀 ) → 𝑀 ∈ ( PrmIdeal ‘ 𝑅 ) ) |
| 86 |
83 85
|
eqeltrd |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝑗 = 𝑀 ) → 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) |
| 87 |
|
ssidd |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝑗 = 𝑀 ) → 𝑗 ⊆ 𝑗 ) |
| 88 |
83 87
|
eqsstrrd |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝑗 = 𝑀 ) → 𝑀 ⊆ 𝑗 ) |
| 89 |
86 88
|
jca |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝑗 = 𝑀 ) → ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ) |
| 90 |
82 89
|
impbida |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) → ( ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ↔ 𝑗 = 𝑀 ) ) |
| 91 |
90
|
alrimiv |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) → ∀ 𝑗 ( ( 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∧ 𝑀 ⊆ 𝑗 ) ↔ 𝑗 = 𝑀 ) ) |
| 92 |
91 33
|
sylibr |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) → { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑀 ⊆ 𝑗 } = { 𝑀 } ) |
| 93 |
68 92
|
eqtr2d |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) → { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) |
| 94 |
93
|
adantlr |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) → { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ) |
| 95 |
62 94
|
impbida |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) → ( { 𝑀 } = ( 𝑉 ‘ 𝑀 ) ↔ 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ) ) |