Theorem phlbase

Description: The base set 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	phlbase	⊢ ( 𝐵 ∈ 𝑋 → 𝐵 = ( Base ‘ 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		baseid	⊢ Base = Slot ( Base ‘ ndx )
4		snsstp1	⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ } ⊆ { ⟨ ( 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	⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ } ⊆ 𝐻
8	2 3 7	strfv	⊢ ( 𝐵 ∈ 𝑋 → 𝐵 = ( Base ‘ 𝐻 ) )