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Theorem hstcl 29921
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 ∈ CHStates ∧ 𝐴C ) → (𝑆𝐴) ∈ ℋ)

Proof of Theorem hstcl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ishst 29918 . . 3 (𝑆 ∈ CHStates ↔ (𝑆: C ⟶ ℋ ∧ (norm‘(𝑆‘ ℋ)) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦))))))
21simp1bi 1137 . 2 (𝑆 ∈ CHStates → 𝑆: C ⟶ ℋ)
32ffvelrnda 6843 1 ((𝑆 ∈ CHStates ∧ 𝐴C ) → (𝑆𝐴) ∈ ℋ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  wss 3933  wf 6344  cfv 6348  (class class class)co 7145  0cc0 10525  1c1 10526  chba 28623   + cva 28624   ·ih csp 28626  normcno 28627   C cch 28633  cort 28634   chj 28637  CHStateschst 28667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-hilex 28703
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8397  df-sh 28911  df-ch 28925  df-hst 29916
This theorem is referenced by:  hstnmoc  29927  hstle1  29930  hst1h  29931  hst0h  29932  hstpyth  29933  hstle  29934  hstles  29935  hstoh  29936  hstrlem6  29968
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