Theorem cntrss

Description: The center is a subset of the base field. (Contributed by Thierry Arnoux, 21-Aug-2023)

	Ref	Expression
Hypothesis	cntrss.1	⊢ 𝐵 = ( Base ‘ 𝑀 )
Assertion	cntrss	⊢ ( Cntr ‘ 𝑀 ) ⊆ 𝐵

Proof

Step	Hyp	Ref	Expression
1		cntrss.1	⊢ 𝐵 = ( Base ‘ 𝑀 )
2		eqid	⊢ ( Cntz ‘ 𝑀 ) = ( Cntz ‘ 𝑀 )
3	1 2	cntrval	⊢ ( ( Cntz ‘ 𝑀 ) ‘ 𝐵 ) = ( Cntr ‘ 𝑀 )
4	1 2	cntzssv	⊢ ( ( Cntz ‘ 𝑀 ) ‘ 𝐵 ) ⊆ 𝐵
5	3 4	eqsstrri	⊢ ( Cntr ‘ 𝑀 ) ⊆ 𝐵