Step |
Hyp |
Ref |
Expression |
1 |
|
mxidlval.1 |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
2 |
1
|
mxidlval |
⊢ ( 𝑅 ∈ Ring → ( MaxIdeal ‘ 𝑅 ) = { 𝑖 ∈ ( LIdeal ‘ 𝑅 ) ∣ ( 𝑖 ≠ 𝐵 ∧ ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑖 ⊆ 𝑗 → ( 𝑗 = 𝑖 ∨ 𝑗 = 𝐵 ) ) ) } ) |
3 |
2
|
eleq2d |
⊢ ( 𝑅 ∈ Ring → ( 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ↔ 𝑀 ∈ { 𝑖 ∈ ( LIdeal ‘ 𝑅 ) ∣ ( 𝑖 ≠ 𝐵 ∧ ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑖 ⊆ 𝑗 → ( 𝑗 = 𝑖 ∨ 𝑗 = 𝐵 ) ) ) } ) ) |
4 |
|
neeq1 |
⊢ ( 𝑖 = 𝑀 → ( 𝑖 ≠ 𝐵 ↔ 𝑀 ≠ 𝐵 ) ) |
5 |
|
sseq1 |
⊢ ( 𝑖 = 𝑀 → ( 𝑖 ⊆ 𝑗 ↔ 𝑀 ⊆ 𝑗 ) ) |
6 |
|
eqeq2 |
⊢ ( 𝑖 = 𝑀 → ( 𝑗 = 𝑖 ↔ 𝑗 = 𝑀 ) ) |
7 |
6
|
orbi1d |
⊢ ( 𝑖 = 𝑀 → ( ( 𝑗 = 𝑖 ∨ 𝑗 = 𝐵 ) ↔ ( 𝑗 = 𝑀 ∨ 𝑗 = 𝐵 ) ) ) |
8 |
5 7
|
imbi12d |
⊢ ( 𝑖 = 𝑀 → ( ( 𝑖 ⊆ 𝑗 → ( 𝑗 = 𝑖 ∨ 𝑗 = 𝐵 ) ) ↔ ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = 𝐵 ) ) ) ) |
9 |
8
|
ralbidv |
⊢ ( 𝑖 = 𝑀 → ( ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑖 ⊆ 𝑗 → ( 𝑗 = 𝑖 ∨ 𝑗 = 𝐵 ) ) ↔ ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = 𝐵 ) ) ) ) |
10 |
4 9
|
anbi12d |
⊢ ( 𝑖 = 𝑀 → ( ( 𝑖 ≠ 𝐵 ∧ ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑖 ⊆ 𝑗 → ( 𝑗 = 𝑖 ∨ 𝑗 = 𝐵 ) ) ) ↔ ( 𝑀 ≠ 𝐵 ∧ ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = 𝐵 ) ) ) ) ) |
11 |
10
|
elrab |
⊢ ( 𝑀 ∈ { 𝑖 ∈ ( LIdeal ‘ 𝑅 ) ∣ ( 𝑖 ≠ 𝐵 ∧ ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑖 ⊆ 𝑗 → ( 𝑗 = 𝑖 ∨ 𝑗 = 𝐵 ) ) ) } ↔ ( 𝑀 ∈ ( LIdeal ‘ 𝑅 ) ∧ ( 𝑀 ≠ 𝐵 ∧ ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = 𝐵 ) ) ) ) ) |
12 |
|
3anass |
⊢ ( ( 𝑀 ∈ ( LIdeal ‘ 𝑅 ) ∧ 𝑀 ≠ 𝐵 ∧ ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = 𝐵 ) ) ) ↔ ( 𝑀 ∈ ( LIdeal ‘ 𝑅 ) ∧ ( 𝑀 ≠ 𝐵 ∧ ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = 𝐵 ) ) ) ) ) |
13 |
11 12
|
bitr4i |
⊢ ( 𝑀 ∈ { 𝑖 ∈ ( LIdeal ‘ 𝑅 ) ∣ ( 𝑖 ≠ 𝐵 ∧ ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑖 ⊆ 𝑗 → ( 𝑗 = 𝑖 ∨ 𝑗 = 𝐵 ) ) ) } ↔ ( 𝑀 ∈ ( LIdeal ‘ 𝑅 ) ∧ 𝑀 ≠ 𝐵 ∧ ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = 𝐵 ) ) ) ) |
14 |
3 13
|
bitrdi |
⊢ ( 𝑅 ∈ Ring → ( 𝑀 ∈ ( MaxIdeal ‘ 𝑅 ) ↔ ( 𝑀 ∈ ( LIdeal ‘ 𝑅 ) ∧ 𝑀 ≠ 𝐵 ∧ ∀ 𝑗 ∈ ( LIdeal ‘ 𝑅 ) ( 𝑀 ⊆ 𝑗 → ( 𝑗 = 𝑀 ∨ 𝑗 = 𝐵 ) ) ) ) ) |