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Theorem isst 23557
Description: Property of a state. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
isst  |-  ( 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 ) ) ) ) )
Distinct variable group:    x, y, S

Proof of Theorem isst
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 ovex 6038 . . . 4  |-  ( 0 [,] 1 )  e. 
_V
2 chex 22570 . . . 4  |-  CH  e.  _V
31, 2elmap 6971 . . 3  |-  ( S  e.  ( ( 0 [,] 1 )  ^m  CH )  <->  S : CH --> ( 0 [,] 1 ) )
43anbi1i 677 . 2  |-  ( ( S  e.  ( ( 0 [,] 1 )  ^m  CH )  /\  ( ( S `  ~H )  =  1  /\  A. x  e.  CH  A. y  e.  CH  (
x  C_  ( _|_ `  y )  ->  ( S `  ( x  vH  y ) )  =  ( ( S `  x )  +  ( S `  y ) ) ) ) )  <-> 
( 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 ) ) ) ) ) )
5 fveq1 5660 . . . . 5  |-  ( f  =  S  ->  (
f `  ~H )  =  ( S `  ~H ) )
65eqeq1d 2388 . . . 4  |-  ( f  =  S  ->  (
( f `  ~H )  =  1  <->  ( S `  ~H )  =  1 ) )
7 fveq1 5660 . . . . . . 7  |-  ( f  =  S  ->  (
f `  ( x  vH  y ) )  =  ( S `  (
x  vH  y )
) )
8 fveq1 5660 . . . . . . . 8  |-  ( f  =  S  ->  (
f `  x )  =  ( S `  x ) )
9 fveq1 5660 . . . . . . . 8  |-  ( f  =  S  ->  (
f `  y )  =  ( S `  y ) )
108, 9oveq12d 6031 . . . . . . 7  |-  ( f  =  S  ->  (
( f `  x
)  +  ( f `
 y ) )  =  ( ( S `
 x )  +  ( S `  y
) ) )
117, 10eqeq12d 2394 . . . . . 6  |-  ( f  =  S  ->  (
( f `  (
x  vH  y )
)  =  ( ( f `  x )  +  ( f `  y ) )  <->  ( S `  ( x  vH  y
) )  =  ( ( S `  x
)  +  ( S `
 y ) ) ) )
1211imbi2d 308 . . . . 5  |-  ( f  =  S  ->  (
( x  C_  ( _|_ `  y )  -> 
( f `  (
x  vH  y )
)  =  ( ( f `  x )  +  ( f `  y ) ) )  <-> 
( x  C_  ( _|_ `  y )  -> 
( S `  (
x  vH  y )
)  =  ( ( S `  x )  +  ( S `  y ) ) ) ) )
13122ralbidv 2684 . . . 4  |-  ( f  =  S  ->  ( A. x  e.  CH  A. y  e.  CH  ( x 
C_  ( _|_ `  y
)  ->  ( f `  ( x  vH  y
) )  =  ( ( f `  x
)  +  ( f `
 y ) ) )  <->  A. x  e.  CH  A. y  e.  CH  (
x  C_  ( _|_ `  y )  ->  ( S `  ( x  vH  y ) )  =  ( ( S `  x )  +  ( S `  y ) ) ) ) )
146, 13anbi12d 692 . . 3  |-  ( f  =  S  ->  (
( ( f `  ~H )  =  1  /\  A. x  e.  CH  A. y  e.  CH  (
x  C_  ( _|_ `  y )  ->  (
f `  ( x  vH  y ) )  =  ( ( f `  x )  +  ( f `  y ) ) ) )  <->  ( ( S `  ~H )  =  1  /\  A. x  e.  CH  A. y  e.  CH  ( x  C_  ( _|_ `  y )  ->  ( S `  ( x  vH  y
) )  =  ( ( S `  x
)  +  ( S `
 y ) ) ) ) ) )
15 df-st 23555 . . 3  |-  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 ) ) ) ) }
1614, 15elrab2 3030 . 2  |-  ( S  e.  States 
<->  ( S  e.  ( ( 0 [,] 1
)  ^m  CH )  /\  ( ( S `  ~H )  =  1  /\  A. x  e.  CH  A. y  e.  CH  (
x  C_  ( _|_ `  y )  ->  ( S `  ( x  vH  y ) )  =  ( ( S `  x )  +  ( S `  y ) ) ) ) ) )
17 3anass 940 . 2  |-  ( ( 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 ) ) ) )  <->  ( 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 ) ) ) ) ) )
184, 16, 173bitr4i 269 1  |-  ( 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 ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2642    C_ wss 3256   -->wf 5383   ` cfv 5387  (class class class)co 6013    ^m cmap 6947   0cc0 8916   1c1 8917    + caddc 8919   [,]cicc 10844   ~Hchil 22263   CHcch 22273   _|_cort 22274    vH chj 22277   Statescst 22306
This theorem is referenced by:  sticl  23559  sthil  23578  stj  23579  strlem3a  23596
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-hilex 22343
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-map 6949  df-sh 22550  df-ch 22565  df-st 23555
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