Step |
Hyp |
Ref |
Expression |
1 |
|
esumcvg2.j |
⊢ 𝐽 = ( TopOpen ‘ ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) ) |
2 |
|
esumcvg2.a |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → 𝐴 ∈ ( 0 [,] +∞ ) ) |
3 |
|
esumcvg2.l |
⊢ ( 𝑘 = 𝑙 → 𝐴 = 𝐵 ) |
4 |
|
esumcvg2.m |
⊢ ( 𝑘 = 𝑚 → 𝐴 = 𝐶 ) |
5 |
4
|
cbvesumv |
⊢ Σ* 𝑘 ∈ ( 1 ... 𝑖 ) 𝐴 = Σ* 𝑚 ∈ ( 1 ... 𝑖 ) 𝐶 |
6 |
|
oveq2 |
⊢ ( 𝑖 = 𝑛 → ( 1 ... 𝑖 ) = ( 1 ... 𝑛 ) ) |
7 |
|
esumeq1 |
⊢ ( ( 1 ... 𝑖 ) = ( 1 ... 𝑛 ) → Σ* 𝑘 ∈ ( 1 ... 𝑖 ) 𝐴 = Σ* 𝑘 ∈ ( 1 ... 𝑛 ) 𝐴 ) |
8 |
6 7
|
syl |
⊢ ( 𝑖 = 𝑛 → Σ* 𝑘 ∈ ( 1 ... 𝑖 ) 𝐴 = Σ* 𝑘 ∈ ( 1 ... 𝑛 ) 𝐴 ) |
9 |
5 8
|
eqtr3id |
⊢ ( 𝑖 = 𝑛 → Σ* 𝑚 ∈ ( 1 ... 𝑖 ) 𝐶 = Σ* 𝑘 ∈ ( 1 ... 𝑛 ) 𝐴 ) |
10 |
9
|
cbvmptv |
⊢ ( 𝑖 ∈ ℕ ↦ Σ* 𝑚 ∈ ( 1 ... 𝑖 ) 𝐶 ) = ( 𝑛 ∈ ℕ ↦ Σ* 𝑘 ∈ ( 1 ... 𝑛 ) 𝐴 ) |
11 |
1 10 2 3
|
esumcvg |
⊢ ( 𝜑 → ( 𝑖 ∈ ℕ ↦ Σ* 𝑚 ∈ ( 1 ... 𝑖 ) 𝐶 ) ( ⇝𝑡 ‘ 𝐽 ) Σ* 𝑘 ∈ ℕ 𝐴 ) |
12 |
10 11
|
eqbrtrrid |
⊢ ( 𝜑 → ( 𝑛 ∈ ℕ ↦ Σ* 𝑘 ∈ ( 1 ... 𝑛 ) 𝐴 ) ( ⇝𝑡 ‘ 𝐽 ) Σ* 𝑘 ∈ ℕ 𝐴 ) |