Step |
Hyp |
Ref |
Expression |
1 |
|
addccncf2.1 |
⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 + 𝑥 ) ) |
2 |
|
simpl |
⊢ ( ( 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ ) → 𝐴 ⊆ ℂ ) |
3 |
|
simpr |
⊢ ( ( 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ ) → 𝐵 ∈ ℂ ) |
4 |
|
ssidd |
⊢ ( ( 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ ) → ℂ ⊆ ℂ ) |
5 |
2 3 4
|
constcncfg |
⊢ ( ( 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ ( 𝐴 –cn→ ℂ ) ) |
6 |
|
ssid |
⊢ ℂ ⊆ ℂ |
7 |
|
cncfmptid |
⊢ ( ( 𝐴 ⊆ ℂ ∧ ℂ ⊆ ℂ ) → ( 𝑥 ∈ 𝐴 ↦ 𝑥 ) ∈ ( 𝐴 –cn→ ℂ ) ) |
8 |
6 7
|
mpan2 |
⊢ ( 𝐴 ⊆ ℂ → ( 𝑥 ∈ 𝐴 ↦ 𝑥 ) ∈ ( 𝐴 –cn→ ℂ ) ) |
9 |
8
|
adantr |
⊢ ( ( 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝑥 ∈ 𝐴 ↦ 𝑥 ) ∈ ( 𝐴 –cn→ ℂ ) ) |
10 |
5 9
|
addcncf |
⊢ ( ( 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 + 𝑥 ) ) ∈ ( 𝐴 –cn→ ℂ ) ) |
11 |
1 10
|
eqeltrid |
⊢ ( ( 𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ ) → 𝐹 ∈ ( 𝐴 –cn→ ℂ ) ) |