Theorem qqtopn

Description: The topology of the field of the rational numbers. (Contributed by Thierry Arnoux, 29-Aug-2020)

	Ref	Expression
Assertion	qqtopn	$$⊢\mathrm{TopOpen}\left({ℝ}_{\mathrm{fld}}\right){\upharpoonright }_{𝑡}\mathbb{Q}=\mathrm{TopOpen}\left({ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}\mathbb{Q}\right)$$

Proof

Step	Hyp	Ref	Expression
1		retopn	$$⊢\mathrm{topGen}\left(\mathrm{ran}\left(.\right)\right)=\mathrm{TopOpen}\left({ℝ}_{\mathrm{fld}}\right)$$
2	1	oveq1i	$$⊢\mathrm{topGen}\left(\mathrm{ran}\left(.\right)\right){\upharpoonright }_{𝑡}\mathbb{Q}=\mathrm{TopOpen}\left({ℝ}_{\mathrm{fld}}\right){\upharpoonright }_{𝑡}\mathbb{Q}$$
3		df-refld	$$⊢{ℝ}_{\mathrm{fld}}={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}ℝ$$
4	3	oveq1i	$$⊢{ℝ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}\mathbb{Q}=\left({ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}ℝ\right){\upharpoonright }_{𝑠}\mathbb{Q}$$
5		reex	$$⊢ℝ\in \mathrm{V}$$
6		qssre	$$⊢\mathbb{Q}\subseteq ℝ$$
7		ressabs	$$⊢\left(ℝ\in \mathrm{V}\wedge \mathbb{Q}\subseteq ℝ\right)\to \left({ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}ℝ\right){\upharpoonright }_{𝑠}\mathbb{Q}={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}\mathbb{Q}$$
8	5 6 7	mp2an	$$⊢\left({ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}ℝ\right){\upharpoonright }_{𝑠}\mathbb{Q}={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}\mathbb{Q}$$
9	4 8	eqtr2i	$$⊢{ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}\mathbb{Q}={ℝ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}\mathbb{Q}$$
10	9 1	resstopn	$$⊢\mathrm{topGen}\left(\mathrm{ran}\left(.\right)\right){\upharpoonright }_{𝑡}\mathbb{Q}=\mathrm{TopOpen}\left({ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}\mathbb{Q}\right)$$
11	2 10	eqtr3i	$$⊢\mathrm{TopOpen}\left({ℝ}_{\mathrm{fld}}\right){\upharpoonright }_{𝑡}\mathbb{Q}=\mathrm{TopOpen}\left({ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}\mathbb{Q}\right)$$