HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  sticl Structured version   Visualization version   GIF version

Theorem sticl 32243
Description: [0, 1] closure of the value of a state. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
sticl (𝑆 ∈ States → (𝐴C → (𝑆𝐴) ∈ (0[,]1)))

Proof of Theorem sticl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isst 32241 . . 3 (𝑆 ∈ States ↔ (𝑆: C ⟶(0[,]1) ∧ (𝑆‘ ℋ) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))))
21simp1bi 1144 . 2 (𝑆 ∈ States → 𝑆: C ⟶(0[,]1))
3 ffvelcdm 7100 . . 3 ((𝑆: C ⟶(0[,]1) ∧ 𝐴C ) → (𝑆𝐴) ∈ (0[,]1))
43ex 412 . 2 (𝑆: C ⟶(0[,]1) → (𝐴C → (𝑆𝐴) ∈ (0[,]1)))
52, 4syl 17 1 (𝑆 ∈ States → (𝐴C → (𝑆𝐴) ∈ (0[,]1)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wcel 2105  wral 3058  wss 3962  wf 6558  cfv 6562  (class class class)co 7430  0cc0 11152  1c1 11153   + caddc 11155  [,]cicc 13386  chba 30947   C cch 30957  cort 30958   chj 30961  Statescst 30990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-hilex 31027
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-map 8866  df-sh 31235  df-ch 31249  df-st 32239
This theorem is referenced by:  stcl  32244  stge0  32252  stle1  32253
  Copyright terms: Public domain W3C validator