ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pitonn Unicode version

Theorem pitonn 6982
Description: Mapping from  N. to  NN. (Contributed by Jim Kingdon, 22-Apr-2020.)
Assertion
Ref Expression
pitonn  |-  ( N  e.  N.  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  |^| { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1 )  e.  x ) } )
Distinct variable groups:    N, l, u   
y, l, u    x, y
Allowed substitution hints:    N( x, y)

Proof of Theorem pitonn
Dummy variables  w  z  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opeq1 3577 . . . . . . . . . . . . . . 15  |-  ( w  =  1o  ->  <. w ,  1o >.  =  <. 1o ,  1o >. )
21eceq1d 6173 . . . . . . . . . . . . . 14  |-  ( w  =  1o  ->  [ <. w ,  1o >. ]  ~Q  =  [ <. 1o ,  1o >. ]  ~Q  )
32breq2d 3804 . . . . . . . . . . . . 13  |-  ( w  =  1o  ->  (
l  <Q  [ <. w ,  1o >. ]  ~Q  <->  l  <Q  [
<. 1o ,  1o >. ]  ~Q  ) )
43abbidv 2171 . . . . . . . . . . . 12  |-  ( w  =  1o  ->  { l  |  l  <Q  [ <. w ,  1o >. ]  ~Q  }  =  { l  |  l  <Q  [ <. 1o ,  1o >. ]  ~Q  }
)
52breq1d 3802 . . . . . . . . . . . . 13  |-  ( w  =  1o  ->  ( [ <. w ,  1o >. ]  ~Q  <Q  u  <->  [
<. 1o ,  1o >. ]  ~Q  <Q  u )
)
65abbidv 2171 . . . . . . . . . . . 12  |-  ( w  =  1o  ->  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u }  =  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } )
74, 6opeq12d 3585 . . . . . . . . . . 11  |-  ( w  =  1o  ->  <. { l  |  l  <Q  [ <. w ,  1o >. ]  ~Q  } ,  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u } >.  =  <. { l  |  l  <Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >. )
87oveq1d 5555 . . . . . . . . . 10  |-  ( w  =  1o  ->  ( <. { l  |  l 
<Q  [ <. w ,  1o >. ]  ~Q  } ,  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  =  ( <. { l  |  l  <Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) )
98opeq1d 3583 . . . . . . . . 9  |-  ( w  =  1o  ->  <. ( <. { l  |  l 
<Q  [ <. w ,  1o >. ]  ~Q  } ,  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >.  =  <. (
<. { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. )
109eceq1d 6173 . . . . . . . 8  |-  ( w  =  1o  ->  [ <. (
<. { l  |  l 
<Q  [ <. w ,  1o >. ]  ~Q  } ,  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  =  [ <. ( <. { l  |  l  <Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )
1110opeq1d 3583 . . . . . . 7  |-  ( w  =  1o  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. w ,  1o >. ]  ~Q  } ,  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  <. [ <. (
<. { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )
1211eleq1d 2122 . . . . . 6  |-  ( w  =  1o  ->  ( <. [ <. ( <. { l  |  l  <Q  [ <. w ,  1o >. ]  ~Q  } ,  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z  <->  <. [ <. (
<. { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z ) )
1312imbi2d 223 . . . . 5  |-  ( w  =  1o  ->  (
( ( 1  e.  z  /\  A. y  e.  z  ( y  +  1 )  e.  z )  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. w ,  1o >. ]  ~Q  } ,  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z )  <-> 
( ( 1  e.  z  /\  A. y  e.  z  ( y  +  1 )  e.  z )  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z ) ) )
14 opeq1 3577 . . . . . . . . . . . . . . 15  |-  ( w  =  k  ->  <. w ,  1o >.  =  <. k ,  1o >. )
1514eceq1d 6173 . . . . . . . . . . . . . 14  |-  ( w  =  k  ->  [ <. w ,  1o >. ]  ~Q  =  [ <. k ,  1o >. ]  ~Q  )
1615breq2d 3804 . . . . . . . . . . . . 13  |-  ( w  =  k  ->  (
l  <Q  [ <. w ,  1o >. ]  ~Q  <->  l  <Q  [
<. k ,  1o >. ]  ~Q  ) )
1716abbidv 2171 . . . . . . . . . . . 12  |-  ( w  =  k  ->  { l  |  l  <Q  [ <. w ,  1o >. ]  ~Q  }  =  { l  |  l  <Q  [ <. k ,  1o >. ]  ~Q  }
)
1815breq1d 3802 . . . . . . . . . . . . 13  |-  ( w  =  k  ->  ( [ <. w ,  1o >. ]  ~Q  <Q  u  <->  [
<. k ,  1o >. ]  ~Q  <Q  u )
)
1918abbidv 2171 . . . . . . . . . . . 12  |-  ( w  =  k  ->  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u }  =  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } )
2017, 19opeq12d 3585 . . . . . . . . . . 11  |-  ( w  =  k  ->  <. { l  |  l  <Q  [ <. w ,  1o >. ]  ~Q  } ,  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u } >.  =  <. { l  |  l  <Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >. )
2120oveq1d 5555 . . . . . . . . . 10  |-  ( w  =  k  ->  ( <. { l  |  l 
<Q  [ <. w ,  1o >. ]  ~Q  } ,  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  =  ( <. { l  |  l  <Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) )
2221opeq1d 3583 . . . . . . . . 9  |-  ( w  =  k  ->  <. ( <. { l  |  l 
<Q  [ <. w ,  1o >. ]  ~Q  } ,  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >.  =  <. (
<. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. )
2322eceq1d 6173 . . . . . . . 8  |-  ( w  =  k  ->  [ <. (
<. { l  |  l 
<Q  [ <. w ,  1o >. ]  ~Q  } ,  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  =  [ <. ( <. { l  |  l  <Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )
2423opeq1d 3583 . . . . . . 7  |-  ( w  =  k  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. w ,  1o >. ]  ~Q  } ,  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  <. [ <. (
<. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )
2524eleq1d 2122 . . . . . 6  |-  ( w  =  k  ->  ( <. [ <. ( <. { l  |  l  <Q  [ <. w ,  1o >. ]  ~Q  } ,  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z  <->  <. [ <. (
<. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z ) )
2625imbi2d 223 . . . . 5  |-  ( w  =  k  ->  (
( ( 1  e.  z  /\  A. y  e.  z  ( y  +  1 )  e.  z )  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. w ,  1o >. ]  ~Q  } ,  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z )  <-> 
( ( 1  e.  z  /\  A. y  e.  z  ( y  +  1 )  e.  z )  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z ) ) )
27 opeq1 3577 . . . . . . . . . . . . . . 15  |-  ( w  =  ( k  +N  1o )  ->  <. w ,  1o >.  =  <. ( k  +N  1o ) ,  1o >. )
2827eceq1d 6173 . . . . . . . . . . . . . 14  |-  ( w  =  ( k  +N  1o )  ->  [ <. w ,  1o >. ]  ~Q  =  [ <. ( k  +N  1o ) ,  1o >. ]  ~Q  )
2928breq2d 3804 . . . . . . . . . . . . 13  |-  ( w  =  ( k  +N  1o )  ->  (
l  <Q  [ <. w ,  1o >. ]  ~Q  <->  l  <Q  [
<. ( k  +N  1o ) ,  1o >. ]  ~Q  ) )
3029abbidv 2171 . . . . . . . . . . . 12  |-  ( w  =  ( k  +N  1o )  ->  { l  |  l  <Q  [ <. w ,  1o >. ]  ~Q  }  =  { l  |  l  <Q  [ <. (
k  +N  1o ) ,  1o >. ]  ~Q  } )
3128breq1d 3802 . . . . . . . . . . . . 13  |-  ( w  =  ( k  +N  1o )  ->  ( [ <. w ,  1o >. ]  ~Q  <Q  u  <->  [
<. ( k  +N  1o ) ,  1o >. ]  ~Q  <Q  u ) )
3231abbidv 2171 . . . . . . . . . . . 12  |-  ( w  =  ( k  +N  1o )  ->  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u }  =  { u  |  [ <. ( k  +N  1o ) ,  1o >. ]  ~Q  <Q  u } )
3330, 32opeq12d 3585 . . . . . . . . . . 11  |-  ( w  =  ( k  +N  1o )  ->  <. { l  |  l  <Q  [ <. w ,  1o >. ]  ~Q  } ,  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u } >.  =  <. { l  |  l  <Q  [ <. ( k  +N  1o ) ,  1o >. ]  ~Q  } ,  { u  |  [ <. ( k  +N  1o ) ,  1o >. ]  ~Q  <Q  u } >. )
3433oveq1d 5555 . . . . . . . . . 10  |-  ( w  =  ( k  +N  1o )  ->  ( <. { l  |  l 
<Q  [ <. w ,  1o >. ]  ~Q  } ,  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  =  ( <. { l  |  l  <Q  [ <. (
k  +N  1o ) ,  1o >. ]  ~Q  } ,  { u  |  [ <. ( k  +N  1o ) ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )
)
3534opeq1d 3583 . . . . . . . . 9  |-  ( w  =  ( k  +N  1o )  ->  <. ( <. { l  |  l 
<Q  [ <. w ,  1o >. ]  ~Q  } ,  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >.  =  <. (
<. { l  |  l 
<Q  [ <. ( k  +N  1o ) ,  1o >. ]  ~Q  } ,  { u  |  [ <. ( k  +N  1o ) ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. )
3635eceq1d 6173 . . . . . . . 8  |-  ( w  =  ( k  +N  1o )  ->  [ <. (
<. { l  |  l 
<Q  [ <. w ,  1o >. ]  ~Q  } ,  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  =  [ <. ( <. { l  |  l  <Q  [ <. ( k  +N  1o ) ,  1o >. ]  ~Q  } ,  { u  |  [ <. ( k  +N  1o ) ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )
3736opeq1d 3583 . . . . . . 7  |-  ( w  =  ( k  +N  1o )  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. w ,  1o >. ]  ~Q  } ,  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  <. [ <. (
<. { l  |  l 
<Q  [ <. ( k  +N  1o ) ,  1o >. ]  ~Q  } ,  { u  |  [ <. ( k  +N  1o ) ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )
3837eleq1d 2122 . . . . . 6  |-  ( w  =  ( k  +N  1o )  ->  ( <. [ <. ( <. { l  |  l  <Q  [ <. w ,  1o >. ]  ~Q  } ,  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z  <->  <. [ <. (
<. { l  |  l 
<Q  [ <. ( k  +N  1o ) ,  1o >. ]  ~Q  } ,  { u  |  [ <. ( k  +N  1o ) ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z
) )
3938imbi2d 223 . . . . 5  |-  ( w  =  ( k  +N  1o )  ->  (
( ( 1  e.  z  /\  A. y  e.  z  ( y  +  1 )  e.  z )  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. w ,  1o >. ]  ~Q  } ,  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z )  <-> 
( ( 1  e.  z  /\  A. y  e.  z  ( y  +  1 )  e.  z )  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. ( k  +N  1o ) ,  1o >. ]  ~Q  } ,  { u  |  [ <. ( k  +N  1o ) ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z
) ) )
40 opeq1 3577 . . . . . . . . . . . . . . 15  |-  ( w  =  N  ->  <. w ,  1o >.  =  <. N ,  1o >. )
4140eceq1d 6173 . . . . . . . . . . . . . 14  |-  ( w  =  N  ->  [ <. w ,  1o >. ]  ~Q  =  [ <. N ,  1o >. ]  ~Q  )
4241breq2d 3804 . . . . . . . . . . . . 13  |-  ( w  =  N  ->  (
l  <Q  [ <. w ,  1o >. ]  ~Q  <->  l  <Q  [
<. N ,  1o >. ]  ~Q  ) )
4342abbidv 2171 . . . . . . . . . . . 12  |-  ( w  =  N  ->  { l  |  l  <Q  [ <. w ,  1o >. ]  ~Q  }  =  { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  }
)
4441breq1d 3802 . . . . . . . . . . . . 13  |-  ( w  =  N  ->  ( [ <. w ,  1o >. ]  ~Q  <Q  u  <->  [
<. N ,  1o >. ]  ~Q  <Q  u )
)
4544abbidv 2171 . . . . . . . . . . . 12  |-  ( w  =  N  ->  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u }  =  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } )
4643, 45opeq12d 3585 . . . . . . . . . . 11  |-  ( w  =  N  ->  <. { l  |  l  <Q  [ <. w ,  1o >. ]  ~Q  } ,  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u } >.  =  <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >. )
4746oveq1d 5555 . . . . . . . . . 10  |-  ( w  =  N  ->  ( <. { l  |  l 
<Q  [ <. w ,  1o >. ]  ~Q  } ,  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  =  ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) )
4847opeq1d 3583 . . . . . . . . 9  |-  ( w  =  N  ->  <. ( <. { l  |  l 
<Q  [ <. w ,  1o >. ]  ~Q  } ,  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >.  =  <. (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. )
4948eceq1d 6173 . . . . . . . 8  |-  ( w  =  N  ->  [ <. (
<. { l  |  l 
<Q  [ <. w ,  1o >. ]  ~Q  } ,  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  =  [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )
5049opeq1d 3583 . . . . . . 7  |-  ( w  =  N  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. w ,  1o >. ]  ~Q  } ,  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  <. [ <. (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )
5150eleq1d 2122 . . . . . 6  |-  ( w  =  N  ->  ( <. [ <. ( <. { l  |  l  <Q  [ <. w ,  1o >. ]  ~Q  } ,  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z  <->  <. [ <. (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z ) )
5251imbi2d 223 . . . . 5  |-  ( w  =  N  ->  (
( ( 1  e.  z  /\  A. y  e.  z  ( y  +  1 )  e.  z )  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. w ,  1o >. ]  ~Q  } ,  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z )  <-> 
( ( 1  e.  z  /\  A. y  e.  z  ( y  +  1 )  e.  z )  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z ) ) )
53 pitonnlem1 6979 . . . . . . . 8  |-  <. [ <. (
<. { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  1
5453eleq1i 2119 . . . . . . 7  |-  ( <. [ <. ( <. { l  |  l  <Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z  <->  1  e.  z )
5554biimpri 128 . . . . . 6  |-  ( 1  e.  z  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z )
5655adantr 265 . . . . 5  |-  ( ( 1  e.  z  /\  A. y  e.  z  ( y  +  1 )  e.  z )  ->  <. [ <. ( <. { l  |  l  <Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z )
57 oveq1 5547 . . . . . . . . . . 11  |-  ( y  =  <. [ <. ( <. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  ->  ( y  +  1 )  =  ( <. [ <. ( <. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  +  1 ) )
5857eleq1d 2122 . . . . . . . . . 10  |-  ( y  =  <. [ <. ( <. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  ->  ( (
y  +  1 )  e.  z  <->  ( <. [
<. ( <. { l  |  l  <Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  +  1 )  e.  z ) )
5958rspccv 2670 . . . . . . . . 9  |-  ( A. y  e.  z  (
y  +  1 )  e.  z  ->  ( <. [ <. ( <. { l  |  l  <Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z  -> 
( <. [ <. ( <. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  +  1 )  e.  z ) )
6059ad2antll 468 . . . . . . . 8  |-  ( ( k  e.  N.  /\  ( 1  e.  z  /\  A. y  e.  z  ( y  +  1 )  e.  z ) )  ->  ( <. [ <. ( <. { l  |  l  <Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z  -> 
( <. [ <. ( <. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  +  1 )  e.  z ) )
61 pitonnlem2 6981 . . . . . . . . . 10  |-  ( k  e.  N.  ->  ( <. [ <. ( <. { l  |  l  <Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  +  1 )  =  <. [ <. ( <. { l  |  l 
<Q  [ <. ( k  +N  1o ) ,  1o >. ]  ~Q  } ,  { u  |  [ <. ( k  +N  1o ) ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )
6261eleq1d 2122 . . . . . . . . 9  |-  ( k  e.  N.  ->  (
( <. [ <. ( <. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  +  1 )  e.  z  <->  <. [ <. (
<. { l  |  l 
<Q  [ <. ( k  +N  1o ) ,  1o >. ]  ~Q  } ,  { u  |  [ <. ( k  +N  1o ) ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z
) )
6362adantr 265 . . . . . . . 8  |-  ( ( k  e.  N.  /\  ( 1  e.  z  /\  A. y  e.  z  ( y  +  1 )  e.  z ) )  ->  (
( <. [ <. ( <. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  +  1 )  e.  z  <->  <. [ <. (
<. { l  |  l 
<Q  [ <. ( k  +N  1o ) ,  1o >. ]  ~Q  } ,  { u  |  [ <. ( k  +N  1o ) ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z
) )
6460, 63sylibd 142 . . . . . . 7  |-  ( ( k  e.  N.  /\  ( 1  e.  z  /\  A. y  e.  z  ( y  +  1 )  e.  z ) )  ->  ( <. [ <. ( <. { l  |  l  <Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z  ->  <. [ <. ( <. { l  |  l  <Q  [ <. ( k  +N  1o ) ,  1o >. ]  ~Q  } ,  { u  |  [ <. ( k  +N  1o ) ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z ) )
6564ex 112 . . . . . 6  |-  ( k  e.  N.  ->  (
( 1  e.  z  /\  A. y  e.  z  ( y  +  1 )  e.  z )  ->  ( <. [
<. ( <. { l  |  l  <Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z  ->  <. [ <. ( <. { l  |  l  <Q  [ <. ( k  +N  1o ) ,  1o >. ]  ~Q  } ,  { u  |  [ <. ( k  +N  1o ) ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z ) ) )
6665a2d 26 . . . . 5  |-  ( k  e.  N.  ->  (
( ( 1  e.  z  /\  A. y  e.  z  ( y  +  1 )  e.  z )  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. k ,  1o >. ]  ~Q  } ,  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z )  ->  ( ( 1  e.  z  /\  A. y  e.  z  (
y  +  1 )  e.  z )  ->  <. [ <. ( <. { l  |  l  <Q  [ <. ( k  +N  1o ) ,  1o >. ]  ~Q  } ,  { u  |  [ <. ( k  +N  1o ) ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z ) ) )
6713, 26, 39, 52, 56, 66indpi 6498 . . . 4  |-  ( N  e.  N.  ->  (
( 1  e.  z  /\  A. y  e.  z  ( y  +  1 )  e.  z )  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z ) )
6867alrimiv 1770 . . 3  |-  ( N  e.  N.  ->  A. z
( ( 1  e.  z  /\  A. y  e.  z  ( y  +  1 )  e.  z )  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z ) )
69 eleq2 2117 . . . . 5  |-  ( x  =  z  ->  (
1  e.  x  <->  1  e.  z ) )
70 eleq2 2117 . . . . . 6  |-  ( x  =  z  ->  (
( y  +  1 )  e.  x  <->  ( y  +  1 )  e.  z ) )
7170raleqbi1dv 2530 . . . . 5  |-  ( x  =  z  ->  ( A. y  e.  x  ( y  +  1 )  e.  x  <->  A. y  e.  z  ( y  +  1 )  e.  z ) )
7269, 71anbi12d 450 . . . 4  |-  ( x  =  z  ->  (
( 1  e.  x  /\  A. y  e.  x  ( y  +  1 )  e.  x )  <-> 
( 1  e.  z  /\  A. y  e.  z  ( y  +  1 )  e.  z ) ) )
7372ralab 2724 . . 3  |-  ( A. z  e.  { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1 )  e.  x ) } <. [
<. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z  <->  A. z
( ( 1  e.  z  /\  A. y  e.  z  ( y  +  1 )  e.  z )  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z ) )
7468, 73sylibr 141 . 2  |-  ( N  e.  N.  ->  A. z  e.  { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1 )  e.  x ) }
<. [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z )
75 nnprlu 6709 . . . . . . 7  |-  ( N  e.  N.  ->  <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  e.  P. )
76 1pr 6710 . . . . . . 7  |-  1P  e.  P.
77 addclpr 6693 . . . . . . 7  |-  ( (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  e.  P.  /\  1P  e.  P. )  ->  ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  e. 
P. )
7875, 76, 77sylancl 398 . . . . . 6  |-  ( N  e.  N.  ->  ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  e. 
P. )
79 opelxpi 4404 . . . . . 6  |-  ( ( ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  e. 
P.  /\  1P  e.  P. )  ->  <. ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >.  e.  ( P.  X.  P. ) )
8078, 76, 79sylancl 398 . . . . 5  |-  ( N  e.  N.  ->  <. ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >.  e.  ( P.  X.  P. ) )
81 enrex 6880 . . . . . 6  |-  ~R  e.  _V
8281ecelqsi 6191 . . . . 5  |-  ( <.
( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >.  e.  ( P.  X.  P. )  ->  [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  )
)
8380, 82syl 14 . . . 4  |-  ( N  e.  N.  ->  [ <. (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  )
)
84 0r 6893 . . . 4  |-  0R  e.  R.
85 opelxpi 4404 . . . 4  |-  ( ( [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  0R  e.  R. )  -> 
<. [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  ( ( ( P.  X.  P. ) /.  ~R  )  X. 
R. ) )
8683, 84, 85sylancl 398 . . 3  |-  ( N  e.  N.  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  ( ( ( P.  X.  P. ) /.  ~R  )  X. 
R. ) )
87 elintg 3651 . . 3  |-  ( <. [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  ( ( ( P.  X.  P. ) /.  ~R  )  X. 
R. )  ->  ( <. [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  |^| { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1 )  e.  x ) }  <->  A. z  e.  { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1 )  e.  x ) }
<. [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z ) )
8886, 87syl 14 . 2  |-  ( N  e.  N.  ->  ( <. [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  |^| { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1 )  e.  x ) }  <->  A. z  e.  { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1 )  e.  x ) }
<. [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z ) )
8974, 88mpbird 160 1  |-  ( N  e.  N.  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  |^| { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1 )  e.  x ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102   A.wal 1257    = wceq 1259    e. wcel 1409   {cab 2042   A.wral 2323   <.cop 3406   |^|cint 3643   class class class wbr 3792    X. cxp 4371  (class class class)co 5540   1oc1o 6025   [cec 6135   /.cqs 6136   N.cnpi 6428    +N cpli 6429    ~Q ceq 6435    <Q cltq 6441   P.cnp 6447   1Pc1p 6448    +P. cpp 6449    ~R cer 6452   R.cnr 6453   0Rc0r 6454   1c1 6948    + caddc 6950
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-i1p 6623  df-iplp 6624  df-enr 6869  df-nr 6870  df-plr 6871  df-0r 6874  df-1r 6875  df-c 6953  df-1 6955  df-add 6958
This theorem is referenced by:  axarch  7023  axcaucvglemcl  7027  axcaucvglemval  7029  axcaucvglemcau  7030  axcaucvglemres  7031
  Copyright terms: Public domain W3C validator