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Theorem hstcl 30009
 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 30006 . . 3 (𝑆 ∈ CHStates ↔ (𝑆: C ⟶ ℋ ∧ (norm‘(𝑆‘ ℋ)) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (((𝑆𝑥) ·ih (𝑆𝑦)) = 0 ∧ (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦))))))
21simp1bi 1142 . 2 (𝑆 ∈ CHStates → 𝑆: C ⟶ ℋ)
32ffvelrnda 6844 1 ((𝑆 ∈ CHStates ∧ 𝐴C ) → (𝑆𝐴) ∈ ℋ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2115  ∀wral 3133   ⊆ wss 3919  ⟶wf 6341  ‘cfv 6345  (class class class)co 7151  0cc0 10537  1c1 10538   ℋchba 28711   +ℎ cva 28712   ·ih csp 28714  normℎcno 28715   Cℋ cch 28721  ⊥cort 28722   ∨ℋ chj 28725  CHStateschst 28755 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457  ax-hilex 28791 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5054  df-opab 5116  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-fv 6353  df-ov 7154  df-oprab 7155  df-mpo 7156  df-map 8406  df-sh 28999  df-ch 29013  df-hst 30004 This theorem is referenced by:  hstnmoc  30015  hstle1  30018  hst1h  30019  hst0h  30020  hstpyth  30021  hstle  30022  hstles  30023  hstoh  30024  hstrlem6  30056
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