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Metamath Proof Explorer

Theorem phlsca

Description: The ring of scalars of a constructed pre-Hilbert space. (Contributed by Mario Carneiro, 6-Oct-2013) (Revised by Mario Carneiro, 29-Aug-2015)

Ref Expression
Hypothesis phlfn.h ⊢ 𝐻 = ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝑇 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , , ⟩ } )
Assertion phlsca ( 𝑇 ∈ 𝑋 → 𝑇 = ( Scalar ‘ 𝐻 ) )

Proof

Step Hyp Ref Expression
1 phlfn.h ⊢ 𝐻 = ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝑇 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , , ⟩ } )
2 1 phlstr ⊢ 𝐻 Struct ⟨ 1 , 8 ⟩
3 scaid ⊢ Scalar = Slot ( Scalar ‘ ndx )
4 snsstp3 ⊢ { ⟨ ( Scalar ‘ ndx ) , 𝑇 ⟩ } ⊆ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝑇 ⟩ }
5 ssun1 ⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝑇 ⟩ } ⊆ ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝑇 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , , ⟩ } )
6 5 1 sseqtrri ⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝑇 ⟩ } ⊆ 𝐻
7 4 6 sstri ⊢ { ⟨ ( Scalar ‘ ndx ) , 𝑇 ⟩ } ⊆ 𝐻
8 2 3 7 strfv ⊢ ( 𝑇 ∈ 𝑋 → 𝑇 = ( Scalar ‘ 𝐻 ) )