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Theorem isst 22809
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 5899 . . . 4  |-  ( 0 [,] 1 )  e. 
_V
2 chex 21822 . . . 4  |-  CH  e.  _V
31, 2elmap 6812 . . 3  |-  ( S  e.  ( ( 0 [,] 1 )  ^m  CH )  <->  S : CH --> ( 0 [,] 1 ) )
43anbi1i 676 . 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 5540 . . . . 5  |-  ( f  =  S  ->  (
f `  ~H )  =  ( S `  ~H ) )
65eqeq1d 2304 . . . 4  |-  ( f  =  S  ->  (
( f `  ~H )  =  1  <->  ( S `  ~H )  =  1 ) )
7 fveq1 5540 . . . . . . 7  |-  ( f  =  S  ->  (
f `  ( x  vH  y ) )  =  ( S `  (
x  vH  y )
) )
8 fveq1 5540 . . . . . . . 8  |-  ( f  =  S  ->  (
f `  x )  =  ( S `  x ) )
9 fveq1 5540 . . . . . . . 8  |-  ( f  =  S  ->  (
f `  y )  =  ( S `  y ) )
108, 9oveq12d 5892 . . . . . . 7  |-  ( f  =  S  ->  (
( f `  x
)  +  ( f `
 y ) )  =  ( ( S `
 x )  +  ( S `  y
) ) )
117, 10eqeq12d 2310 . . . . . 6  |-  ( f  =  S  ->  (
( f `  (
x  vH  y )
)  =  ( ( f `  x )  +  ( f `  y ) )  <->  ( S `  ( x  vH  y
) )  =  ( ( S `  x
)  +  ( S `
 y ) ) ) )
1211imbi2d 307 . . . . 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 2598 . . . 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 691 . . 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 22807 . . 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 2938 . 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 938 . 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 268 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   0cc0 8753   1c1 8754    + caddc 8756   [,]cicc 10675   ~Hchil 21515   CHcch 21525   _|_cort 21526    vH chj 21529   Statescst 21558
This theorem is referenced by:  sticl  22811  sthil  22830  stj  22831  strlem3a  22848
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-hilex 21595
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-sh 21802  df-ch 21817  df-st 22807
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