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Metamath Proof Explorer

Theorem pwstps

Description: A structure power of a topological space is a topological space. (Contributed by Mario Carneiro, 27-Aug-2015)

Ref Expression
Hypothesis pwstps.y ⊢ 𝑌 = ( 𝑅 ↑s 𝐼 )
Assertion pwstps ( ( 𝑅 ∈ TopSp ∧ 𝐼 ∈ 𝑉 ) → 𝑌 ∈ TopSp )

Proof

Step Hyp Ref Expression
1 pwstps.y ⊢ 𝑌 = ( 𝑅 ↑s 𝐼 )
2 eqid ⊢ ( Scalar ‘ 𝑅 ) = ( Scalar ‘ 𝑅 )
3 1 2 pwsval ⊢ ( ( 𝑅 ∈ TopSp ∧ 𝐼 ∈ 𝑉 ) → 𝑌 = ( ( Scalar ‘ 𝑅 ) Xs ( 𝐼 × { 𝑅 } ) ) )
4 eqid ⊢ ( ( Scalar ‘ 𝑅 ) Xs ( 𝐼 × { 𝑅 } ) ) = ( ( Scalar ‘ 𝑅 ) Xs ( 𝐼 × { 𝑅 } ) )
5 fvexd ⊢ ( ( 𝑅 ∈ TopSp ∧ 𝐼 ∈ 𝑉 ) → ( Scalar ‘ 𝑅 ) ∈ V )
6 simpr ⊢ ( ( 𝑅 ∈ TopSp ∧ 𝐼 ∈ 𝑉 ) → 𝐼 ∈ 𝑉 )
7 fconst6g ⊢ ( 𝑅 ∈ TopSp → ( 𝐼 × { 𝑅 } ) : 𝐼 ⟶ TopSp )
8 7 adantr ⊢ ( ( 𝑅 ∈ TopSp ∧ 𝐼 ∈ 𝑉 ) → ( 𝐼 × { 𝑅 } ) : 𝐼 ⟶ TopSp )
9 4 5 6 8 prdstps ⊢ ( ( 𝑅 ∈ TopSp ∧ 𝐼 ∈ 𝑉 ) → ( ( Scalar ‘ 𝑅 ) Xs ( 𝐼 × { 𝑅 } ) ) ∈ TopSp )
10 3 9 eqeltrd ⊢ ( ( 𝑅 ∈ TopSp ∧ 𝐼 ∈ 𝑉 ) → 𝑌 ∈ TopSp )