Step |
Hyp |
Ref |
Expression |
1 |
|
dmatval.a |
⊢ 𝐴 = ( 𝑁 Mat 𝑅 ) |
2 |
|
dmatval.b |
⊢ 𝐵 = ( Base ‘ 𝐴 ) |
3 |
|
dmatval.0 |
⊢ 0 = ( 0g ‘ 𝑅 ) |
4 |
|
dmatval.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 |
5
|
eleq2d |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → ( 𝑀 ∈ 𝐷 ↔ 𝑀 ∈ { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ) ) |
7 |
|
oveq |
⊢ ( 𝑚 = 𝑀 → ( 𝑖 𝑚 𝑗 ) = ( 𝑖 𝑀 𝑗 ) ) |
8 |
7
|
eqeq1d |
⊢ ( 𝑚 = 𝑀 → ( ( 𝑖 𝑚 𝑗 ) = 0 ↔ ( 𝑖 𝑀 𝑗 ) = 0 ) ) |
9 |
8
|
imbi2d |
⊢ ( 𝑚 = 𝑀 → ( ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) ↔ ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑀 𝑗 ) = 0 ) ) ) |
10 |
9
|
2ralbidv |
⊢ ( 𝑚 = 𝑀 → ( ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) ↔ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑀 𝑗 ) = 0 ) ) ) |
11 |
10
|
elrab |
⊢ ( 𝑀 ∈ { 𝑚 ∈ 𝐵 ∣ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑚 𝑗 ) = 0 ) } ↔ ( 𝑀 ∈ 𝐵 ∧ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑀 𝑗 ) = 0 ) ) ) |
12 |
6 11
|
bitrdi |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ) → ( 𝑀 ∈ 𝐷 ↔ ( 𝑀 ∈ 𝐵 ∧ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑀 𝑗 ) = 0 ) ) ) ) |