Step |
Hyp |
Ref |
Expression |
1 |
|
iblidicc.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
iblidicc.b |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
3 |
|
iccssre |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 [,] 𝐵 ) ⊆ ℝ ) |
4 |
1 2 3
|
syl2anc |
⊢ ( 𝜑 → ( 𝐴 [,] 𝐵 ) ⊆ ℝ ) |
5 |
|
ax-resscn |
⊢ ℝ ⊆ ℂ |
6 |
4 5
|
sstrdi |
⊢ ( 𝜑 → ( 𝐴 [,] 𝐵 ) ⊆ ℂ ) |
7 |
|
ssid |
⊢ ℂ ⊆ ℂ |
8 |
|
cncfmptid |
⊢ ( ( ( 𝐴 [,] 𝐵 ) ⊆ ℂ ∧ ℂ ⊆ ℂ ) → ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ 𝑥 ) ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) |
9 |
6 7 8
|
sylancl |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ 𝑥 ) ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) |
10 |
|
cniccibl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ 𝑥 ) ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) → ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ 𝑥 ) ∈ 𝐿1 ) |
11 |
1 2 9 10
|
syl3anc |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ 𝑥 ) ∈ 𝐿1 ) |