Theorem ipsbase

Description: The base set of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014) (Revised by Mario Carneiro, 29-Aug-2015) (Revised by Thierry Arnoux, 16-Jun-2019)

	Ref	Expression
Hypothesis	ipspart.a	⊢ 𝐴 = ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , × ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , 𝐼 ⟩ } )
Assertion	ipsbase	⊢ ( 𝐵 ∈ 𝑉 → 𝐵 = ( Base ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		ipspart.a	⊢ 𝐴 = ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , × ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , 𝐼 ⟩ } )
2	1	ipsstr	⊢ 𝐴 Struct ⟨ 1 , 8 ⟩
3		baseid	⊢ Base = Slot ( Base ‘ ndx )
4		snsstp1	⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ } ⊆ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , × ⟩ }
5		ssun1	⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , × ⟩ } ⊆ ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , × ⟩ } ∪ { ⟨ ( Scalar ‘ ndx ) , 𝑆 ⟩ , ⟨ ( ·𝑠 ‘ ndx ) , · ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , 𝐼 ⟩ } )
6	5 1	sseqtrri	⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , × ⟩ } ⊆ 𝐴
7	4 6	sstri	⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ } ⊆ 𝐴
8	2 3 7	strfv	⊢ ( 𝐵 ∈ 𝑉 → 𝐵 = ( Base ‘ 𝐴 ) )