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Theorem hst1a 23682
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 23678 . 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 973 1  |-  ( S  e.  CHStates  ->  ( normh `  ( S `  ~H )
)  =  1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2674    C_ wss 3288   -->wf 5417   ` cfv 5421  (class class class)co 6048   0cc0 8954   1c1 8955   ~Hchil 22383    +h cva 22384    .ih csp 22386   normhcno 22387   CHcch 22393   _|_cort 22394    vH chj 22397   CHStateschst 22427
This theorem is referenced by:  hstnmoc  23687  hst1h  23691
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-hilex 22463
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-map 6987  df-sh 22670  df-ch 22685  df-hst 23676
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