Metamath Proof Explorer


Theorem hlhilsmul

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 hlhilslem.h 𝐻 = ( LHyp ‘ 𝐾 )
hlhilslem.e 𝐸 = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 )
hlhilslem.u 𝑈 = ( ( HLHil ‘ 𝐾 ) ‘ 𝑊 )
hlhilslem.r 𝑅 = ( Scalar ‘ 𝑈 )
hlhilslem.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
hlhilsmul.m · = ( .r𝐸 )
Assertion hlhilsmul ( 𝜑· = ( .r𝑅 ) )

Proof

Step Hyp Ref Expression
1 hlhilslem.h 𝐻 = ( LHyp ‘ 𝐾 )
2 hlhilslem.e 𝐸 = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 )
3 hlhilslem.u 𝑈 = ( ( HLHil ‘ 𝐾 ) ‘ 𝑊 )
4 hlhilslem.r 𝑅 = ( Scalar ‘ 𝑈 )
5 hlhilslem.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
6 hlhilsmul.m · = ( .r𝐸 )
7 df-mulr .r = Slot 3
8 3nn 3 ∈ ℕ
9 3lt4 3 < 4
10 1 2 3 4 5 7 8 9 6 hlhilslem ( 𝜑· = ( .r𝑅 ) )