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Theorem sticl 31963
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 31961 . . 3 (𝑆 ∈ States ↔ (𝑆: C ⟶(0[,]1) ∧ (𝑆‘ ℋ) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))))
21simp1bi 1142 . 2 (𝑆 ∈ States → 𝑆: C ⟶(0[,]1))
3 ffvelcdm 7074 . . 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 1533  wcel 2098  wral 3053  wss 3941  wf 6530  cfv 6534  (class class class)co 7402  0cc0 11107  1c1 11108   + caddc 11110  [,]cicc 13328  chba 30667   C cch 30677  cort 30678   chj 30681  Statescst 30710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-hilex 30747
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-map 8819  df-sh 30955  df-ch 30969  df-st 31959
This theorem is referenced by:  stcl  31964  stge0  31972  stle1  31973
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