Metamath Proof Explorer


Theorem hlhilnvl

Description: The involution operation of the star division ring for the final constructed Hilbert space. (Contributed by NM, 20-Jun-2015) (Revised by Mario Carneiro, 28-Jun-2015)

Ref Expression
Hypotheses hlhilnvl.h 𝐻 = ( LHyp ‘ 𝐾 )
hlhilnvl.u 𝑈 = ( ( HLHil ‘ 𝐾 ) ‘ 𝑊 )
hlhilnvl.r 𝑅 = ( Scalar ‘ 𝑈 )
hlhilnvl.i = ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 )
hlhilnvl.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
Assertion hlhilnvl ( 𝜑 = ( *𝑟𝑅 ) )

Proof

Step Hyp Ref Expression
1 hlhilnvl.h 𝐻 = ( LHyp ‘ 𝐾 )
2 hlhilnvl.u 𝑈 = ( ( HLHil ‘ 𝐾 ) ‘ 𝑊 )
3 hlhilnvl.r 𝑅 = ( Scalar ‘ 𝑈 )
4 hlhilnvl.i = ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 )
5 hlhilnvl.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
6 fvex ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ∈ V
7 4 fvexi ∈ V
8 starvid *𝑟 = Slot ( *𝑟 ‘ ndx )
9 8 setsid ( ( ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ∈ V ∧ ∈ V ) → = ( *𝑟 ‘ ( ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) sSet ⟨ ( *𝑟 ‘ ndx ) , ⟩ ) ) )
10 6 7 9 mp2an = ( *𝑟 ‘ ( ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) sSet ⟨ ( *𝑟 ‘ ndx ) , ⟩ ) )
11 eqid ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 )
12 eqid ( ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) sSet ⟨ ( *𝑟 ‘ ndx ) , ⟩ ) = ( ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) sSet ⟨ ( *𝑟 ‘ ndx ) , ⟩ )
13 1 2 5 11 4 12 hlhilsca ( 𝜑 → ( ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) sSet ⟨ ( *𝑟 ‘ ndx ) , ⟩ ) = ( Scalar ‘ 𝑈 ) )
14 13 3 eqtr4di ( 𝜑 → ( ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) sSet ⟨ ( *𝑟 ‘ ndx ) , ⟩ ) = 𝑅 )
15 14 fveq2d ( 𝜑 → ( *𝑟 ‘ ( ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) sSet ⟨ ( *𝑟 ‘ ndx ) , ⟩ ) ) = ( *𝑟𝑅 ) )
16 10 15 syl5eq ( 𝜑 = ( *𝑟𝑅 ) )