Step |
Hyp |
Ref |
Expression |
1 |
|
df-sumge0 |
⊢ Σ^ = ( 𝑥 ∈ V ↦ if ( +∞ ∈ ran 𝑥 , +∞ , sup ( ran ( 𝑦 ∈ ( 𝒫 dom 𝑥 ∩ Fin ) ↦ Σ 𝑤 ∈ 𝑦 ( 𝑥 ‘ 𝑤 ) ) , ℝ* , < ) ) ) |
2 |
1
|
a1i |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) → Σ^ = ( 𝑥 ∈ V ↦ if ( +∞ ∈ ran 𝑥 , +∞ , sup ( ran ( 𝑦 ∈ ( 𝒫 dom 𝑥 ∩ Fin ) ↦ Σ 𝑤 ∈ 𝑦 ( 𝑥 ‘ 𝑤 ) ) , ℝ* , < ) ) ) ) |
3 |
|
rneq |
⊢ ( 𝑥 = 𝐹 → ran 𝑥 = ran 𝐹 ) |
4 |
3
|
eleq2d |
⊢ ( 𝑥 = 𝐹 → ( +∞ ∈ ran 𝑥 ↔ +∞ ∈ ran 𝐹 ) ) |
5 |
4
|
adantl |
⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) ∧ 𝑥 = 𝐹 ) → ( +∞ ∈ ran 𝑥 ↔ +∞ ∈ ran 𝐹 ) ) |
6 |
|
dmeq |
⊢ ( 𝑥 = 𝐹 → dom 𝑥 = dom 𝐹 ) |
7 |
6
|
adantl |
⊢ ( ( 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ∧ 𝑥 = 𝐹 ) → dom 𝑥 = dom 𝐹 ) |
8 |
|
fdm |
⊢ ( 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) → dom 𝐹 = 𝑋 ) |
9 |
8
|
adantr |
⊢ ( ( 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ∧ 𝑥 = 𝐹 ) → dom 𝐹 = 𝑋 ) |
10 |
7 9
|
eqtrd |
⊢ ( ( 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ∧ 𝑥 = 𝐹 ) → dom 𝑥 = 𝑋 ) |
11 |
10
|
pweqd |
⊢ ( ( 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ∧ 𝑥 = 𝐹 ) → 𝒫 dom 𝑥 = 𝒫 𝑋 ) |
12 |
11
|
ineq1d |
⊢ ( ( 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ∧ 𝑥 = 𝐹 ) → ( 𝒫 dom 𝑥 ∩ Fin ) = ( 𝒫 𝑋 ∩ Fin ) ) |
13 |
12
|
mpteq1d |
⊢ ( ( 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ∧ 𝑥 = 𝐹 ) → ( 𝑦 ∈ ( 𝒫 dom 𝑥 ∩ Fin ) ↦ Σ 𝑤 ∈ 𝑦 ( 𝑥 ‘ 𝑤 ) ) = ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑤 ∈ 𝑦 ( 𝑥 ‘ 𝑤 ) ) ) |
14 |
13
|
adantll |
⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) ∧ 𝑥 = 𝐹 ) → ( 𝑦 ∈ ( 𝒫 dom 𝑥 ∩ Fin ) ↦ Σ 𝑤 ∈ 𝑦 ( 𝑥 ‘ 𝑤 ) ) = ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑤 ∈ 𝑦 ( 𝑥 ‘ 𝑤 ) ) ) |
15 |
|
fveq1 |
⊢ ( 𝑥 = 𝐹 → ( 𝑥 ‘ 𝑤 ) = ( 𝐹 ‘ 𝑤 ) ) |
16 |
15
|
sumeq2sdv |
⊢ ( 𝑥 = 𝐹 → Σ 𝑤 ∈ 𝑦 ( 𝑥 ‘ 𝑤 ) = Σ 𝑤 ∈ 𝑦 ( 𝐹 ‘ 𝑤 ) ) |
17 |
16
|
mpteq2dv |
⊢ ( 𝑥 = 𝐹 → ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑤 ∈ 𝑦 ( 𝑥 ‘ 𝑤 ) ) = ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑤 ∈ 𝑦 ( 𝐹 ‘ 𝑤 ) ) ) |
18 |
17
|
adantl |
⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) ∧ 𝑥 = 𝐹 ) → ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑤 ∈ 𝑦 ( 𝑥 ‘ 𝑤 ) ) = ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑤 ∈ 𝑦 ( 𝐹 ‘ 𝑤 ) ) ) |
19 |
14 18
|
eqtrd |
⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) ∧ 𝑥 = 𝐹 ) → ( 𝑦 ∈ ( 𝒫 dom 𝑥 ∩ Fin ) ↦ Σ 𝑤 ∈ 𝑦 ( 𝑥 ‘ 𝑤 ) ) = ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑤 ∈ 𝑦 ( 𝐹 ‘ 𝑤 ) ) ) |
20 |
19
|
rneqd |
⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) ∧ 𝑥 = 𝐹 ) → ran ( 𝑦 ∈ ( 𝒫 dom 𝑥 ∩ Fin ) ↦ Σ 𝑤 ∈ 𝑦 ( 𝑥 ‘ 𝑤 ) ) = ran ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑤 ∈ 𝑦 ( 𝐹 ‘ 𝑤 ) ) ) |
21 |
20
|
supeq1d |
⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) ∧ 𝑥 = 𝐹 ) → sup ( ran ( 𝑦 ∈ ( 𝒫 dom 𝑥 ∩ Fin ) ↦ Σ 𝑤 ∈ 𝑦 ( 𝑥 ‘ 𝑤 ) ) , ℝ* , < ) = sup ( ran ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑤 ∈ 𝑦 ( 𝐹 ‘ 𝑤 ) ) , ℝ* , < ) ) |
22 |
5 21
|
ifbieq2d |
⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) ∧ 𝑥 = 𝐹 ) → if ( +∞ ∈ ran 𝑥 , +∞ , sup ( ran ( 𝑦 ∈ ( 𝒫 dom 𝑥 ∩ Fin ) ↦ Σ 𝑤 ∈ 𝑦 ( 𝑥 ‘ 𝑤 ) ) , ℝ* , < ) ) = if ( +∞ ∈ ran 𝐹 , +∞ , sup ( ran ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑤 ∈ 𝑦 ( 𝐹 ‘ 𝑤 ) ) , ℝ* , < ) ) ) |
23 |
|
simpr |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) → 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) |
24 |
|
simpl |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) → 𝑋 ∈ 𝑉 ) |
25 |
23 24
|
fexd |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) → 𝐹 ∈ V ) |
26 |
|
pnfxr |
⊢ +∞ ∈ ℝ* |
27 |
26
|
a1i |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) → +∞ ∈ ℝ* ) |
28 |
|
xrltso |
⊢ < Or ℝ* |
29 |
28
|
supex |
⊢ sup ( ran ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑤 ∈ 𝑦 ( 𝐹 ‘ 𝑤 ) ) , ℝ* , < ) ∈ V |
30 |
29
|
a1i |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) → sup ( ran ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑤 ∈ 𝑦 ( 𝐹 ‘ 𝑤 ) ) , ℝ* , < ) ∈ V ) |
31 |
27 30
|
ifexd |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) → if ( +∞ ∈ ran 𝐹 , +∞ , sup ( ran ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑤 ∈ 𝑦 ( 𝐹 ‘ 𝑤 ) ) , ℝ* , < ) ) ∈ V ) |
32 |
2 22 25 31
|
fvmptd |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) → ( Σ^ ‘ 𝐹 ) = if ( +∞ ∈ ran 𝐹 , +∞ , sup ( ran ( 𝑦 ∈ ( 𝒫 𝑋 ∩ Fin ) ↦ Σ 𝑤 ∈ 𝑦 ( 𝐹 ‘ 𝑤 ) ) , ℝ* , < ) ) ) |