Theorem hlpr

Description: The scalar field of a subcomplex Hilbert space is either RR or CC . (Contributed by Mario Carneiro, 15-Oct-2015)

	Ref	Expression
Hypotheses	hlress.f	$$⊢F=\mathrm{Scalar}\left(W\right)$$
	hlress.k	$$⊢K={\mathrm{Base}}_F$$
Assertion	hlpr	$$⊢W\in \mathrm{ℂHil}\to K\in \left\{ℝ,ℂ\right\}$$

Proof

Step	Hyp	Ref	Expression
1		hlress.f	$$⊢F=\mathrm{Scalar}\left(W\right)$$
2		hlress.k	$$⊢K={\mathrm{Base}}_F$$
3	1 2	hlprlem	$$⊢W\in \mathrm{ℂHil}\to \left(K\in \mathrm{SubRing}\left({ℂ}_{\mathrm{fld}}\right)\wedge {ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}K\in \mathrm{DivRing}\wedge {ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}K\in \mathrm{CMetSp}\right)$$
4		eqid	$$⊢{ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}K={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}K$$
5	4	cncdrg	$$⊢\left(K\in \mathrm{SubRing}\left({ℂ}_{\mathrm{fld}}\right)\wedge {ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}K\in \mathrm{DivRing}\wedge {ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}K\in \mathrm{CMetSp}\right)\to K\in \left\{ℝ,ℂ\right\}$$
6	3 5	syl	$$⊢W\in \mathrm{ℂHil}\to K\in \left\{ℝ,ℂ\right\}$$