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Metamath Proof Explorer

Theorem aspsubrg

Description: The algebraic span of a set of vectors is a subring of the algebra. (Contributed by Mario Carneiro, 7-Jan-2015)

Ref Expression
Hypotheses aspval.a ⊢ 𝐴 = ( AlgSpan ‘ 𝑊 )
aspval.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
Assertion aspsubrg ( ( 𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉 ) → ( 𝐴 ‘ 𝑆 ) ∈ ( SubRing ‘ 𝑊 ) )

Proof

Step Hyp Ref Expression
1 aspval.a ⊢ 𝐴 = ( AlgSpan ‘ 𝑊 )
2 aspval.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
3 eqid ⊢ ( LSubSp ‘ 𝑊 ) = ( LSubSp ‘ 𝑊 )
4 1 2 3 aspval ⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉 ) → ( 𝐴 ‘ 𝑆 ) = ∩ { 𝑡 ∈ ( ( SubRing ‘ 𝑊 ) ∩ ( LSubSp ‘ 𝑊 ) ) ∣ 𝑆 ⊆ 𝑡 } )
5 ssrab2 ⊢ { 𝑡 ∈ ( ( SubRing ‘ 𝑊 ) ∩ ( LSubSp ‘ 𝑊 ) ) ∣ 𝑆 ⊆ 𝑡 } ⊆ ( ( SubRing ‘ 𝑊 ) ∩ ( LSubSp ‘ 𝑊 ) )
6 inss1 ⊢ ( ( SubRing ‘ 𝑊 ) ∩ ( LSubSp ‘ 𝑊 ) ) ⊆ ( SubRing ‘ 𝑊 )
7 5 6 sstri ⊢ { 𝑡 ∈ ( ( SubRing ‘ 𝑊 ) ∩ ( LSubSp ‘ 𝑊 ) ) ∣ 𝑆 ⊆ 𝑡 } ⊆ ( SubRing ‘ 𝑊 )
8 fvex ⊢ ( 𝐴 ‘ 𝑆 ) ∈ V
9 4 8 eqeltrrdi ⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉 ) → ∩ { 𝑡 ∈ ( ( SubRing ‘ 𝑊 ) ∩ ( LSubSp ‘ 𝑊 ) ) ∣ 𝑆 ⊆ 𝑡 } ∈ V )
10 intex ⊢ ( { 𝑡 ∈ ( ( SubRing ‘ 𝑊 ) ∩ ( LSubSp ‘ 𝑊 ) ) ∣ 𝑆 ⊆ 𝑡 } ≠ ∅ ↔ ∩ { 𝑡 ∈ ( ( SubRing ‘ 𝑊 ) ∩ ( LSubSp ‘ 𝑊 ) ) ∣ 𝑆 ⊆ 𝑡 } ∈ V )
11 9 10 sylibr ⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉 ) → { 𝑡 ∈ ( ( SubRing ‘ 𝑊 ) ∩ ( LSubSp ‘ 𝑊 ) ) ∣ 𝑆 ⊆ 𝑡 } ≠ ∅ )
12 subrgint ⊢ ( ( { 𝑡 ∈ ( ( SubRing ‘ 𝑊 ) ∩ ( LSubSp ‘ 𝑊 ) ) ∣ 𝑆 ⊆ 𝑡 } ⊆ ( SubRing ‘ 𝑊 ) ∧ { 𝑡 ∈ ( ( SubRing ‘ 𝑊 ) ∩ ( LSubSp ‘ 𝑊 ) ) ∣ 𝑆 ⊆ 𝑡 } ≠ ∅ ) → ∩ { 𝑡 ∈ ( ( SubRing ‘ 𝑊 ) ∩ ( LSubSp ‘ 𝑊 ) ) ∣ 𝑆 ⊆ 𝑡 } ∈ ( SubRing ‘ 𝑊 ) )
13 7 11 12 sylancr ⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉 ) → ∩ { 𝑡 ∈ ( ( SubRing ‘ 𝑊 ) ∩ ( LSubSp ‘ 𝑊 ) ) ∣ 𝑆 ⊆ 𝑡 } ∈ ( SubRing ‘ 𝑊 ) )
14 4 13 eqeltrd ⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉 ) → ( 𝐴 ‘ 𝑆 ) ∈ ( SubRing ‘ 𝑊 ) )