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Metamath Proof Explorer

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 ) ) )