Metamath Proof Explorer


Theorem sub2cncf

Description: Subtraction from a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009) (Proof shortened by Mario Carneiro, 12-Sep-2015)

Ref Expression
Hypothesis sub2cncf.1 𝐹 = ( 𝑥 ∈ ℂ ↦ ( 𝐴𝑥 ) )
Assertion sub2cncf ( 𝐴 ∈ ℂ → 𝐹 ∈ ( ℂ –cn→ ℂ ) )

Proof

Step Hyp Ref Expression
1 sub2cncf.1 𝐹 = ( 𝑥 ∈ ℂ ↦ ( 𝐴𝑥 ) )
2 eqid ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld )
3 2 subcn − ∈ ( ( ( TopOpen ‘ ℂfld ) ×t ( TopOpen ‘ ℂfld ) ) Cn ( TopOpen ‘ ℂfld ) )
4 3 a1i ( 𝐴 ∈ ℂ → − ∈ ( ( ( TopOpen ‘ ℂfld ) ×t ( TopOpen ‘ ℂfld ) ) Cn ( TopOpen ‘ ℂfld ) ) )
5 ssid ℂ ⊆ ℂ
6 cncfmptc ( ( 𝐴 ∈ ℂ ∧ ℂ ⊆ ℂ ∧ ℂ ⊆ ℂ ) → ( 𝑥 ∈ ℂ ↦ 𝐴 ) ∈ ( ℂ –cn→ ℂ ) )
7 5 5 6 mp3an23 ( 𝐴 ∈ ℂ → ( 𝑥 ∈ ℂ ↦ 𝐴 ) ∈ ( ℂ –cn→ ℂ ) )
8 eqid ( 𝑥 ∈ ℂ ↦ 𝑥 ) = ( 𝑥 ∈ ℂ ↦ 𝑥 )
9 8 idcncf ( 𝑥 ∈ ℂ ↦ 𝑥 ) ∈ ( ℂ –cn→ ℂ )
10 9 a1i ( 𝐴 ∈ ℂ → ( 𝑥 ∈ ℂ ↦ 𝑥 ) ∈ ( ℂ –cn→ ℂ ) )
11 2 4 7 10 cncfmpt2f ( 𝐴 ∈ ℂ → ( 𝑥 ∈ ℂ ↦ ( 𝐴𝑥 ) ) ∈ ( ℂ –cn→ ℂ ) )
12 1 11 eqeltrid ( 𝐴 ∈ ℂ → 𝐹 ∈ ( ℂ –cn→ ℂ ) )