Theorem hlhilsmul2

Description: Scalar multiplication for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015) (Revised by Mario Carneiro, 28-Jun-2015)

	Ref	Expression
Hypotheses	hlhilsbase.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	hlhilsbase.l	⊢ 𝐿 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	hlhilsbase.s	⊢ 𝑆 = ( Scalar ‘ 𝐿 )
	hlhilsbase.u	⊢ 𝑈 = ( ( HLHil ‘ 𝐾 ) ‘ 𝑊 )
	hlhilsbase.r	⊢ 𝑅 = ( Scalar ‘ 𝑈 )
	hlhilsbase.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	hlhilsmul2.m	⊢ · = ( .r ‘ 𝑆 )
Assertion	hlhilsmul2	⊢ ( 𝜑 → · = ( .r ‘ 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		hlhilsbase.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		hlhilsbase.l	⊢ 𝐿 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		hlhilsbase.s	⊢ 𝑆 = ( Scalar ‘ 𝐿 )
4		hlhilsbase.u	⊢ 𝑈 = ( ( HLHil ‘ 𝐾 ) ‘ 𝑊 )
5		hlhilsbase.r	⊢ 𝑅 = ( Scalar ‘ 𝑈 )
6		hlhilsbase.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
7		hlhilsmul2.m	⊢ · = ( .r ‘ 𝑆 )
8		eqid	⊢ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 )
9	1 8 2 3	dvhsca	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑆 = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) )
10	6 9	syl	⊢ ( 𝜑 → 𝑆 = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) )
11	10	fveq2d	⊢ ( 𝜑 → ( .r ‘ 𝑆 ) = ( .r ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) )
12	7 11	eqtrid	⊢ ( 𝜑 → · = ( .r ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) )
13		eqid	⊢ ( .r ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) = ( .r ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) )
14	1 8 4 5 6 13	hlhilsmul	⊢ ( 𝜑 → ( .r ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) = ( .r ‘ 𝑅 ) )
15	12 14	eqtrd	⊢ ( 𝜑 → · = ( .r ‘ 𝑅 ) )