Theorem cnfldsrngmul

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

	Ref	Expression
Hypothesis	cnfldsrngbas.r	⊢ 𝑅 = ( ℂfld ↾s 𝑆 )
Assertion	cnfldsrngmul	⊢ ( 𝑆 ∈ 𝑉 → · = ( .r ‘ 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		cnfldsrngbas.r	⊢ 𝑅 = ( ℂfld ↾s 𝑆 )
2		cnfldmul	⊢ · = ( .r ‘ ℂfld )
3	1 2	ressmulr	⊢ ( 𝑆 ∈ 𝑉 → · = ( .r ‘ 𝑅 ) )