This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem phlip

Description: The inner product (Hermitian form) operation 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 phlip ( , ∈ 𝑋 → , = ( ·𝑖 ‘ 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 ipid ⊢ ·𝑖 = Slot ( ·𝑖 ‘ ndx )
4 snsspr2 ⊢ { ⟨ ( ·𝑖 ‘ ndx ) , , ⟩ } ⊆ { ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , , ⟩ }
5 ssun2 ⊢ { ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , , ⟩ } ⊆ ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝑇 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , , ⟩ } )
6 5 1 sseqtrri ⊢ { ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , , ⟩ } ⊆ 𝐻
7 4 6 sstri ⊢ { ⟨ ( ·𝑖 ‘ ndx ) , , ⟩ } ⊆ 𝐻
8 2 3 7 strfv ⊢ ( , ∈ 𝑋 → , = ( ·𝑖 ‘ 𝐻 ) )