Step |
Hyp |
Ref |
Expression |
1 |
|
dmatALTval.a |
⊢ 𝐴 = ( 𝑁 Mat 𝑅 ) |
2 |
|
dmatALTval.b |
⊢ 𝐵 = ( Base ‘ 𝐴 ) |
3 |
|
dmatALTval.0 |
⊢ 0 = ( 0g ‘ 𝑅 ) |
4 |
|
dmatALTval.d |
⊢ 𝐷 = ( 𝑁 DMatALT 𝑅 ) |
5 |
1 2 3 4
|
dmatALTval |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) → 𝐷 = ( 𝐴 ↾s { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ) ) |
6 |
5
|
fveq2d |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) → ( Base ‘ 𝐷 ) = ( Base ‘ ( 𝐴 ↾s { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ) ) ) |
7 |
2
|
fvexi |
⊢ 𝐵 ∈ V |
8 |
|
rabexg |
⊢ ( 𝐵 ∈ V → { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ∈ V ) |
9 |
7 8
|
mp1i |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) → { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ∈ V ) |
10 |
|
eqid |
⊢ ( 𝐴 ↾s { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ) = ( 𝐴 ↾s { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ) |
11 |
10 2
|
ressbas |
⊢ ( { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ∈ V → ( { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ∩ 𝐵 ) = ( Base ‘ ( 𝐴 ↾s { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ) ) ) |
12 |
9 11
|
syl |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) → ( { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ∩ 𝐵 ) = ( Base ‘ ( 𝐴 ↾s { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ) ) ) |
13 |
|
inrab2 |
⊢ ( { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ∩ 𝐵 ) = { 𝑚 ∈ ( 𝐵 ∩ 𝐵 ) ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } |
14 |
|
inidm |
⊢ ( 𝐵 ∩ 𝐵 ) = 𝐵 |
15 |
|
rabeq |
⊢ ( ( 𝐵 ∩ 𝐵 ) = 𝐵 → { 𝑚 ∈ ( 𝐵 ∩ 𝐵 ) ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } = { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ) |
16 |
14 15
|
mp1i |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) → { 𝑚 ∈ ( 𝐵 ∩ 𝐵 ) ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } = { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ) |
17 |
13 16
|
syl5eq |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) → ( { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ∩ 𝐵 ) = { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ) |
18 |
6 12 17
|
3eqtr2d |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) → ( Base ‘ 𝐷 ) = { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ) |