Theorem sub1cncf

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

	Ref	Expression
Hypothesis	sub1cncf.1	$$⊢F=\left(x\in ℂ⟼x-A\right)$$
Assertion	sub1cncf	$$⊢A\in ℂ\to F:ℂ\underset{cn}{⟶}ℂ$$

Proof

Step	Hyp	Ref	Expression
1		sub1cncf.1	$$⊢F=\left(x\in ℂ⟼x-A\right)$$
2		eqid	$$⊢\mathrm{TopOpen}\left({ℂ}_{\mathrm{fld}}\right)=\mathrm{TopOpen}\left({ℂ}_{\mathrm{fld}}\right)$$
3	2	subcn	$$⊢-\in \left(\left(\mathrm{TopOpen}\left({ℂ}_{\mathrm{fld}}\right){\times }_t\mathrm{TopOpen}\left({ℂ}_{\mathrm{fld}}\right)\right)\mathrm{Cn}\mathrm{TopOpen}\left({ℂ}_{\mathrm{fld}}\right)\right)$$
4	3	a1i	$$⊢A\in ℂ\to -\in \left(\left(\mathrm{TopOpen}\left({ℂ}_{\mathrm{fld}}\right){\times }_t\mathrm{TopOpen}\left({ℂ}_{\mathrm{fld}}\right)\right)\mathrm{Cn}\mathrm{TopOpen}\left({ℂ}_{\mathrm{fld}}\right)\right)$$
5		eqid	$$⊢\left(x\in ℂ⟼x\right)=\left(x\in ℂ⟼x\right)$$
6	5	idcncf	$$⊢\left(x\in ℂ⟼x\right):ℂ\underset{cn}{⟶}ℂ$$
7	6	a1i	$$⊢A\in ℂ\to \left(x\in ℂ⟼x\right):ℂ\underset{cn}{⟶}ℂ$$
8		ssid	$$⊢ℂ\subseteq ℂ$$
9		cncfmptc	$$⊢\left(A\in ℂ\wedge ℂ\subseteq ℂ\wedge ℂ\subseteq ℂ\right)\to \left(x\in ℂ⟼A\right):ℂ\underset{cn}{⟶}ℂ$$
10	8 8 9	mp3an23	$$⊢A\in ℂ\to \left(x\in ℂ⟼A\right):ℂ\underset{cn}{⟶}ℂ$$
11	2 4 7 10	cncfmpt2f	$$⊢A\in ℂ\to \left(x\in ℂ⟼x-A\right):ℂ\underset{cn}{⟶}ℂ$$
12	1 11	eqeltrid	$$⊢A\in ℂ\to F:ℂ\underset{cn}{⟶}ℂ$$