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Theorem pitonn 6896
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 3546 . . . . . . . . . . . . . . 15  |-  ( w  =  1o  ->  <. w ,  1o >.  =  <. 1o ,  1o >. )
21eceq1d 6120 . . . . . . . . . . . . . 14  |-  ( w  =  1o  ->  [ <. w ,  1o >. ]  ~Q  =  [ <. 1o ,  1o >. ]  ~Q  )
32breq2d 3773 . . . . . . . . . . . . 13  |-  ( w  =  1o  ->  (
l  <Q  [ <. w ,  1o >. ]  ~Q  <->  l  <Q  [
<. 1o ,  1o >. ]  ~Q  ) )
43abbidv 2155 . . . . . . . . . . . 12  |-  ( w  =  1o  ->  { l  |  l  <Q  [ <. w ,  1o >. ]  ~Q  }  =  { l  |  l  <Q  [ <. 1o ,  1o >. ]  ~Q  }
)
52breq1d 3771 . . . . . . . . . . . . 13  |-  ( w  =  1o  ->  ( [ <. w ,  1o >. ]  ~Q  <Q  u  <->  [
<. 1o ,  1o >. ]  ~Q  <Q  u )
)
65abbidv 2155 . . . . . . . . . . . 12  |-  ( w  =  1o  ->  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u }  =  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } )
74, 6opeq12d 3554 . . . . . . . . . . 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 5505 . . . . . . . . . 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 3552 . . . . . . . . 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 6120 . . . . . . . 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 3552 . . . . . . 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 2106 . . . . . 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 219 . . . . 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 3546 . . . . . . . . . . . . . . 15  |-  ( w  =  k  ->  <. w ,  1o >.  =  <. k ,  1o >. )
1514eceq1d 6120 . . . . . . . . . . . . . 14  |-  ( w  =  k  ->  [ <. w ,  1o >. ]  ~Q  =  [ <. k ,  1o >. ]  ~Q  )
1615breq2d 3773 . . . . . . . . . . . . 13  |-  ( w  =  k  ->  (
l  <Q  [ <. w ,  1o >. ]  ~Q  <->  l  <Q  [
<. k ,  1o >. ]  ~Q  ) )
1716abbidv 2155 . . . . . . . . . . . 12  |-  ( w  =  k  ->  { l  |  l  <Q  [ <. w ,  1o >. ]  ~Q  }  =  { l  |  l  <Q  [ <. k ,  1o >. ]  ~Q  }
)
1815breq1d 3771 . . . . . . . . . . . . 13  |-  ( w  =  k  ->  ( [ <. w ,  1o >. ]  ~Q  <Q  u  <->  [
<. k ,  1o >. ]  ~Q  <Q  u )
)
1918abbidv 2155 . . . . . . . . . . . 12  |-  ( w  =  k  ->  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u }  =  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } )
2017, 19opeq12d 3554 . . . . . . . . . . 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 5505 . . . . . . . . . 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 3552 . . . . . . . . 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 6120 . . . . . . . 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 3552 . . . . . . 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 2106 . . . . . 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 219 . . . . 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 3546 . . . . . . . . . . . . . . 15  |-  ( w  =  ( k  +N  1o )  ->  <. w ,  1o >.  =  <. ( k  +N  1o ) ,  1o >. )
2827eceq1d 6120 . . . . . . . . . . . . . 14  |-  ( w  =  ( k  +N  1o )  ->  [ <. w ,  1o >. ]  ~Q  =  [ <. ( k  +N  1o ) ,  1o >. ]  ~Q  )
2928breq2d 3773 . . . . . . . . . . . . 13  |-  ( w  =  ( k  +N  1o )  ->  (
l  <Q  [ <. w ,  1o >. ]  ~Q  <->  l  <Q  [
<. ( k  +N  1o ) ,  1o >. ]  ~Q  ) )
3029abbidv 2155 . . . . . . . . . . . 12  |-  ( w  =  ( k  +N  1o )  ->  { l  |  l  <Q  [ <. w ,  1o >. ]  ~Q  }  =  { l  |  l  <Q  [ <. (
k  +N  1o ) ,  1o >. ]  ~Q  } )
3128breq1d 3771 . . . . . . . . . . . . 13  |-  ( w  =  ( k  +N  1o )  ->  ( [ <. w ,  1o >. ]  ~Q  <Q  u  <->  [
<. ( k  +N  1o ) ,  1o >. ]  ~Q  <Q  u ) )
3231abbidv 2155 . . . . . . . . . . . 12  |-  ( w  =  ( k  +N  1o )  ->  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u }  =  { u  |  [ <. ( k  +N  1o ) ,  1o >. ]  ~Q  <Q  u } )
3330, 32opeq12d 3554 . . . . . . . . . . 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 5505 . . . . . . . . . 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 3552 . . . . . . . . 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 6120 . . . . . . . 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 3552 . . . . . . 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 2106 . . . . . 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 219 . . . . 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 3546 . . . . . . . . . . . . . . 15  |-  ( w  =  N  ->  <. w ,  1o >.  =  <. N ,  1o >. )
4140eceq1d 6120 . . . . . . . . . . . . . 14  |-  ( w  =  N  ->  [ <. w ,  1o >. ]  ~Q  =  [ <. N ,  1o >. ]  ~Q  )
4241breq2d 3773 . . . . . . . . . . . . 13  |-  ( w  =  N  ->  (
l  <Q  [ <. w ,  1o >. ]  ~Q  <->  l  <Q  [
<. N ,  1o >. ]  ~Q  ) )
4342abbidv 2155 . . . . . . . . . . . 12  |-  ( w  =  N  ->  { l  |  l  <Q  [ <. w ,  1o >. ]  ~Q  }  =  { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  }
)
4441breq1d 3771 . . . . . . . . . . . . 13  |-  ( w  =  N  ->  ( [ <. w ,  1o >. ]  ~Q  <Q  u  <->  [
<. N ,  1o >. ]  ~Q  <Q  u )
)
4544abbidv 2155 . . . . . . . . . . . 12  |-  ( w  =  N  ->  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u }  =  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } )
4643, 45opeq12d 3554 . . . . . . . . . . 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 5505 . . . . . . . . . 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 3552 . . . . . . . . 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 6120 . . . . . . . 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 3552 . . . . . . 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 2106 . . . . . 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 219 . . . . 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 6893 . . . . . . . 8  |-  <. [ <. (
<. { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  1
5453eleq1i 2103 . . . . . . 7  |-  ( <. [ <. ( <. { l  |  l  <Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z  <->  1  e.  z )
5554biimpri 124 . . . . . 6  |-  ( 1  e.  z  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z )
5655adantr 261 . . . . 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 5497 . . . . . . . . . . 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 2106 . . . . . . . . . 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 2650 . . . . . . . . 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 460 . . . . . . . 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 6895 . . . . . . . . . 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 2106 . . . . . . . . 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 261 . . . . . . . 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 138 . . . . . . 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 108 . . . . . 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 23 . . . . 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 6412 . . . 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 1754 . . 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 2101 . . . . 5  |-  ( x  =  z  ->  (
1  e.  x  <->  1  e.  z ) )
70 eleq2 2101 . . . . . 6  |-  ( x  =  z  ->  (
( y  +  1 )  e.  x  <->  ( y  +  1 )  e.  z ) )
7170raleqbi1dv 2510 . . . . 5  |-  ( x  =  z  ->  ( A. y  e.  x  ( y  +  1 )  e.  x  <->  A. y  e.  z  ( y  +  1 )  e.  z ) )
7269, 71anbi12d 442 . . . 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 2698 . . 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 137 . 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 6623 . . . . . . 7  |-  ( N  e.  N.  ->  <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  e.  P. )
76 1pr 6624 . . . . . . 7  |-  1P  e.  P.
77 addclpr 6607 . . . . . . 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 392 . . . . . 6  |-  ( N  e.  N.  ->  ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  e. 
P. )
79 opelxpi 4354 . . . . . 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 392 . . . . 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 6794 . . . . . 6  |-  ~R  e.  _V
8281ecelqsi 6138 . . . . 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 6807 . . . 4  |-  0R  e.  R.
85 opelxpi 4354 . . . 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 392 . . 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 3620 . . 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 156 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 97    <-> wb 98   A.wal 1241    = wceq 1243    e. wcel 1393   {cab 2026   A.wral 2303   <.cop 3375   |^|cint 3612   class class class wbr 3761    X. cxp 4321  (class class class)co 5490   1oc1o 5972   [cec 6082   /.cqs 6083   N.cnpi 6342    +N cpli 6343    ~Q ceq 6349    <Q cltq 6355   P.cnp 6361   1Pc1p 6362    +P. cpp 6363    ~R cer 6366   R.cnr 6367   0Rc0r 6368   1c1 6862    + caddc 6864
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3869  ax-sep 3872  ax-nul 3880  ax-pow 3924  ax-pr 3941  ax-un 4157  ax-setind 4247  ax-iinf 4289
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2308  df-rex 2309  df-reu 2310  df-rab 2312  df-v 2556  df-sbc 2762  df-csb 2850  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-int 3613  df-iun 3656  df-br 3762  df-opab 3816  df-mpt 3817  df-tr 3852  df-eprel 4023  df-id 4027  df-po 4030  df-iso 4031  df-iord 4090  df-on 4092  df-suc 4095  df-iom 4292  df-xp 4329  df-rel 4330  df-cnv 4331  df-co 4332  df-dm 4333  df-rn 4334  df-res 4335  df-ima 4336  df-iota 4845  df-fun 4882  df-fn 4883  df-f 4884  df-f1 4885  df-fo 4886  df-f1o 4887  df-fv 4888  df-ov 5493  df-oprab 5494  df-mpt2 5495  df-1st 5745  df-2nd 5746  df-recs 5898  df-irdg 5935  df-1o 5979  df-2o 5980  df-oadd 5983  df-omul 5984  df-er 6084  df-ec 6086  df-qs 6090  df-ni 6374  df-pli 6375  df-mi 6376  df-lti 6377  df-plpq 6414  df-mpq 6415  df-enq 6417  df-nqqs 6418  df-plqqs 6419  df-mqqs 6420  df-1nqqs 6421  df-rq 6422  df-ltnqqs 6423  df-enq0 6494  df-nq0 6495  df-0nq0 6496  df-plq0 6497  df-mq0 6498  df-inp 6536  df-i1p 6537  df-iplp 6538  df-enr 6783  df-nr 6784  df-plr 6785  df-0r 6788  df-1r 6789  df-c 6867  df-1 6869  df-add 6872
This theorem is referenced by:  axarch  6937  axcaucvglemcl  6941  axcaucvglemval  6943  axcaucvglemcau  6944  axcaucvglemres  6945
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