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Theorem sticl 27744
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  e.  States  ->  ( A  e. 
CH  ->  ( S `  A )  e.  ( 0 [,] 1 ) ) )

Proof of Theorem sticl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isst 27742 . . 3  |-  ( S  e.  States 
<->  ( S : CH --> ( 0 [,] 1
)  /\  ( S `  ~H )  =  1  /\  A. x  e. 
CH  A. y  e.  CH  ( x  C_  ( _|_ `  y )  ->  ( S `  ( x  vH  y ) )  =  ( ( S `  x )  +  ( S `  y ) ) ) ) )
21simp1bi 1020 . 2  |-  ( S  e.  States  ->  S : CH --> ( 0 [,] 1
) )
3 ffvelrn 6026 . . 3  |-  ( ( S : CH --> ( 0 [,] 1 )  /\  A  e.  CH )  ->  ( S `  A
)  e.  ( 0 [,] 1 ) )
43ex 435 . 2  |-  ( S : CH --> ( 0 [,] 1 )  -> 
( A  e.  CH  ->  ( S `  A
)  e.  ( 0 [,] 1 ) ) )
52, 4syl 17 1  |-  ( S  e.  States  ->  ( A  e. 
CH  ->  ( S `  A )  e.  ( 0 [,] 1 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1867   A.wral 2773    C_ wss 3433   -->wf 5588   ` cfv 5592  (class class class)co 6296   0cc0 9528   1c1 9529    + caddc 9531   [,]cicc 11627   ~Hchil 26448   CHcch 26458   _|_cort 26459    vH chj 26462   Statescst 26491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-hilex 26528
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-map 7473  df-sh 26736  df-ch 26750  df-st 27740
This theorem is referenced by:  stcl  27745  stge0  27753  stle1  27754
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