Theorem cntzel

Description: Membership in a centralizer. (Contributed by Stefan O'Rear, 6-Sep-2015)

	Ref	Expression
Hypotheses	cntzfval.b	⊢ 𝐵 = ( Base ‘ 𝑀 )
	cntzfval.p	⊢ + = ( +g ‘ 𝑀 )
	cntzfval.z	⊢ 𝑍 = ( Cntz ‘ 𝑀 )
Assertion	cntzel	⊢ ( ( 𝑆 ⊆ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∈ ( 𝑍 ‘ 𝑆 ) ↔ ∀ 𝑦 ∈ 𝑆 ( 𝑋 + 𝑦 ) = ( 𝑦 + 𝑋 ) ) )

Proof

Step	Hyp	Ref	Expression
1		cntzfval.b	⊢ 𝐵 = ( Base ‘ 𝑀 )
2		cntzfval.p	⊢ + = ( +g ‘ 𝑀 )
3		cntzfval.z	⊢ 𝑍 = ( Cntz ‘ 𝑀 )
4	1 2 3	elcntz	⊢ ( 𝑆 ⊆ 𝐵 → ( 𝑋 ∈ ( 𝑍 ‘ 𝑆 ) ↔ ( 𝑋 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝑆 ( 𝑋 + 𝑦 ) = ( 𝑦 + 𝑋 ) ) ) )
5	4	baibd	⊢ ( ( 𝑆 ⊆ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∈ ( 𝑍 ‘ 𝑆 ) ↔ ∀ 𝑦 ∈ 𝑆 ( 𝑋 + 𝑦 ) = ( 𝑦 + 𝑋 ) ) )