Step |
Hyp |
Ref |
Expression |
1 |
|
sge0pnfmpt.k |
⊢ Ⅎ 𝑘 𝜑 |
2 |
|
sge0pnfmpt.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
3 |
|
sge0pnfmpt.b |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ( 0 [,] +∞ ) ) |
4 |
|
sge0pnfmpt.p |
⊢ ( 𝜑 → ∃ 𝑘 ∈ 𝐴 𝐵 = +∞ ) |
5 |
|
eqid |
⊢ ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) |
6 |
1 3 5
|
fmptdf |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ( 0 [,] +∞ ) ) |
7 |
|
eqcom |
⊢ ( 𝐵 = +∞ ↔ +∞ = 𝐵 ) |
8 |
7
|
rexbii |
⊢ ( ∃ 𝑘 ∈ 𝐴 𝐵 = +∞ ↔ ∃ 𝑘 ∈ 𝐴 +∞ = 𝐵 ) |
9 |
4 8
|
sylib |
⊢ ( 𝜑 → ∃ 𝑘 ∈ 𝐴 +∞ = 𝐵 ) |
10 |
|
pnfex |
⊢ +∞ ∈ V |
11 |
10
|
a1i |
⊢ ( 𝜑 → +∞ ∈ V ) |
12 |
5 9 11
|
elrnmptd |
⊢ ( 𝜑 → +∞ ∈ ran ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) |
13 |
2 6 12
|
sge0pnfval |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) = +∞ ) |