Theorem elcntz

Description: Elementhood in the centralizer. (Contributed by Mario Carneiro, 22-Sep-2015)

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

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	cntzval	⊢ ( 𝑆 ⊆ 𝐵 → ( 𝑍 ‘ 𝑆 ) = { 𝑥 ∈ 𝐵 ∣ ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) } )
5	4	eleq2d	⊢ ( 𝑆 ⊆ 𝐵 → ( 𝐴 ∈ ( 𝑍 ‘ 𝑆 ) ↔ 𝐴 ∈ { 𝑥 ∈ 𝐵 ∣ ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) } ) )
6		oveq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 + 𝑦 ) = ( 𝐴 + 𝑦 ) )
7		oveq2	⊢ ( 𝑥 = 𝐴 → ( 𝑦 + 𝑥 ) = ( 𝑦 + 𝐴 ) )
8	6 7	eqeq12d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) ↔ ( 𝐴 + 𝑦 ) = ( 𝑦 + 𝐴 ) ) )
9	8	ralbidv	⊢ ( 𝑥 = 𝐴 → ( ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) ↔ ∀ 𝑦 ∈ 𝑆 ( 𝐴 + 𝑦 ) = ( 𝑦 + 𝐴 ) ) )
10	9	elrab	⊢ ( 𝐴 ∈ { 𝑥 ∈ 𝐵 ∣ ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) } ↔ ( 𝐴 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝑆 ( 𝐴 + 𝑦 ) = ( 𝑦 + 𝐴 ) ) )
11	5 10	bitrdi	⊢ ( 𝑆 ⊆ 𝐵 → ( 𝐴 ∈ ( 𝑍 ‘ 𝑆 ) ↔ ( 𝐴 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝑆 ( 𝐴 + 𝑦 ) = ( 𝑦 + 𝐴 ) ) ) )