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Metamath Proof Explorer

Theorem cntzidss

Description: If the elements of S commute, the elements of a subset T also commute. (Contributed by Mario Carneiro, 25-Apr-2016)

Ref Expression
Hypothesis cntzmhm.z ⊢ 𝑍 = ( Cntz ‘ 𝐺 )
Assertion cntzidss ( ( 𝑆 ⊆ ( 𝑍 ‘ 𝑆 ) ∧ 𝑇 ⊆ 𝑆 ) → 𝑇 ⊆ ( 𝑍 ‘ 𝑇 ) )

Proof

Step Hyp Ref Expression
1 cntzmhm.z ⊢ 𝑍 = ( Cntz ‘ 𝐺 )
2 simpr ⊢ ( ( 𝑆 ⊆ ( 𝑍 ‘ 𝑆 ) ∧ 𝑇 ⊆ 𝑆 ) → 𝑇 ⊆ 𝑆 )
3 simpl ⊢ ( ( 𝑆 ⊆ ( 𝑍 ‘ 𝑆 ) ∧ 𝑇 ⊆ 𝑆 ) → 𝑆 ⊆ ( 𝑍 ‘ 𝑆 ) )
4 eqid ⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 )
5 4 1 cntzssv ⊢ ( 𝑍 ‘ 𝑆 ) ⊆ ( Base ‘ 𝐺 )
6 3 5 sstrdi ⊢ ( ( 𝑆 ⊆ ( 𝑍 ‘ 𝑆 ) ∧ 𝑇 ⊆ 𝑆 ) → 𝑆 ⊆ ( Base ‘ 𝐺 ) )
7 4 1 cntz2ss ⊢ ( ( 𝑆 ⊆ ( Base ‘ 𝐺 ) ∧ 𝑇 ⊆ 𝑆 ) → ( 𝑍 ‘ 𝑆 ) ⊆ ( 𝑍 ‘ 𝑇 ) )
8 6 7 sylancom ⊢ ( ( 𝑆 ⊆ ( 𝑍 ‘ 𝑆 ) ∧ 𝑇 ⊆ 𝑆 ) → ( 𝑍 ‘ 𝑆 ) ⊆ ( 𝑍 ‘ 𝑇 ) )
9 3 8 sstrd ⊢ ( ( 𝑆 ⊆ ( 𝑍 ‘ 𝑆 ) ∧ 𝑇 ⊆ 𝑆 ) → 𝑆 ⊆ ( 𝑍 ‘ 𝑇 ) )
10 2 9 sstrd ⊢ ( ( 𝑆 ⊆ ( 𝑍 ‘ 𝑆 ) ∧ 𝑇 ⊆ 𝑆 ) → 𝑇 ⊆ ( 𝑍 ‘ 𝑇 ) )