Step |
Hyp |
Ref |
Expression |
1 |
|
zarclsx.1 |
⊢ 𝑉 = ( 𝑖 ∈ ( LIdeal ‘ 𝑅 ) ↦ { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖 ⊆ 𝑗 } ) |
2 |
|
zarcls1.1 |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
3 |
|
simplr |
⊢ ( ( ( ( 𝑅 ∈ CRing ∧ 𝐼 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ ( 𝑉 ‘ 𝐼 ) = ∅ ) ∧ 𝐼 ≠ 𝐵 ) → ( 𝑉 ‘ 𝐼 ) = ∅ ) |
4 |
|
sseq2 |
⊢ ( 𝑗 = 𝑚 → ( 𝐼 ⊆ 𝑗 ↔ 𝐼 ⊆ 𝑚 ) ) |
5 |
|
eqid |
⊢ ( LSSum ‘ ( mulGrp ‘ 𝑅 ) ) = ( LSSum ‘ ( mulGrp ‘ 𝑅 ) ) |
6 |
5
|
mxidlprm |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝑚 ∈ ( MaxIdeal ‘ 𝑅 ) ) → 𝑚 ∈ ( PrmIdeal ‘ 𝑅 ) ) |
7 |
6
|
ad5ant14 |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝐼 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝐼 ≠ 𝐵 ) ∧ 𝑚 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝐼 ⊆ 𝑚 ) → 𝑚 ∈ ( PrmIdeal ‘ 𝑅 ) ) |
8 |
|
simpr |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝐼 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝐼 ≠ 𝐵 ) ∧ 𝑚 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝐼 ⊆ 𝑚 ) → 𝐼 ⊆ 𝑚 ) |
9 |
4 7 8
|
elrabd |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝐼 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝐼 ≠ 𝐵 ) ∧ 𝑚 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝐼 ⊆ 𝑚 ) → 𝑚 ∈ { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝐼 ⊆ 𝑗 } ) |
10 |
1
|
a1i |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝐼 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝐼 ≠ 𝐵 ) ∧ 𝑚 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝐼 ⊆ 𝑚 ) → 𝑉 = ( 𝑖 ∈ ( LIdeal ‘ 𝑅 ) ↦ { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖 ⊆ 𝑗 } ) ) |
11 |
|
sseq1 |
⊢ ( 𝑖 = 𝐼 → ( 𝑖 ⊆ 𝑗 ↔ 𝐼 ⊆ 𝑗 ) ) |
12 |
11
|
rabbidv |
⊢ ( 𝑖 = 𝐼 → { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖 ⊆ 𝑗 } = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝐼 ⊆ 𝑗 } ) |
13 |
12
|
adantl |
⊢ ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝐼 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝐼 ≠ 𝐵 ) ∧ 𝑚 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝐼 ⊆ 𝑚 ) ∧ 𝑖 = 𝐼 ) → { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖 ⊆ 𝑗 } = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝐼 ⊆ 𝑗 } ) |
14 |
|
simp-4r |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝐼 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝐼 ≠ 𝐵 ) ∧ 𝑚 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝐼 ⊆ 𝑚 ) → 𝐼 ∈ ( LIdeal ‘ 𝑅 ) ) |
15 |
|
fvex |
⊢ ( PrmIdeal ‘ 𝑅 ) ∈ V |
16 |
15
|
rabex |
⊢ { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝐼 ⊆ 𝑗 } ∈ V |
17 |
16
|
a1i |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝐼 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝐼 ≠ 𝐵 ) ∧ 𝑚 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝐼 ⊆ 𝑚 ) → { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝐼 ⊆ 𝑗 } ∈ V ) |
18 |
10 13 14 17
|
fvmptd |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝐼 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝐼 ≠ 𝐵 ) ∧ 𝑚 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝐼 ⊆ 𝑚 ) → ( 𝑉 ‘ 𝐼 ) = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝐼 ⊆ 𝑗 } ) |
19 |
9 18
|
eleqtrrd |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝐼 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝐼 ≠ 𝐵 ) ∧ 𝑚 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝐼 ⊆ 𝑚 ) → 𝑚 ∈ ( 𝑉 ‘ 𝐼 ) ) |
20 |
|
ne0i |
⊢ ( 𝑚 ∈ ( 𝑉 ‘ 𝐼 ) → ( 𝑉 ‘ 𝐼 ) ≠ ∅ ) |
21 |
19 20
|
syl |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝐼 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝐼 ≠ 𝐵 ) ∧ 𝑚 ∈ ( MaxIdeal ‘ 𝑅 ) ) ∧ 𝐼 ⊆ 𝑚 ) → ( 𝑉 ‘ 𝐼 ) ≠ ∅ ) |
22 |
|
crngring |
⊢ ( 𝑅 ∈ CRing → 𝑅 ∈ Ring ) |
23 |
2
|
ssmxidl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ ( LIdeal ‘ 𝑅 ) ∧ 𝐼 ≠ 𝐵 ) → ∃ 𝑚 ∈ ( MaxIdeal ‘ 𝑅 ) 𝐼 ⊆ 𝑚 ) |
24 |
23
|
3expa |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝐼 ≠ 𝐵 ) → ∃ 𝑚 ∈ ( MaxIdeal ‘ 𝑅 ) 𝐼 ⊆ 𝑚 ) |
25 |
22 24
|
sylanl1 |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝐼 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝐼 ≠ 𝐵 ) → ∃ 𝑚 ∈ ( MaxIdeal ‘ 𝑅 ) 𝐼 ⊆ 𝑚 ) |
26 |
21 25
|
r19.29a |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝐼 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝐼 ≠ 𝐵 ) → ( 𝑉 ‘ 𝐼 ) ≠ ∅ ) |
27 |
26
|
adantlr |
⊢ ( ( ( ( 𝑅 ∈ CRing ∧ 𝐼 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ ( 𝑉 ‘ 𝐼 ) = ∅ ) ∧ 𝐼 ≠ 𝐵 ) → ( 𝑉 ‘ 𝐼 ) ≠ ∅ ) |
28 |
27
|
neneqd |
⊢ ( ( ( ( 𝑅 ∈ CRing ∧ 𝐼 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ ( 𝑉 ‘ 𝐼 ) = ∅ ) ∧ 𝐼 ≠ 𝐵 ) → ¬ ( 𝑉 ‘ 𝐼 ) = ∅ ) |
29 |
3 28
|
pm2.65da |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝐼 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ ( 𝑉 ‘ 𝐼 ) = ∅ ) → ¬ 𝐼 ≠ 𝐵 ) |
30 |
|
nne |
⊢ ( ¬ 𝐼 ≠ 𝐵 ↔ 𝐼 = 𝐵 ) |
31 |
29 30
|
sylib |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝐼 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ ( 𝑉 ‘ 𝐼 ) = ∅ ) → 𝐼 = 𝐵 ) |
32 |
|
fveq2 |
⊢ ( 𝐼 = 𝐵 → ( 𝑉 ‘ 𝐼 ) = ( 𝑉 ‘ 𝐵 ) ) |
33 |
32
|
adantl |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝐼 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝐼 = 𝐵 ) → ( 𝑉 ‘ 𝐼 ) = ( 𝑉 ‘ 𝐵 ) ) |
34 |
1
|
a1i |
⊢ ( 𝑅 ∈ Ring → 𝑉 = ( 𝑖 ∈ ( LIdeal ‘ 𝑅 ) ↦ { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖 ⊆ 𝑗 } ) ) |
35 |
|
sseq1 |
⊢ ( 𝑖 = 𝐵 → ( 𝑖 ⊆ 𝑗 ↔ 𝐵 ⊆ 𝑗 ) ) |
36 |
35
|
adantl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑖 = 𝐵 ) → ( 𝑖 ⊆ 𝑗 ↔ 𝐵 ⊆ 𝑗 ) ) |
37 |
36
|
rabbidv |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑖 = 𝐵 ) → { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖 ⊆ 𝑗 } = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝐵 ⊆ 𝑗 } ) |
38 |
|
eqid |
⊢ ( LIdeal ‘ 𝑅 ) = ( LIdeal ‘ 𝑅 ) |
39 |
38 2
|
lidl1 |
⊢ ( 𝑅 ∈ Ring → 𝐵 ∈ ( LIdeal ‘ 𝑅 ) ) |
40 |
15
|
rabex |
⊢ { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝐵 ⊆ 𝑗 } ∈ V |
41 |
40
|
a1i |
⊢ ( 𝑅 ∈ Ring → { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝐵 ⊆ 𝑗 } ∈ V ) |
42 |
34 37 39 41
|
fvmptd |
⊢ ( 𝑅 ∈ Ring → ( 𝑉 ‘ 𝐵 ) = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝐵 ⊆ 𝑗 } ) |
43 |
|
prmidlidl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) → 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) |
44 |
2 38
|
lidlss |
⊢ ( 𝑗 ∈ ( LIdeal ‘ 𝑅 ) → 𝑗 ⊆ 𝐵 ) |
45 |
43 44
|
syl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) → 𝑗 ⊆ 𝐵 ) |
46 |
45
|
adantr |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) ∧ 𝐵 ⊆ 𝑗 ) → 𝑗 ⊆ 𝐵 ) |
47 |
|
simpr |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) ∧ 𝐵 ⊆ 𝑗 ) → 𝐵 ⊆ 𝑗 ) |
48 |
46 47
|
eqssd |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) ∧ 𝐵 ⊆ 𝑗 ) → 𝑗 = 𝐵 ) |
49 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
50 |
2 49
|
prmidlnr |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) → 𝑗 ≠ 𝐵 ) |
51 |
50
|
adantr |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) ∧ 𝐵 ⊆ 𝑗 ) → 𝑗 ≠ 𝐵 ) |
52 |
51
|
neneqd |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) ∧ 𝐵 ⊆ 𝑗 ) → ¬ 𝑗 = 𝐵 ) |
53 |
48 52
|
pm2.65da |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) → ¬ 𝐵 ⊆ 𝑗 ) |
54 |
53
|
ralrimiva |
⊢ ( 𝑅 ∈ Ring → ∀ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ¬ 𝐵 ⊆ 𝑗 ) |
55 |
|
rabeq0 |
⊢ ( { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝐵 ⊆ 𝑗 } = ∅ ↔ ∀ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ¬ 𝐵 ⊆ 𝑗 ) |
56 |
54 55
|
sylibr |
⊢ ( 𝑅 ∈ Ring → { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝐵 ⊆ 𝑗 } = ∅ ) |
57 |
42 56
|
eqtrd |
⊢ ( 𝑅 ∈ Ring → ( 𝑉 ‘ 𝐵 ) = ∅ ) |
58 |
22 57
|
syl |
⊢ ( 𝑅 ∈ CRing → ( 𝑉 ‘ 𝐵 ) = ∅ ) |
59 |
58
|
ad2antrr |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝐼 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝐼 = 𝐵 ) → ( 𝑉 ‘ 𝐵 ) = ∅ ) |
60 |
33 59
|
eqtrd |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝐼 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ 𝐼 = 𝐵 ) → ( 𝑉 ‘ 𝐼 ) = ∅ ) |
61 |
31 60
|
impbida |
⊢ ( ( 𝑅 ∈ CRing ∧ 𝐼 ∈ ( LIdeal ‘ 𝑅 ) ) → ( ( 𝑉 ‘ 𝐼 ) = ∅ ↔ 𝐼 = 𝐵 ) ) |