Theorem 2zrngmul

Description: The ring multiplication operation of R is the multiplication on complex numbers. (Contributed by AV, 31-Jan-2020)

	Ref	Expression
Hypotheses	2zrng.e	$$⊢E=\left\{z\in \mathbb{Z}|\exists x\in \mathbb{Z}z=2x\right\}$$
	2zrngbas.r	$$⊢R={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}E$$
Assertion	2zrngmul	$$⊢\times ={\cdot }_R$$

Proof

Step	Hyp	Ref	Expression
1		2zrng.e	$$⊢E=\left\{z\in \mathbb{Z}|\exists x\in \mathbb{Z}z=2x\right\}$$
2		2zrngbas.r	$$⊢R={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}E$$
3		zex	$$⊢\mathbb{Z}\in \mathrm{V}$$
4	1 3	rabex2	$$⊢E\in \mathrm{V}$$
5	2	cnfldsrngmul	$$⊢E\in \mathrm{V}\to \times ={\cdot }_R$$
6	4 5	ax-mp	$$⊢\times ={\cdot }_R$$