Step |
Hyp |
Ref |
Expression |
1 |
|
esumel.1 |
⊢ Ⅎ 𝑘 𝜑 |
2 |
|
esumel.2 |
⊢ Ⅎ 𝑘 𝐴 |
3 |
|
esumel.3 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
4 |
|
esumel.4 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ( 0 [,] +∞ ) ) |
5 |
4
|
ex |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 → 𝐵 ∈ ( 0 [,] +∞ ) ) ) |
6 |
1 5
|
ralrimi |
⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ( 0 [,] +∞ ) ) |
7 |
2
|
esumcl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ∀ 𝑘 ∈ 𝐴 𝐵 ∈ ( 0 [,] +∞ ) ) → Σ* 𝑘 ∈ 𝐴 𝐵 ∈ ( 0 [,] +∞ ) ) |
8 |
3 6 7
|
syl2anc |
⊢ ( 𝜑 → Σ* 𝑘 ∈ 𝐴 𝐵 ∈ ( 0 [,] +∞ ) ) |
9 |
|
snidg |
⊢ ( Σ* 𝑘 ∈ 𝐴 𝐵 ∈ ( 0 [,] +∞ ) → Σ* 𝑘 ∈ 𝐴 𝐵 ∈ { Σ* 𝑘 ∈ 𝐴 𝐵 } ) |
10 |
8 9
|
syl |
⊢ ( 𝜑 → Σ* 𝑘 ∈ 𝐴 𝐵 ∈ { Σ* 𝑘 ∈ 𝐴 𝐵 } ) |
11 |
|
eqid |
⊢ ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) = ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) |
12 |
|
nfcv |
⊢ Ⅎ 𝑘 ( 0 [,] +∞ ) |
13 |
|
eqid |
⊢ ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) |
14 |
1 2 12 4 13
|
fmptdF |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ( 0 [,] +∞ ) ) |
15 |
|
inss1 |
⊢ ( 𝒫 𝐴 ∩ Fin ) ⊆ 𝒫 𝐴 |
16 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) |
17 |
15 16
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → 𝑥 ∈ 𝒫 𝐴 ) |
18 |
17
|
elpwid |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → 𝑥 ⊆ 𝐴 ) |
19 |
|
nfcv |
⊢ Ⅎ 𝑘 𝑥 |
20 |
2 19
|
resmptf |
⊢ ( 𝑥 ⊆ 𝐴 → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ↾ 𝑥 ) = ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) |
21 |
18 20
|
syl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ↾ 𝑥 ) = ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) |
22 |
21
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) = ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ↾ 𝑥 ) ) |
23 |
22
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ) → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( 𝑘 ∈ 𝑥 ↦ 𝐵 ) ) = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ↾ 𝑥 ) ) ) |
24 |
1 2 3 4 23
|
esumval |
⊢ ( 𝜑 → Σ* 𝑘 ∈ 𝐴 𝐵 = sup ( ran ( 𝑥 ∈ ( 𝒫 𝐴 ∩ Fin ) ↦ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) Σg ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ↾ 𝑥 ) ) ) , ℝ* , < ) ) |
25 |
11 3 14 24
|
xrge0tsmsd |
⊢ ( 𝜑 → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) tsums ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) = { Σ* 𝑘 ∈ 𝐴 𝐵 } ) |
26 |
10 25
|
eleqtrrd |
⊢ ( 𝜑 → Σ* 𝑘 ∈ 𝐴 𝐵 ∈ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) tsums ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) ) |