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 W H
Assertion hlhilnvl φ ˙ = * R

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 W H
6 fvex EDRing K W V
7 4 fvexi ˙ V
8 starvid 𝑟 = Slot * ndx
9 8 setsid EDRing K W V ˙ V ˙ = * EDRing K W sSet * ndx ˙
10 6 7 9 mp2an ˙ = * EDRing K W sSet * ndx ˙
11 eqid EDRing K W = EDRing K W
12 eqid EDRing K W sSet * ndx ˙ = EDRing K W sSet * ndx ˙
13 1 2 5 11 4 12 hlhilsca φ EDRing K W sSet * ndx ˙ = Scalar U
14 13 3 eqtr4di φ EDRing K W sSet * ndx ˙ = R
15 14 fveq2d φ * EDRing K W sSet * ndx ˙ = * R
16 10 15 syl5eq φ ˙ = * R