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Metamath Proof Explorer

Theorem dvcl

Description: The derivative function takes values in the complex numbers. (Contributed by Mario Carneiro, 7-Aug-2014) (Revised by Mario Carneiro, 9-Feb-2015)

Ref Expression
Hypotheses dvcl.s ⊢ ( 𝜑 → 𝑆 ⊆ ℂ )
dvcl.f ⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ ℂ )
dvcl.a ⊢ ( 𝜑 → 𝐴 ⊆ 𝑆 )
Assertion dvcl ( ( 𝜑 ∧ 𝐵 ( 𝑆 D 𝐹 ) 𝐶 ) → 𝐶 ∈ ℂ )

Proof

Step Hyp Ref Expression
1 dvcl.s ⊢ ( 𝜑 → 𝑆 ⊆ ℂ )
2 dvcl.f ⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ ℂ )
3 dvcl.a ⊢ ( 𝜑 → 𝐴 ⊆ 𝑆 )
4 limccl ⊢ ( ( 𝑧 ∈ ( 𝐴 ∖ { 𝐵 } ) ↦ ( ( ( 𝐹 ‘ 𝑧 ) − ( 𝐹 ‘ 𝐵 ) ) / ( 𝑧 − 𝐵 ) ) ) limℂ 𝐵 ) ⊆ ℂ
5 eqid ⊢ ( ( TopOpen ‘ ℂfld ) ↾t 𝑆 ) = ( ( TopOpen ‘ ℂfld ) ↾t 𝑆 )
6 eqid ⊢ ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld )
7 eqid ⊢ ( 𝑧 ∈ ( 𝐴 ∖ { 𝐵 } ) ↦ ( ( ( 𝐹 ‘ 𝑧 ) − ( 𝐹 ‘ 𝐵 ) ) / ( 𝑧 − 𝐵 ) ) ) = ( 𝑧 ∈ ( 𝐴 ∖ { 𝐵 } ) ↦ ( ( ( 𝐹 ‘ 𝑧 ) − ( 𝐹 ‘ 𝐵 ) ) / ( 𝑧 − 𝐵 ) ) )
8 5 6 7 1 2 3 eldv ⊢ ( 𝜑 → ( 𝐵 ( 𝑆 D 𝐹 ) 𝐶 ↔ ( 𝐵 ∈ ( ( int ‘ ( ( TopOpen ‘ ℂfld ) ↾t 𝑆 ) ) ‘ 𝐴 ) ∧ 𝐶 ∈ ( ( 𝑧 ∈ ( 𝐴 ∖ { 𝐵 } ) ↦ ( ( ( 𝐹 ‘ 𝑧 ) − ( 𝐹 ‘ 𝐵 ) ) / ( 𝑧 − 𝐵 ) ) ) limℂ 𝐵 ) ) ) )
9 8 simplbda ⊢ ( ( 𝜑 ∧ 𝐵 ( 𝑆 D 𝐹 ) 𝐶 ) → 𝐶 ∈ ( ( 𝑧 ∈ ( 𝐴 ∖ { 𝐵 } ) ↦ ( ( ( 𝐹 ‘ 𝑧 ) − ( 𝐹 ‘ 𝐵 ) ) / ( 𝑧 − 𝐵 ) ) ) limℂ 𝐵 ) )
10 4 9 sselid ⊢ ( ( 𝜑 ∧ 𝐵 ( 𝑆 D 𝐹 ) 𝐶 ) → 𝐶 ∈ ℂ )