Metamath Proof Explorer


Theorem dvsubcncf

Description: A sufficient condition for the derivative of a product to be continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Hypotheses dvsubcncf.s ( 𝜑𝑆 ∈ { ℝ , ℂ } )
dvsubcncf.f ( 𝜑𝐹 : 𝑋 ⟶ ℂ )
dvsubcncf.g ( 𝜑𝐺 : 𝑋 ⟶ ℂ )
dvsubcncf.fdv ( 𝜑 → ( 𝑆 D 𝐹 ) ∈ ( 𝑋cn→ ℂ ) )
dvsubcncf.gdv ( 𝜑 → ( 𝑆 D 𝐺 ) ∈ ( 𝑋cn→ ℂ ) )
Assertion dvsubcncf ( 𝜑 → ( 𝑆 D ( 𝐹f𝐺 ) ) ∈ ( 𝑋cn→ ℂ ) )

Proof

Step Hyp Ref Expression
1 dvsubcncf.s ( 𝜑𝑆 ∈ { ℝ , ℂ } )
2 dvsubcncf.f ( 𝜑𝐹 : 𝑋 ⟶ ℂ )
3 dvsubcncf.g ( 𝜑𝐺 : 𝑋 ⟶ ℂ )
4 dvsubcncf.fdv ( 𝜑 → ( 𝑆 D 𝐹 ) ∈ ( 𝑋cn→ ℂ ) )
5 dvsubcncf.gdv ( 𝜑 → ( 𝑆 D 𝐺 ) ∈ ( 𝑋cn→ ℂ ) )
6 cncff ( ( 𝑆 D 𝐹 ) ∈ ( 𝑋cn→ 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 ) ∈ ( 𝑋cn→ 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 ( 𝐹f𝐺 ) ) = ( ( 𝑆 D 𝐹 ) ∘f − ( 𝑆 D 𝐺 ) ) )
13 4 5 subcncff ( 𝜑 → ( ( 𝑆 D 𝐹 ) ∘f − ( 𝑆 D 𝐺 ) ) ∈ ( 𝑋cn→ ℂ ) )
14 12 13 eqeltrd ( 𝜑 → ( 𝑆 D ( 𝐹f𝐺 ) ) ∈ ( 𝑋cn→ ℂ ) )