Theorem hlhilsrng

Description: The star division ring for the final constructed Hilbert space is a division ring. (Contributed by NM, 21-Jun-2015)

	Ref	Expression
Hypotheses	hlhillvec.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	hlhillvec.u	⊢ 𝑈 = ( ( HLHil ‘ 𝐾 ) ‘ 𝑊 )
	hlhillvec.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	hlhildrng.r	⊢ 𝑅 = ( Scalar ‘ 𝑈 )
Assertion	hlhilsrng	⊢ ( 𝜑 → 𝑅 ∈ *-Ring )

Proof

Step	Hyp	Ref	Expression
1		hlhillvec.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		hlhillvec.u	⊢ 𝑈 = ( ( HLHil ‘ 𝐾 ) ‘ 𝑊 )
3		hlhillvec.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
4		hlhildrng.r	⊢ 𝑅 = ( Scalar ‘ 𝑈 )
5		eqid	⊢ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
6		eqid	⊢ ( Scalar ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( Scalar ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) )
7		eqid	⊢ ( Base ‘ ( Scalar ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) = ( Base ‘ ( Scalar ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) )
8		eqid	⊢ ( +g ‘ ( Scalar ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) = ( +g ‘ ( Scalar ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) )
9		eqid	⊢ ( .r ‘ ( Scalar ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) = ( .r ‘ ( Scalar ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) )
10		eqid	⊢ ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 ) = ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 )
11	1 2 3 4 5 6 7 8 9 10	hlhilsrnglem	⊢ ( 𝜑 → 𝑅 ∈ *-Ring )