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Metamath Proof Explorer

Theorem sscntz

Description: A centralizer expression for two sets elementwise commuting. (Contributed by Stefan O'Rear, 5-Sep-2015)

Ref Expression
Hypotheses cntzfval.b ⊢ 𝐵 = ( Base ‘ 𝑀 )
cntzfval.p ⊢ + = ( +g ‘ 𝑀 )
cntzfval.z ⊢ 𝑍 = ( Cntz ‘ 𝑀 )
Assertion sscntz ( ( 𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝐵 ) → ( 𝑆 ⊆ ( 𝑍 ‘ 𝑇 ) ↔ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑇 ( 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 cntzval ⊢ ( 𝑇 ⊆ 𝐵 → ( 𝑍 ‘ 𝑇 ) = { 𝑥 ∈ 𝐵 ∣ ∀ 𝑦 ∈ 𝑇 ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) } )
5 4 sseq2d ⊢ ( 𝑇 ⊆ 𝐵 → ( 𝑆 ⊆ ( 𝑍 ‘ 𝑇 ) ↔ 𝑆 ⊆ { 𝑥 ∈ 𝐵 ∣ ∀ 𝑦 ∈ 𝑇 ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) } ) )
6 ssrab ⊢ ( 𝑆 ⊆ { 𝑥 ∈ 𝐵 ∣ ∀ 𝑦 ∈ 𝑇 ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) } ↔ ( 𝑆 ⊆ 𝐵 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑇 ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) ) )
7 5 6 bitrdi ⊢ ( 𝑇 ⊆ 𝐵 → ( 𝑆 ⊆ ( 𝑍 ‘ 𝑇 ) ↔ ( 𝑆 ⊆ 𝐵 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑇 ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) ) ) )
8 ibar ⊢ ( 𝑆 ⊆ 𝐵 → ( ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑇 ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) ↔ ( 𝑆 ⊆ 𝐵 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑇 ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) ) ) )
9 8 bicomd ⊢ ( 𝑆 ⊆ 𝐵 → ( ( 𝑆 ⊆ 𝐵 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑇 ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) ) ↔ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑇 ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) ) )
10 7 9 sylan9bbr ⊢ ( ( 𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝐵 ) → ( 𝑆 ⊆ ( 𝑍 ‘ 𝑇 ) ↔ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑇 ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) ) )