Step |
Hyp |
Ref |
Expression |
1 |
|
mvmulfval.x |
⊢ × = ( 𝑅 maVecMul 〈 𝑀 , 𝑁 〉 ) |
2 |
|
mvmulfval.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
3 |
|
mvmulfval.t |
⊢ · = ( .r ‘ 𝑅 ) |
4 |
|
mvmulfval.r |
⊢ ( 𝜑 → 𝑅 ∈ 𝑉 ) |
5 |
|
mvmulfval.m |
⊢ ( 𝜑 → 𝑀 ∈ Fin ) |
6 |
|
mvmulfval.n |
⊢ ( 𝜑 → 𝑁 ∈ Fin ) |
7 |
|
mvmulval.x |
⊢ ( 𝜑 → 𝑋 ∈ ( 𝐵 ↑m ( 𝑀 × 𝑁 ) ) ) |
8 |
|
mvmulval.y |
⊢ ( 𝜑 → 𝑌 ∈ ( 𝐵 ↑m 𝑁 ) ) |
9 |
|
mvmulfv.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑀 ) |
10 |
1 2 3 4 5 6 7 8
|
mvmulval |
⊢ ( 𝜑 → ( 𝑋 × 𝑌 ) = ( 𝑖 ∈ 𝑀 ↦ ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑋 𝑗 ) · ( 𝑌 ‘ 𝑗 ) ) ) ) ) ) |
11 |
|
oveq1 |
⊢ ( 𝑖 = 𝐼 → ( 𝑖 𝑋 𝑗 ) = ( 𝐼 𝑋 𝑗 ) ) |
12 |
11
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 = 𝐼 ) → ( 𝑖 𝑋 𝑗 ) = ( 𝐼 𝑋 𝑗 ) ) |
13 |
12
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑖 = 𝐼 ) → ( ( 𝑖 𝑋 𝑗 ) · ( 𝑌 ‘ 𝑗 ) ) = ( ( 𝐼 𝑋 𝑗 ) · ( 𝑌 ‘ 𝑗 ) ) ) |
14 |
13
|
mpteq2dv |
⊢ ( ( 𝜑 ∧ 𝑖 = 𝐼 ) → ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑋 𝑗 ) · ( 𝑌 ‘ 𝑗 ) ) ) = ( 𝑗 ∈ 𝑁 ↦ ( ( 𝐼 𝑋 𝑗 ) · ( 𝑌 ‘ 𝑗 ) ) ) ) |
15 |
14
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑖 = 𝐼 ) → ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝑖 𝑋 𝑗 ) · ( 𝑌 ‘ 𝑗 ) ) ) ) = ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝐼 𝑋 𝑗 ) · ( 𝑌 ‘ 𝑗 ) ) ) ) ) |
16 |
|
ovexd |
⊢ ( 𝜑 → ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝐼 𝑋 𝑗 ) · ( 𝑌 ‘ 𝑗 ) ) ) ) ∈ V ) |
17 |
10 15 9 16
|
fvmptd |
⊢ ( 𝜑 → ( ( 𝑋 × 𝑌 ) ‘ 𝐼 ) = ( 𝑅 Σg ( 𝑗 ∈ 𝑁 ↦ ( ( 𝐼 𝑋 𝑗 ) · ( 𝑌 ‘ 𝑗 ) ) ) ) ) |