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 ) ) )