Step |
Hyp |
Ref |
Expression |
1 |
|
esumid.p |
⊢ Ⅎ 𝑘 𝜑 |
2 |
|
esumid.0 |
⊢ Ⅎ 𝑘 𝐴 |
3 |
|
esumid.1 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
4 |
|
esumid.2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ( 0 [,] +∞ ) ) |
5 |
|
esumid.3 |
⊢ ( 𝜑 → 𝐶 ∈ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) tsums ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) ) |
6 |
|
df-esum |
⊢ Σ* 𝑘 ∈ 𝐴 𝐵 = ∪ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) tsums ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) |
7 |
|
eqid |
⊢ ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) = ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) |
8 |
|
nfcv |
⊢ Ⅎ 𝑘 ( 0 [,] +∞ ) |
9 |
|
eqid |
⊢ ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) |
10 |
1 2 8 4 9
|
fmptdF |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ( 0 [,] +∞ ) ) |
11 |
7 3 10 5
|
xrge0tsmseq |
⊢ ( 𝜑 → 𝐶 = ∪ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) tsums ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) ) |
12 |
6 11
|
eqtr4id |
⊢ ( 𝜑 → Σ* 𝑘 ∈ 𝐴 𝐵 = 𝐶 ) |