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Theorem isst 23708
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 6098 . . . 4  |-  ( 0 [,] 1 )  e. 
_V
2 chex 22721 . . . 4  |-  CH  e.  _V
31, 2elmap 7034 . . 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 5719 . . . . 5  |-  ( f  =  S  ->  (
f `  ~H )  =  ( S `  ~H ) )
65eqeq1d 2443 . . . 4  |-  ( f  =  S  ->  (
( f `  ~H )  =  1  <->  ( S `  ~H )  =  1 ) )
7 fveq1 5719 . . . . . . 7  |-  ( f  =  S  ->  (
f `  ( x  vH  y ) )  =  ( S `  (
x  vH  y )
) )
8 fveq1 5719 . . . . . . . 8  |-  ( f  =  S  ->  (
f `  x )  =  ( S `  x ) )
9 fveq1 5719 . . . . . . . 8  |-  ( f  =  S  ->  (
f `  y )  =  ( S `  y ) )
108, 9oveq12d 6091 . . . . . . 7  |-  ( f  =  S  ->  (
( f `  x
)  +  ( f `
 y ) )  =  ( ( S `
 x )  +  ( S `  y
) ) )
117, 10eqeq12d 2449 . . . . . 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 2739 . . . 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 23706 . . 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 3086 . 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 1652    e. wcel 1725   A.wral 2697    C_ wss 3312   -->wf 5442   ` cfv 5446  (class class class)co 6073    ^m cmap 7010   0cc0 8982   1c1 8983    + caddc 8985   [,]cicc 10911   ~Hchil 22414   CHcch 22424   _|_cort 22425    vH chj 22428   Statescst 22457
This theorem is referenced by:  sticl  23710  sthil  23729  stj  23730  strlem3a  23747
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-hilex 22494
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-sh 22701  df-ch 22716  df-st 23706
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