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Theorem pitonn 7668
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 3705 . . . . . . . . . . . . . . 15  |-  ( w  =  1o  ->  <. w ,  1o >.  =  <. 1o ,  1o >. )
21eceq1d 6465 . . . . . . . . . . . . . 14  |-  ( w  =  1o  ->  [ <. w ,  1o >. ]  ~Q  =  [ <. 1o ,  1o >. ]  ~Q  )
32breq2d 3941 . . . . . . . . . . . . 13  |-  ( w  =  1o  ->  (
l  <Q  [ <. w ,  1o >. ]  ~Q  <->  l  <Q  [
<. 1o ,  1o >. ]  ~Q  ) )
43abbidv 2257 . . . . . . . . . . . 12  |-  ( w  =  1o  ->  { l  |  l  <Q  [ <. w ,  1o >. ]  ~Q  }  =  { l  |  l  <Q  [ <. 1o ,  1o >. ]  ~Q  }
)
52breq1d 3939 . . . . . . . . . . . . 13  |-  ( w  =  1o  ->  ( [ <. w ,  1o >. ]  ~Q  <Q  u  <->  [
<. 1o ,  1o >. ]  ~Q  <Q  u )
)
65abbidv 2257 . . . . . . . . . . . 12  |-  ( w  =  1o  ->  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u }  =  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } )
74, 6opeq12d 3713 . . . . . . . . . . 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 5789 . . . . . . . . . 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 3711 . . . . . . . . 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 6465 . . . . . . . 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 3711 . . . . . . 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 2208 . . . . . 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 229 . . . . 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 3705 . . . . . . . . . . . . . . 15  |-  ( w  =  k  ->  <. w ,  1o >.  =  <. k ,  1o >. )
1514eceq1d 6465 . . . . . . . . . . . . . 14  |-  ( w  =  k  ->  [ <. w ,  1o >. ]  ~Q  =  [ <. k ,  1o >. ]  ~Q  )
1615breq2d 3941 . . . . . . . . . . . . 13  |-  ( w  =  k  ->  (
l  <Q  [ <. w ,  1o >. ]  ~Q  <->  l  <Q  [
<. k ,  1o >. ]  ~Q  ) )
1716abbidv 2257 . . . . . . . . . . . 12  |-  ( w  =  k  ->  { l  |  l  <Q  [ <. w ,  1o >. ]  ~Q  }  =  { l  |  l  <Q  [ <. k ,  1o >. ]  ~Q  }
)
1815breq1d 3939 . . . . . . . . . . . . 13  |-  ( w  =  k  ->  ( [ <. w ,  1o >. ]  ~Q  <Q  u  <->  [
<. k ,  1o >. ]  ~Q  <Q  u )
)
1918abbidv 2257 . . . . . . . . . . . 12  |-  ( w  =  k  ->  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u }  =  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } )
2017, 19opeq12d 3713 . . . . . . . . . . 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 5789 . . . . . . . . . 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 3711 . . . . . . . . 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 6465 . . . . . . . 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 3711 . . . . . . 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 2208 . . . . . 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 229 . . . . 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 3705 . . . . . . . . . . . . . . 15  |-  ( w  =  ( k  +N  1o )  ->  <. w ,  1o >.  =  <. ( k  +N  1o ) ,  1o >. )
2827eceq1d 6465 . . . . . . . . . . . . . 14  |-  ( w  =  ( k  +N  1o )  ->  [ <. w ,  1o >. ]  ~Q  =  [ <. ( k  +N  1o ) ,  1o >. ]  ~Q  )
2928breq2d 3941 . . . . . . . . . . . . 13  |-  ( w  =  ( k  +N  1o )  ->  (
l  <Q  [ <. w ,  1o >. ]  ~Q  <->  l  <Q  [
<. ( k  +N  1o ) ,  1o >. ]  ~Q  ) )
3029abbidv 2257 . . . . . . . . . . . 12  |-  ( w  =  ( k  +N  1o )  ->  { l  |  l  <Q  [ <. w ,  1o >. ]  ~Q  }  =  { l  |  l  <Q  [ <. (
k  +N  1o ) ,  1o >. ]  ~Q  } )
3128breq1d 3939 . . . . . . . . . . . . 13  |-  ( w  =  ( k  +N  1o )  ->  ( [ <. w ,  1o >. ]  ~Q  <Q  u  <->  [
<. ( k  +N  1o ) ,  1o >. ]  ~Q  <Q  u ) )
3231abbidv 2257 . . . . . . . . . . . 12  |-  ( w  =  ( k  +N  1o )  ->  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u }  =  { u  |  [ <. ( k  +N  1o ) ,  1o >. ]  ~Q  <Q  u } )
3330, 32opeq12d 3713 . . . . . . . . . . 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 5789 . . . . . . . . . 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 3711 . . . . . . . . 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 6465 . . . . . . . 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 3711 . . . . . . 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 2208 . . . . . 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 229 . . . . 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 3705 . . . . . . . . . . . . . . 15  |-  ( w  =  N  ->  <. w ,  1o >.  =  <. N ,  1o >. )
4140eceq1d 6465 . . . . . . . . . . . . . 14  |-  ( w  =  N  ->  [ <. w ,  1o >. ]  ~Q  =  [ <. N ,  1o >. ]  ~Q  )
4241breq2d 3941 . . . . . . . . . . . . 13  |-  ( w  =  N  ->  (
l  <Q  [ <. w ,  1o >. ]  ~Q  <->  l  <Q  [
<. N ,  1o >. ]  ~Q  ) )
4342abbidv 2257 . . . . . . . . . . . 12  |-  ( w  =  N  ->  { l  |  l  <Q  [ <. w ,  1o >. ]  ~Q  }  =  { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  }
)
4441breq1d 3939 . . . . . . . . . . . . 13  |-  ( w  =  N  ->  ( [ <. w ,  1o >. ]  ~Q  <Q  u  <->  [
<. N ,  1o >. ]  ~Q  <Q  u )
)
4544abbidv 2257 . . . . . . . . . . . 12  |-  ( w  =  N  ->  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u }  =  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } )
4643, 45opeq12d 3713 . . . . . . . . . . 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 5789 . . . . . . . . . 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 3711 . . . . . . . . 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 6465 . . . . . . . 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 3711 . . . . . . 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 2208 . . . . . 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 229 . . . . 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 7665 . . . . . . . 8  |-  <. [ <. (
<. { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  1
5453eleq1i 2205 . . . . . . 7  |-  ( <. [ <. ( <. { l  |  l  <Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z  <->  1  e.  z )
5554biimpri 132 . . . . . 6  |-  ( 1  e.  z  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z )
5655adantr 274 . . . . 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 5781 . . . . . . . . . . 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 2208 . . . . . . . . . 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 2786 . . . . . . . . 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 482 . . . . . . . 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 7667 . . . . . . . . . 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 2208 . . . . . . . . 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 274 . . . . . . . 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 148 . . . . . . 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 114 . . . . . 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 7162 . . . 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 1846 . . 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 2203 . . . . 5  |-  ( x  =  z  ->  (
1  e.  x  <->  1  e.  z ) )
70 eleq2 2203 . . . . . 6  |-  ( x  =  z  ->  (
( y  +  1 )  e.  x  <->  ( y  +  1 )  e.  z ) )
7170raleqbi1dv 2634 . . . . 5  |-  ( x  =  z  ->  ( A. y  e.  x  ( y  +  1 )  e.  x  <->  A. y  e.  z  ( y  +  1 )  e.  z ) )
7269, 71anbi12d 464 . . . 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 2844 . . 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 133 . 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 7373 . . . . . . 7  |-  ( N  e.  N.  ->  <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  e.  P. )
76 1pr 7374 . . . . . . 7  |-  1P  e.  P.
77 addclpr 7357 . . . . . . 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 409 . . . . . 6  |-  ( N  e.  N.  ->  ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  e. 
P. )
79 opelxpi 4571 . . . . . 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 409 . . . . 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 7557 . . . . . 6  |-  ~R  e.  _V
8281ecelqsi 6483 . . . . 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 7570 . . . 4  |-  0R  e.  R.
85 opelxpi 4571 . . . 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 409 . . 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 3779 . . 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 166 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 103    <-> wb 104   A.wal 1329    = wceq 1331    e. wcel 1480   {cab 2125   A.wral 2416   <.cop 3530   |^|cint 3771   class class class wbr 3929    X. cxp 4537  (class class class)co 5774   1oc1o 6306   [cec 6427   /.cqs 6428   N.cnpi 7092    +N cpli 7093    ~Q ceq 7099    <Q cltq 7105   P.cnp 7111   1Pc1p 7112    +P. cpp 7113    ~R cer 7116   R.cnr 7117   0Rc0r 7118   1c1 7633    + caddc 7635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7124  df-pli 7125  df-mi 7126  df-lti 7127  df-plpq 7164  df-mpq 7165  df-enq 7167  df-nqqs 7168  df-plqqs 7169  df-mqqs 7170  df-1nqqs 7171  df-rq 7172  df-ltnqqs 7173  df-enq0 7244  df-nq0 7245  df-0nq0 7246  df-plq0 7247  df-mq0 7248  df-inp 7286  df-i1p 7287  df-iplp 7288  df-enr 7546  df-nr 7547  df-plr 7548  df-0r 7551  df-1r 7552  df-c 7638  df-1 7640  df-add 7643
This theorem is referenced by:  axarch  7711  axcaucvglemcl  7715  axcaucvglemval  7717  axcaucvglemcau  7718  axcaucvglemres  7719
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