Step |
Hyp |
Ref |
Expression |
1 |
|
dvsubcncf.s |
⊢ ( 𝜑 → 𝑆 ∈ { ℝ , ℂ } ) |
2 |
|
dvsubcncf.f |
⊢ ( 𝜑 → 𝐹 : 𝑋 ⟶ ℂ ) |
3 |
|
dvsubcncf.g |
⊢ ( 𝜑 → 𝐺 : 𝑋 ⟶ ℂ ) |
4 |
|
dvsubcncf.fdv |
⊢ ( 𝜑 → ( 𝑆 D 𝐹 ) ∈ ( 𝑋 –cn→ ℂ ) ) |
5 |
|
dvsubcncf.gdv |
⊢ ( 𝜑 → ( 𝑆 D 𝐺 ) ∈ ( 𝑋 –cn→ ℂ ) ) |
6 |
|
cncff |
⊢ ( ( 𝑆 D 𝐹 ) ∈ ( 𝑋 –cn→ ℂ ) → ( 𝑆 D 𝐹 ) : 𝑋 ⟶ ℂ ) |
7 |
|
fdm |
⊢ ( ( 𝑆 D 𝐹 ) : 𝑋 ⟶ ℂ → dom ( 𝑆 D 𝐹 ) = 𝑋 ) |
8 |
4 6 7
|
3syl |
⊢ ( 𝜑 → dom ( 𝑆 D 𝐹 ) = 𝑋 ) |
9 |
|
cncff |
⊢ ( ( 𝑆 D 𝐺 ) ∈ ( 𝑋 –cn→ ℂ ) → ( 𝑆 D 𝐺 ) : 𝑋 ⟶ ℂ ) |
10 |
|
fdm |
⊢ ( ( 𝑆 D 𝐺 ) : 𝑋 ⟶ ℂ → dom ( 𝑆 D 𝐺 ) = 𝑋 ) |
11 |
5 9 10
|
3syl |
⊢ ( 𝜑 → dom ( 𝑆 D 𝐺 ) = 𝑋 ) |
12 |
1 2 3 8 11
|
dvsubf |
⊢ ( 𝜑 → ( 𝑆 D ( 𝐹 ∘f − 𝐺 ) ) = ( ( 𝑆 D 𝐹 ) ∘f − ( 𝑆 D 𝐺 ) ) ) |
13 |
4 5
|
subcncff |
⊢ ( 𝜑 → ( ( 𝑆 D 𝐹 ) ∘f − ( 𝑆 D 𝐺 ) ) ∈ ( 𝑋 –cn→ ℂ ) ) |
14 |
12 13
|
eqeltrd |
⊢ ( 𝜑 → ( 𝑆 D ( 𝐹 ∘f − 𝐺 ) ) ∈ ( 𝑋 –cn→ ℂ ) ) |