Step |
Hyp |
Ref |
Expression |
1 |
|
dmatbas.a |
⊢ 𝐴 = ( 𝑁 Mat 𝑅 ) |
2 |
|
dmatbas.b |
⊢ 𝐵 = ( Base ‘ 𝐴 ) |
3 |
|
dmatbas.0 |
⊢ 0 = ( 0g ‘ 𝑅 ) |
4 |
|
dmatbas.d |
⊢ 𝐷 = ( 𝑁 DMat 𝑅 ) |
5 |
1 2 3 4
|
dmatval |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → 𝐷 = { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ) |
6 |
|
elex |
⊢ ( 𝑅 ∈ 𝑉 → 𝑅 ∈ V ) |
7 |
|
eqid |
⊢ ( 𝑁 DMatALT 𝑅 ) = ( 𝑁 DMatALT 𝑅 ) |
8 |
1 2 3 7
|
dmatALTbas |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) → ( Base ‘ ( 𝑁 DMatALT 𝑅 ) ) = { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ) |
9 |
6 8
|
sylan2 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → ( Base ‘ ( 𝑁 DMatALT 𝑅 ) ) = { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ) |
10 |
5 9
|
eqtr4d |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → 𝐷 = ( Base ‘ ( 𝑁 DMatALT 𝑅 ) ) ) |