Theorem negcncf

Description: The negative function is continuous. (Contributed by Mario Carneiro, 30-Dec-2016)

	Ref	Expression
Hypothesis	negcncf.1	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ - 𝑥 )
Assertion	negcncf	⊢ ( 𝐴 ⊆ ℂ → 𝐹 ∈ ( 𝐴 –cn→ ℂ ) )

Proof

Step	Hyp	Ref	Expression
1		negcncf.1	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ - 𝑥 )
2		ssel2	⊢ ( ( 𝐴 ⊆ ℂ ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ ℂ )
3	2	mulm1d	⊢ ( ( 𝐴 ⊆ ℂ ∧ 𝑥 ∈ 𝐴 ) → ( - 1 · 𝑥 ) = - 𝑥 )
4	3	mpteq2dva	⊢ ( 𝐴 ⊆ ℂ → ( 𝑥 ∈ 𝐴 ↦ ( - 1 · 𝑥 ) ) = ( 𝑥 ∈ 𝐴 ↦ - 𝑥 ) )
5	4 1	eqtr4di	⊢ ( 𝐴 ⊆ ℂ → ( 𝑥 ∈ 𝐴 ↦ ( - 1 · 𝑥 ) ) = 𝐹 )
6		eqid	⊢ ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld )
7	6	mulcn	⊢ · ∈ ( ( ( TopOpen ‘ ℂfld ) ×t ( TopOpen ‘ ℂfld ) ) Cn ( TopOpen ‘ ℂfld ) )
8	7	a1i	⊢ ( 𝐴 ⊆ ℂ → · ∈ ( ( ( TopOpen ‘ ℂfld ) ×t ( TopOpen ‘ ℂfld ) ) Cn ( TopOpen ‘ ℂfld ) ) )
9		neg1cn	⊢ - 1 ∈ ℂ
10		ssid	⊢ ℂ ⊆ ℂ
11		cncfmptc	⊢ ( ( - 1 ∈ ℂ ∧ 𝐴 ⊆ ℂ ∧ ℂ ⊆ ℂ ) → ( 𝑥 ∈ 𝐴 ↦ - 1 ) ∈ ( 𝐴 –cn→ ℂ ) )
12	9 10 11	mp3an13	⊢ ( 𝐴 ⊆ ℂ → ( 𝑥 ∈ 𝐴 ↦ - 1 ) ∈ ( 𝐴 –cn→ ℂ ) )
13		cncfmptid	⊢ ( ( 𝐴 ⊆ ℂ ∧ ℂ ⊆ ℂ ) → ( 𝑥 ∈ 𝐴 ↦ 𝑥 ) ∈ ( 𝐴 –cn→ ℂ ) )
14	10 13	mpan2	⊢ ( 𝐴 ⊆ ℂ → ( 𝑥 ∈ 𝐴 ↦ 𝑥 ) ∈ ( 𝐴 –cn→ ℂ ) )
15	6 8 12 14	cncfmpt2f	⊢ ( 𝐴 ⊆ ℂ → ( 𝑥 ∈ 𝐴 ↦ ( - 1 · 𝑥 ) ) ∈ ( 𝐴 –cn→ ℂ ) )
16	5 15	eqeltrrd	⊢ ( 𝐴 ⊆ ℂ → 𝐹 ∈ ( 𝐴 –cn→ ℂ ) )