Theorem cntzcmnss

Description: Any subset in a commutative monoid is a subset of its centralizer. (Contributed by AV, 12-Jan-2019)

	Ref	Expression
Hypotheses	cntzcmnss.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	cntzcmnss.z	⊢ 𝑍 = ( Cntz ‘ 𝐺 )
Assertion	cntzcmnss	⊢ ( ( 𝐺 ∈ CMnd ∧ 𝑆 ⊆ 𝐵 ) → 𝑆 ⊆ ( 𝑍 ‘ 𝑆 ) )

Proof

Step	Hyp	Ref	Expression
1		cntzcmnss.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		cntzcmnss.z	⊢ 𝑍 = ( Cntz ‘ 𝐺 )
3	1 2	cntzcmn	⊢ ( ( 𝐺 ∈ CMnd ∧ 𝑆 ⊆ 𝐵 ) → ( 𝑍 ‘ 𝑆 ) = 𝐵 )
4		sseq2	⊢ ( 𝐵 = ( 𝑍 ‘ 𝑆 ) → ( 𝑆 ⊆ 𝐵 ↔ 𝑆 ⊆ ( 𝑍 ‘ 𝑆 ) ) )
5	4	eqcoms	⊢ ( ( 𝑍 ‘ 𝑆 ) = 𝐵 → ( 𝑆 ⊆ 𝐵 ↔ 𝑆 ⊆ ( 𝑍 ‘ 𝑆 ) ) )
6	5	biimpd	⊢ ( ( 𝑍 ‘ 𝑆 ) = 𝐵 → ( 𝑆 ⊆ 𝐵 → 𝑆 ⊆ ( 𝑍 ‘ 𝑆 ) ) )
7	6	adantld	⊢ ( ( 𝑍 ‘ 𝑆 ) = 𝐵 → ( ( 𝐺 ∈ CMnd ∧ 𝑆 ⊆ 𝐵 ) → 𝑆 ⊆ ( 𝑍 ‘ 𝑆 ) ) )
8	3 7	mpcom	⊢ ( ( 𝐺 ∈ CMnd ∧ 𝑆 ⊆ 𝐵 ) → 𝑆 ⊆ ( 𝑍 ‘ 𝑆 ) )