Step |
Hyp |
Ref |
Expression |
1 |
|
esumf1o.0 |
⊢ Ⅎ 𝑛 𝜑 |
2 |
|
esumf1o.b |
⊢ Ⅎ 𝑛 𝐵 |
3 |
|
esumf1o.d |
⊢ Ⅎ 𝑘 𝐷 |
4 |
|
esumf1o.a |
⊢ Ⅎ 𝑛 𝐴 |
5 |
|
esumf1o.c |
⊢ Ⅎ 𝑛 𝐶 |
6 |
|
esumf1o.f |
⊢ Ⅎ 𝑛 𝐹 |
7 |
|
esumf1o.1 |
⊢ ( 𝑘 = 𝐺 → 𝐵 = 𝐷 ) |
8 |
|
esumf1o.2 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
9 |
|
esumf1o.3 |
⊢ ( 𝜑 → 𝐹 : 𝐶 –1-1-onto→ 𝐴 ) |
10 |
|
esumf1o.4 |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝐶 ) → ( 𝐹 ‘ 𝑛 ) = 𝐺 ) |
11 |
|
esumf1o.5 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ( 0 [,] +∞ ) ) |
12 |
|
xrge0base |
⊢ ( 0 [,] +∞ ) = ( Base ‘ ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) ) |
13 |
|
xrge0cmn |
⊢ ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) ∈ CMnd |
14 |
13
|
a1i |
⊢ ( 𝜑 → ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) ∈ CMnd ) |
15 |
|
xrge0tps |
⊢ ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) ∈ TopSp |
16 |
15
|
a1i |
⊢ ( 𝜑 → ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) ∈ TopSp ) |
17 |
11
|
fmpttd |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ( 0 [,] +∞ ) ) |
18 |
12 14 16 8 17 9
|
tsmsf1o |
⊢ ( 𝜑 → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) tsums ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) tsums ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝐹 ) ) ) |
19 |
|
f1of |
⊢ ( 𝐹 : 𝐶 –1-1-onto→ 𝐴 → 𝐹 : 𝐶 ⟶ 𝐴 ) |
20 |
9 19
|
syl |
⊢ ( 𝜑 → 𝐹 : 𝐶 ⟶ 𝐴 ) |
21 |
20
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝐶 ) → ( 𝐹 ‘ 𝑛 ) ∈ 𝐴 ) |
22 |
10 21
|
eqeltrrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝐶 ) → 𝐺 ∈ 𝐴 ) |
23 |
22
|
ex |
⊢ ( 𝜑 → ( 𝑛 ∈ 𝐶 → 𝐺 ∈ 𝐴 ) ) |
24 |
1 23
|
ralrimi |
⊢ ( 𝜑 → ∀ 𝑛 ∈ 𝐶 𝐺 ∈ 𝐴 ) |
25 |
5 6 20
|
feqmptdf |
⊢ ( 𝜑 → 𝐹 = ( 𝑛 ∈ 𝐶 ↦ ( 𝐹 ‘ 𝑛 ) ) ) |
26 |
1 10
|
mpteq2da |
⊢ ( 𝜑 → ( 𝑛 ∈ 𝐶 ↦ ( 𝐹 ‘ 𝑛 ) ) = ( 𝑛 ∈ 𝐶 ↦ 𝐺 ) ) |
27 |
25 26
|
eqtrd |
⊢ ( 𝜑 → 𝐹 = ( 𝑛 ∈ 𝐶 ↦ 𝐺 ) ) |
28 |
|
eqidd |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) |
29 |
2 3 5 4 1 24 27 28 7
|
fmptcof2 |
⊢ ( 𝜑 → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝐹 ) = ( 𝑛 ∈ 𝐶 ↦ 𝐷 ) ) |
30 |
29
|
oveq2d |
⊢ ( 𝜑 → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) tsums ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝐹 ) ) = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) tsums ( 𝑛 ∈ 𝐶 ↦ 𝐷 ) ) ) |
31 |
18 30
|
eqtrd |
⊢ ( 𝜑 → ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) tsums ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) = ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) tsums ( 𝑛 ∈ 𝐶 ↦ 𝐷 ) ) ) |
32 |
31
|
unieqd |
⊢ ( 𝜑 → ∪ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) tsums ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) = ∪ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) tsums ( 𝑛 ∈ 𝐶 ↦ 𝐷 ) ) ) |
33 |
|
df-esum |
⊢ Σ* 𝑘 ∈ 𝐴 𝐵 = ∪ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) tsums ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) |
34 |
|
df-esum |
⊢ Σ* 𝑛 ∈ 𝐶 𝐷 = ∪ ( ( ℝ*𝑠 ↾s ( 0 [,] +∞ ) ) tsums ( 𝑛 ∈ 𝐶 ↦ 𝐷 ) ) |
35 |
32 33 34
|
3eqtr4g |
⊢ ( 𝜑 → Σ* 𝑘 ∈ 𝐴 𝐵 = Σ* 𝑛 ∈ 𝐶 𝐷 ) |