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Theorem sticl 29984
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 29982 . . 3 (𝑆 ∈ States ↔ (𝑆: C ⟶(0[,]1) ∧ (𝑆‘ ℋ) = 1 ∧ ∀𝑥C𝑦C (𝑥 ⊆ (⊥‘𝑦) → (𝑆‘(𝑥 𝑦)) = ((𝑆𝑥) + (𝑆𝑦)))))
21simp1bi 1140 . 2 (𝑆 ∈ States → 𝑆: C ⟶(0[,]1))
3 ffvelrn 6842 . . 3 ((𝑆: C ⟶(0[,]1) ∧ 𝐴C ) → (𝑆𝐴) ∈ (0[,]1))
43ex 415 . 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 1531  wcel 2108  wral 3136  wss 3934  wf 6344  cfv 6348  (class class class)co 7148  0cc0 10529  1c1 10530   + caddc 10532  [,]cicc 12733  chba 28688   C cch 28698  cort 28699   chj 28702  Statescst 28731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-hilex 28768
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  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 7151  df-oprab 7152  df-mpo 7153  df-map 8400  df-sh 28976  df-ch 28990  df-st 29980
This theorem is referenced by:  stcl  29985  stge0  29993  stle1  29994
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