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Theorem hst1a 23759
Description: Unit value of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
Assertion
Ref Expression
hst1a  |-  ( S  e.  CHStates  ->  ( normh `  ( S `  ~H )
)  =  1 )

Proof of Theorem hst1a
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ishst 23755 . 2  |-  ( S  e.  CHStates 
<->  ( S : CH --> ~H  /\  ( normh `  ( S `  ~H )
)  =  1  /\ 
A. x  e.  CH  A. y  e.  CH  (
x  C_  ( _|_ `  y )  ->  (
( ( S `  x )  .ih  ( S `  y )
)  =  0  /\  ( S `  (
x  vH  y )
)  =  ( ( S `  x )  +h  ( S `  y ) ) ) ) ) )
21simp2bi 974 1  |-  ( S  e.  CHStates  ->  ( normh `  ( S `  ~H )
)  =  1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1654    e. wcel 1728   A.wral 2712    C_ wss 3309   -->wf 5485   ` cfv 5489  (class class class)co 6117   0cc0 9028   1c1 9029   ~Hchil 22460    +h cva 22461    .ih csp 22463   normhcno 22464   CHcch 22470   _|_cort 22471    vH chj 22474   CHStateschst 22504
This theorem is referenced by:  hstnmoc  23764  hst1h  23768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736  ax-hilex 22540
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-sbc 3171  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-br 4244  df-opab 4298  df-id 4533  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-fv 5497  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-map 7056  df-sh 22747  df-ch 22762  df-hst 23753
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