Step |
Hyp |
Ref |
Expression |
1 |
|
fsumrp0cl.1 |
⊢ ( 𝜑 → 𝐴 ∈ Fin ) |
2 |
|
fsumrp0cl.2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ( 0 [,) +∞ ) ) |
3 |
|
rge0ssre |
⊢ ( 0 [,) +∞ ) ⊆ ℝ |
4 |
|
ax-resscn |
⊢ ℝ ⊆ ℂ |
5 |
3 4
|
sstri |
⊢ ( 0 [,) +∞ ) ⊆ ℂ |
6 |
5
|
a1i |
⊢ ( 𝜑 → ( 0 [,) +∞ ) ⊆ ℂ ) |
7 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 0 [,) +∞ ) ∧ 𝑦 ∈ ( 0 [,) +∞ ) ) ) → 𝑥 ∈ ( 0 [,) +∞ ) ) |
8 |
3 7
|
sselid |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 0 [,) +∞ ) ∧ 𝑦 ∈ ( 0 [,) +∞ ) ) ) → 𝑥 ∈ ℝ ) |
9 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 0 [,) +∞ ) ∧ 𝑦 ∈ ( 0 [,) +∞ ) ) ) → 𝑦 ∈ ( 0 [,) +∞ ) ) |
10 |
3 9
|
sselid |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 0 [,) +∞ ) ∧ 𝑦 ∈ ( 0 [,) +∞ ) ) ) → 𝑦 ∈ ℝ ) |
11 |
8 10
|
readdcld |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 0 [,) +∞ ) ∧ 𝑦 ∈ ( 0 [,) +∞ ) ) ) → ( 𝑥 + 𝑦 ) ∈ ℝ ) |
12 |
11
|
rexrd |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 0 [,) +∞ ) ∧ 𝑦 ∈ ( 0 [,) +∞ ) ) ) → ( 𝑥 + 𝑦 ) ∈ ℝ* ) |
13 |
|
0xr |
⊢ 0 ∈ ℝ* |
14 |
|
pnfxr |
⊢ +∞ ∈ ℝ* |
15 |
|
elico1 |
⊢ ( ( 0 ∈ ℝ* ∧ +∞ ∈ ℝ* ) → ( 𝑥 ∈ ( 0 [,) +∞ ) ↔ ( 𝑥 ∈ ℝ* ∧ 0 ≤ 𝑥 ∧ 𝑥 < +∞ ) ) ) |
16 |
13 14 15
|
mp2an |
⊢ ( 𝑥 ∈ ( 0 [,) +∞ ) ↔ ( 𝑥 ∈ ℝ* ∧ 0 ≤ 𝑥 ∧ 𝑥 < +∞ ) ) |
17 |
16
|
simp2bi |
⊢ ( 𝑥 ∈ ( 0 [,) +∞ ) → 0 ≤ 𝑥 ) |
18 |
7 17
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 0 [,) +∞ ) ∧ 𝑦 ∈ ( 0 [,) +∞ ) ) ) → 0 ≤ 𝑥 ) |
19 |
|
elico1 |
⊢ ( ( 0 ∈ ℝ* ∧ +∞ ∈ ℝ* ) → ( 𝑦 ∈ ( 0 [,) +∞ ) ↔ ( 𝑦 ∈ ℝ* ∧ 0 ≤ 𝑦 ∧ 𝑦 < +∞ ) ) ) |
20 |
13 14 19
|
mp2an |
⊢ ( 𝑦 ∈ ( 0 [,) +∞ ) ↔ ( 𝑦 ∈ ℝ* ∧ 0 ≤ 𝑦 ∧ 𝑦 < +∞ ) ) |
21 |
20
|
simp2bi |
⊢ ( 𝑦 ∈ ( 0 [,) +∞ ) → 0 ≤ 𝑦 ) |
22 |
9 21
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 0 [,) +∞ ) ∧ 𝑦 ∈ ( 0 [,) +∞ ) ) ) → 0 ≤ 𝑦 ) |
23 |
8 10 18 22
|
addge0d |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 0 [,) +∞ ) ∧ 𝑦 ∈ ( 0 [,) +∞ ) ) ) → 0 ≤ ( 𝑥 + 𝑦 ) ) |
24 |
|
ltpnf |
⊢ ( ( 𝑥 + 𝑦 ) ∈ ℝ → ( 𝑥 + 𝑦 ) < +∞ ) |
25 |
11 24
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 0 [,) +∞ ) ∧ 𝑦 ∈ ( 0 [,) +∞ ) ) ) → ( 𝑥 + 𝑦 ) < +∞ ) |
26 |
|
elico1 |
⊢ ( ( 0 ∈ ℝ* ∧ +∞ ∈ ℝ* ) → ( ( 𝑥 + 𝑦 ) ∈ ( 0 [,) +∞ ) ↔ ( ( 𝑥 + 𝑦 ) ∈ ℝ* ∧ 0 ≤ ( 𝑥 + 𝑦 ) ∧ ( 𝑥 + 𝑦 ) < +∞ ) ) ) |
27 |
13 14 26
|
mp2an |
⊢ ( ( 𝑥 + 𝑦 ) ∈ ( 0 [,) +∞ ) ↔ ( ( 𝑥 + 𝑦 ) ∈ ℝ* ∧ 0 ≤ ( 𝑥 + 𝑦 ) ∧ ( 𝑥 + 𝑦 ) < +∞ ) ) |
28 |
12 23 25 27
|
syl3anbrc |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 0 [,) +∞ ) ∧ 𝑦 ∈ ( 0 [,) +∞ ) ) ) → ( 𝑥 + 𝑦 ) ∈ ( 0 [,) +∞ ) ) |
29 |
|
0e0icopnf |
⊢ 0 ∈ ( 0 [,) +∞ ) |
30 |
29
|
a1i |
⊢ ( 𝜑 → 0 ∈ ( 0 [,) +∞ ) ) |
31 |
6 28 1 2 30
|
fsumcllem |
⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐵 ∈ ( 0 [,) +∞ ) ) |