Theorem hlhilsca

Description: The scalar of the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015) (Revised by Mario Carneiro, 28-Jun-2015)

	Ref	Expression
Hypotheses	hlhilbase.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	hlhilbase.u	⊢ 𝑈 = ( ( HLHil ‘ 𝐾 ) ‘ 𝑊 )
	hlhilbase.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	hlhilsca.e	⊢ 𝐸 = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 )
	hlhilsca.g	⊢ 𝐺 = ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 )
	hlhilsca.r	⊢ 𝑅 = ( 𝐸 sSet ⟨ ( *𝑟 ‘ ndx ) , 𝐺 ⟩ )
Assertion	hlhilsca	⊢ ( 𝜑 → 𝑅 = ( Scalar ‘ 𝑈 ) )

Proof

Step	Hyp	Ref	Expression
1		hlhilbase.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		hlhilbase.u	⊢ 𝑈 = ( ( HLHil ‘ 𝐾 ) ‘ 𝑊 )
3		hlhilbase.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
4		hlhilsca.e	⊢ 𝐸 = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 )
5		hlhilsca.g	⊢ 𝐺 = ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 )
6		hlhilsca.r	⊢ 𝑅 = ( 𝐸 sSet ⟨ ( *𝑟 ‘ ndx ) , 𝐺 ⟩ )
7		ovex	⊢ ( 𝐸 sSet ⟨ ( *𝑟 ‘ ndx ) , 𝐺 ⟩ ) ∈ V
8	6 7	eqeltri	⊢ 𝑅 ∈ V
9		eqid	⊢ ( { ⟨ ( Base ‘ ndx ) , ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( ·𝑠 ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) , 𝑦 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ( ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑦 ) ‘ 𝑥 ) ) ⟩ } ) = ( { ⟨ ( Base ‘ ndx ) , ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( ·𝑠 ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) , 𝑦 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ( ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑦 ) ‘ 𝑥 ) ) ⟩ } )
10	9	phlsca	⊢ ( 𝑅 ∈ V → 𝑅 = ( Scalar ‘ ( { ⟨ ( Base ‘ ndx ) , ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( ·𝑠 ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) , 𝑦 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ( ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑦 ) ‘ 𝑥 ) ) ⟩ } ) ) )
11	8 10	ax-mp	⊢ 𝑅 = ( Scalar ‘ ( { ⟨ ( Base ‘ ndx ) , ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( ·𝑠 ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) , 𝑦 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ( ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑦 ) ‘ 𝑥 ) ) ⟩ } ) )
12		eqid	⊢ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
13		eqid	⊢ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) )
14		eqid	⊢ ( +g ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( +g ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) )
15		eqid	⊢ ( ·𝑠 ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ·𝑠 ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) )
16		eqid	⊢ ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
17		eqid	⊢ ( 𝑥 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) , 𝑦 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ( ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑦 ) ‘ 𝑥 ) ) = ( 𝑥 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) , 𝑦 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ( ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑦 ) ‘ 𝑥 ) )
18	1 2 12 13 14 4 5 6 15 16 17 3	hlhilset	⊢ ( 𝜑 → 𝑈 = ( { ⟨ ( Base ‘ ndx ) , ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( ·𝑠 ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) , 𝑦 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ( ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑦 ) ‘ 𝑥 ) ) ⟩ } ) )
19	18	fveq2d	⊢ ( 𝜑 → ( Scalar ‘ 𝑈 ) = ( Scalar ‘ ( { ⟨ ( Base ‘ ndx ) , ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ , ⟨ ( +g ‘ ndx ) , ( +g ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ , ⟨ ( Scalar ‘ ndx ) , 𝑅 ⟩ } ∪ { ⟨ ( ·𝑠 ‘ ndx ) , ( ·𝑠 ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ , ⟨ ( ·𝑖 ‘ ndx ) , ( 𝑥 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) , 𝑦 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ↦ ( ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑦 ) ‘ 𝑥 ) ) ⟩ } ) ) )
20	11 19	eqtr4id	⊢ ( 𝜑 → 𝑅 = ( Scalar ‘ 𝑈 ) )