Step |
Hyp |
Ref |
Expression |
1 |
|
zarclsx.1 |
⊢ 𝑉 = ( 𝑖 ∈ ( LIdeal ‘ 𝑅 ) ↦ { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖 ⊆ 𝑗 } ) |
2 |
|
zarcls0.1 |
⊢ 𝑃 = ( PrmIdeal ‘ 𝑅 ) |
3 |
|
zarcls0.2 |
⊢ 0 = ( 0g ‘ 𝑅 ) |
4 |
1
|
a1i |
⊢ ( 𝑅 ∈ Ring → 𝑉 = ( 𝑖 ∈ ( LIdeal ‘ 𝑅 ) ↦ { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖 ⊆ 𝑗 } ) ) |
5 |
|
simplr |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑖 = { 0 } ) ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) → 𝑖 = { 0 } ) |
6 |
|
simpll |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑖 = { 0 } ) ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) → 𝑅 ∈ Ring ) |
7 |
|
prmidlidl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) → 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) |
8 |
6 7
|
sylancom |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑖 = { 0 } ) ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) → 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) |
9 |
|
eqid |
⊢ ( LIdeal ‘ 𝑅 ) = ( LIdeal ‘ 𝑅 ) |
10 |
9 3
|
lidl0cl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ) → 0 ∈ 𝑗 ) |
11 |
6 8 10
|
syl2anc |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑖 = { 0 } ) ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) → 0 ∈ 𝑗 ) |
12 |
11
|
snssd |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑖 = { 0 } ) ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) → { 0 } ⊆ 𝑗 ) |
13 |
5 12
|
eqsstrd |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑖 = { 0 } ) ∧ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ) → 𝑖 ⊆ 𝑗 ) |
14 |
13
|
ralrimiva |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑖 = { 0 } ) → ∀ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) 𝑖 ⊆ 𝑗 ) |
15 |
|
rabid2 |
⊢ ( ( PrmIdeal ‘ 𝑅 ) = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖 ⊆ 𝑗 } ↔ ∀ 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) 𝑖 ⊆ 𝑗 ) |
16 |
14 15
|
sylibr |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑖 = { 0 } ) → ( PrmIdeal ‘ 𝑅 ) = { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖 ⊆ 𝑗 } ) |
17 |
2 16
|
eqtr2id |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑖 = { 0 } ) → { 𝑗 ∈ ( PrmIdeal ‘ 𝑅 ) ∣ 𝑖 ⊆ 𝑗 } = 𝑃 ) |
18 |
9 3
|
lidl0 |
⊢ ( 𝑅 ∈ Ring → { 0 } ∈ ( LIdeal ‘ 𝑅 ) ) |
19 |
2
|
fvexi |
⊢ 𝑃 ∈ V |
20 |
19
|
a1i |
⊢ ( 𝑅 ∈ Ring → 𝑃 ∈ V ) |
21 |
4 17 18 20
|
fvmptd |
⊢ ( 𝑅 ∈ Ring → ( 𝑉 ‘ { 0 } ) = 𝑃 ) |