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Definition df-st 22791
Description: Define the set of states on a Hilbert lattice. Definition of [Kalmbach] p. 266. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)
Assertion
Ref Expression
df-st  |-  States  =  {
f  e.  ( ( 0 [,] 1 )  ^m  CH )  |  ( ( f `  ~H )  =  1  /\  A. x  e.  CH  A. y  e.  CH  (
x  C_  ( _|_ `  y )  ->  (
f `  ( x  vH  y ) )  =  ( ( f `  x )  +  ( f `  y ) ) ) ) }
Distinct variable group:    x, f, y

Detailed syntax breakdown of Definition df-st
StepHypRef Expression
1 cst 21542 . 2  class  States
2 chil 21499 . . . . . 6  class  ~H
3 vf . . . . . . 7  set  f
43cv 1622 . . . . . 6  class  f
52, 4cfv 5255 . . . . 5  class  ( f `
 ~H )
6 c1 8738 . . . . 5  class  1
75, 6wceq 1623 . . . 4  wff  ( f `
 ~H )  =  1
8 vx . . . . . . . . 9  set  x
98cv 1622 . . . . . . . 8  class  x
10 vy . . . . . . . . . 10  set  y
1110cv 1622 . . . . . . . . 9  class  y
12 cort 21510 . . . . . . . . 9  class  _|_
1311, 12cfv 5255 . . . . . . . 8  class  ( _|_ `  y )
149, 13wss 3152 . . . . . . 7  wff  x  C_  ( _|_ `  y )
15 chj 21513 . . . . . . . . . 10  class  vH
169, 11, 15co 5858 . . . . . . . . 9  class  ( x  vH  y )
1716, 4cfv 5255 . . . . . . . 8  class  ( f `
 ( x  vH  y ) )
189, 4cfv 5255 . . . . . . . . 9  class  ( f `
 x )
1911, 4cfv 5255 . . . . . . . . 9  class  ( f `
 y )
20 caddc 8740 . . . . . . . . 9  class  +
2118, 19, 20co 5858 . . . . . . . 8  class  ( ( f `  x )  +  ( f `  y ) )
2217, 21wceq 1623 . . . . . . 7  wff  ( f `
 ( x  vH  y ) )  =  ( ( f `  x )  +  ( f `  y ) )
2314, 22wi 4 . . . . . 6  wff  ( x 
C_  ( _|_ `  y
)  ->  ( f `  ( x  vH  y
) )  =  ( ( f `  x
)  +  ( f `
 y ) ) )
24 cch 21509 . . . . . 6  class  CH
2523, 10, 24wral 2543 . . . . 5  wff  A. y  e.  CH  ( x  C_  ( _|_ `  y )  ->  ( f `  ( x  vH  y
) )  =  ( ( f `  x
)  +  ( f `
 y ) ) )
2625, 8, 24wral 2543 . . . 4  wff  A. x  e.  CH  A. y  e. 
CH  ( x  C_  ( _|_ `  y )  ->  ( f `  ( x  vH  y
) )  =  ( ( f `  x
)  +  ( f `
 y ) ) )
277, 26wa 358 . . 3  wff  ( ( f `  ~H )  =  1  /\  A. x  e.  CH  A. y  e.  CH  ( x  C_  ( _|_ `  y )  ->  ( f `  ( x  vH  y
) )  =  ( ( f `  x
)  +  ( f `
 y ) ) ) )
28 cc0 8737 . . . . 5  class  0
29 cicc 10659 . . . . 5  class  [,]
3028, 6, 29co 5858 . . . 4  class  ( 0 [,] 1 )
31 cmap 6772 . . . 4  class  ^m
3230, 24, 31co 5858 . . 3  class  ( ( 0 [,] 1 )  ^m  CH )
3327, 3, 32crab 2547 . 2  class  { f  e.  ( ( 0 [,] 1 )  ^m  CH )  |  ( ( f `  ~H )  =  1  /\  A. x  e.  CH  A. y  e.  CH  ( x  C_  ( _|_ `  y )  ->  ( f `  ( x  vH  y
) )  =  ( ( f `  x
)  +  ( f `
 y ) ) ) ) }
341, 33wceq 1623 1  wff  States  =  {
f  e.  ( ( 0 [,] 1 )  ^m  CH )  |  ( ( f `  ~H )  =  1  /\  A. x  e.  CH  A. y  e.  CH  (
x  C_  ( _|_ `  y )  ->  (
f `  ( x  vH  y ) )  =  ( ( f `  x )  +  ( f `  y ) ) ) ) }
Colors of variables: wff set class
This definition is referenced by:  isst  22793
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