Step |
Hyp |
Ref |
Expression |
1 |
|
fsum2mul.1 |
⊢ ( 𝜑 → 𝐴 ∈ Fin ) |
2 |
|
fsum2mul.2 |
⊢ ( 𝜑 → 𝐵 ∈ Fin ) |
3 |
|
fsum2mul.3 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → 𝐶 ∈ ℂ ) |
4 |
|
fsum2mul.4 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → 𝐷 ∈ ℂ ) |
5 |
2 4
|
fsumcl |
⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐵 𝐷 ∈ ℂ ) |
6 |
1 5 3
|
fsummulc1 |
⊢ ( 𝜑 → ( Σ 𝑗 ∈ 𝐴 𝐶 · Σ 𝑘 ∈ 𝐵 𝐷 ) = Σ 𝑗 ∈ 𝐴 ( 𝐶 · Σ 𝑘 ∈ 𝐵 𝐷 ) ) |
7 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → 𝐵 ∈ Fin ) |
8 |
4
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑘 ∈ 𝐵 ) → 𝐷 ∈ ℂ ) |
9 |
7 3 8
|
fsummulc2 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → ( 𝐶 · Σ 𝑘 ∈ 𝐵 𝐷 ) = Σ 𝑘 ∈ 𝐵 ( 𝐶 · 𝐷 ) ) |
10 |
9
|
sumeq2dv |
⊢ ( 𝜑 → Σ 𝑗 ∈ 𝐴 ( 𝐶 · Σ 𝑘 ∈ 𝐵 𝐷 ) = Σ 𝑗 ∈ 𝐴 Σ 𝑘 ∈ 𝐵 ( 𝐶 · 𝐷 ) ) |
11 |
6 10
|
eqtr2d |
⊢ ( 𝜑 → Σ 𝑗 ∈ 𝐴 Σ 𝑘 ∈ 𝐵 ( 𝐶 · 𝐷 ) = ( Σ 𝑗 ∈ 𝐴 𝐶 · Σ 𝑘 ∈ 𝐵 𝐷 ) ) |