This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) ) ) )