Theorem hlhildrng

Description: The star division ring for the final constructed Hilbert space is a division ring. (Contributed by NM, 20-Jun-2015) (Revised by Mario Carneiro, 28-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	hlhildrng	⊢ ( 𝜑 → 𝑅 ∈ DivRing )

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	⊢ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 )
6	1 5	erngdv	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ∈ DivRing )
7	3 6	syl	⊢ ( 𝜑 → ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ∈ DivRing )
8		eqidd	⊢ ( 𝜑 → ( Base ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) = ( Base ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) )
9		eqid	⊢ ( Base ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) = ( Base ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) )
10	1 5 2 4 3 9	hlhilsbase	⊢ ( 𝜑 → ( Base ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) = ( Base ‘ 𝑅 ) )
11		eqid	⊢ ( +g ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) = ( +g ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) )
12	1 5 2 4 3 11	hlhilsplus	⊢ ( 𝜑 → ( +g ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) = ( +g ‘ 𝑅 ) )
13	12	oveqdr	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( Base ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) → ( 𝑥 ( +g ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) )
14		eqid	⊢ ( .r ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) = ( .r ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) )
15	1 5 2 4 3 14	hlhilsmul	⊢ ( 𝜑 → ( .r ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) = ( .r ‘ 𝑅 ) )
16	15	oveqdr	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) ∧ 𝑦 ∈ ( Base ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) ) ) → ( 𝑥 ( .r ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) 𝑦 ) = ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) )
17	8 10 13 16	drngpropd	⊢ ( 𝜑 → ( ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ∈ DivRing ↔ 𝑅 ∈ DivRing ) )
18	7 17	mpbid	⊢ ( 𝜑 → 𝑅 ∈ DivRing )