Theorem cnfldsrngbas

Description: The base set of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020)

	Ref	Expression
Hypothesis	cnfldsrngbas.r	$$⊢R={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}S$$
Assertion	cnfldsrngbas	$$⊢S\subseteq ℂ\to S={\mathrm{Base}}_R$$

Proof

Step	Hyp	Ref	Expression
1		cnfldsrngbas.r	$$⊢R={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}S$$
2		cnfldbas	$$⊢ℂ={\mathrm{Base}}_{{ℂ}_{\mathrm{fld}}}$$
3	1 2	ressbas2	$$⊢S\subseteq ℂ\to S={\mathrm{Base}}_R$$