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Theorem hstcl 27429
Description: Closure of the value of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
Assertion
Ref Expression
hstcl  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  ( S `  A )  e.  ~H )

Proof of Theorem hstcl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ishst 27426 . . 3  |-  ( 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 ) ) ) ) ) )
21simp1bi 1012 . 2  |-  ( S  e.  CHStates  ->  S : CH --> ~H )
32ffvelrnda 5965 1  |-  ( ( S  e.  CHStates  /\  A  e.  CH )  ->  ( S `  A )  e.  ~H )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2753    C_ wss 3413   -->wf 5521   ` cfv 5525  (class class class)co 6234   0cc0 9442   1c1 9443   ~Hchil 26130    +h cva 26131    .ih csp 26133   normhcno 26134   CHcch 26140   _|_cort 26141    vH chj 26144   CHStateschst 26174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-hilex 26210
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-fv 5533  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-map 7379  df-sh 26418  df-ch 26433  df-hst 27424
This theorem is referenced by:  hstnmoc  27435  hstle1  27438  hst1h  27439  hst0h  27440  hstpyth  27441  hstle  27442  hstles  27443  hstoh  27444  hstrlem6  27476
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