HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  isst Structured version   Unicode version

Theorem isst 25615
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 6114 . . . 4  |-  ( 0 [,] 1 )  e. 
_V
2 chex 24627 . . . 4  |-  CH  e.  _V
31, 2elmap 7239 . . 3  |-  ( S  e.  ( ( 0 [,] 1 )  ^m  CH )  <->  S : CH --> ( 0 [,] 1 ) )
43anbi1i 695 . 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 5688 . . . . 5  |-  ( f  =  S  ->  (
f `  ~H )  =  ( S `  ~H ) )
65eqeq1d 2449 . . . 4  |-  ( f  =  S  ->  (
( f `  ~H )  =  1  <->  ( S `  ~H )  =  1 ) )
7 fveq1 5688 . . . . . . 7  |-  ( f  =  S  ->  (
f `  ( x  vH  y ) )  =  ( S `  (
x  vH  y )
) )
8 fveq1 5688 . . . . . . . 8  |-  ( f  =  S  ->  (
f `  x )  =  ( S `  x ) )
9 fveq1 5688 . . . . . . . 8  |-  ( f  =  S  ->  (
f `  y )  =  ( S `  y ) )
108, 9oveq12d 6107 . . . . . . 7  |-  ( f  =  S  ->  (
( f `  x
)  +  ( f `
 y ) )  =  ( ( S `
 x )  +  ( S `  y
) ) )
117, 10eqeq12d 2455 . . . . . 6  |-  ( f  =  S  ->  (
( f `  (
x  vH  y )
)  =  ( ( f `  x )  +  ( f `  y ) )  <->  ( S `  ( x  vH  y
) )  =  ( ( S `  x
)  +  ( S `
 y ) ) ) )
1211imbi2d 316 . . . . 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 2755 . . . 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 710 . . 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 25613 . . 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 3117 . 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 969 . 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 277 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2713    C_ wss 3326   -->wf 5412   ` cfv 5416  (class class class)co 6089    ^m cmap 7212   0cc0 9280   1c1 9281    + caddc 9283   [,]cicc 11301   ~Hchil 24319   CHcch 24329   _|_cort 24330    vH chj 24333   Statescst 24362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-hilex 24399
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-map 7214  df-sh 24607  df-ch 24622  df-st 25613
This theorem is referenced by:  sticl  25617  sthil  25636  stj  25637  strlem3a  25654
  Copyright terms: Public domain W3C validator