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Theorem pitonn 7849
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 3780 . . . . . . . . . . . . . . 15  |-  ( w  =  1o  ->  <. w ,  1o >.  =  <. 1o ,  1o >. )
21eceq1d 6573 . . . . . . . . . . . . . 14  |-  ( w  =  1o  ->  [ <. w ,  1o >. ]  ~Q  =  [ <. 1o ,  1o >. ]  ~Q  )
32breq2d 4017 . . . . . . . . . . . . 13  |-  ( w  =  1o  ->  (
l  <Q  [ <. w ,  1o >. ]  ~Q  <->  l  <Q  [
<. 1o ,  1o >. ]  ~Q  ) )
43abbidv 2295 . . . . . . . . . . . 12  |-  ( w  =  1o  ->  { l  |  l  <Q  [ <. w ,  1o >. ]  ~Q  }  =  { l  |  l  <Q  [ <. 1o ,  1o >. ]  ~Q  }
)
52breq1d 4015 . . . . . . . . . . . . 13  |-  ( w  =  1o  ->  ( [ <. w ,  1o >. ]  ~Q  <Q  u  <->  [
<. 1o ,  1o >. ]  ~Q  <Q  u )
)
65abbidv 2295 . . . . . . . . . . . 12  |-  ( w  =  1o  ->  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u }  =  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } )
74, 6opeq12d 3788 . . . . . . . . . . 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 5892 . . . . . . . . . 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 3786 . . . . . . . . 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 6573 . . . . . . . 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 3786 . . . . . . 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 2246 . . . . . 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 230 . . . . 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 3780 . . . . . . . . . . . . . . 15  |-  ( w  =  k  ->  <. w ,  1o >.  =  <. k ,  1o >. )
1514eceq1d 6573 . . . . . . . . . . . . . 14  |-  ( w  =  k  ->  [ <. w ,  1o >. ]  ~Q  =  [ <. k ,  1o >. ]  ~Q  )
1615breq2d 4017 . . . . . . . . . . . . 13  |-  ( w  =  k  ->  (
l  <Q  [ <. w ,  1o >. ]  ~Q  <->  l  <Q  [
<. k ,  1o >. ]  ~Q  ) )
1716abbidv 2295 . . . . . . . . . . . 12  |-  ( w  =  k  ->  { l  |  l  <Q  [ <. w ,  1o >. ]  ~Q  }  =  { l  |  l  <Q  [ <. k ,  1o >. ]  ~Q  }
)
1815breq1d 4015 . . . . . . . . . . . . 13  |-  ( w  =  k  ->  ( [ <. w ,  1o >. ]  ~Q  <Q  u  <->  [
<. k ,  1o >. ]  ~Q  <Q  u )
)
1918abbidv 2295 . . . . . . . . . . . 12  |-  ( w  =  k  ->  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u }  =  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } )
2017, 19opeq12d 3788 . . . . . . . . . . 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 5892 . . . . . . . . . 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 3786 . . . . . . . . 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 6573 . . . . . . . 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 3786 . . . . . . 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 2246 . . . . . 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 230 . . . . 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 3780 . . . . . . . . . . . . . . 15  |-  ( w  =  ( k  +N  1o )  ->  <. w ,  1o >.  =  <. ( k  +N  1o ) ,  1o >. )
2827eceq1d 6573 . . . . . . . . . . . . . 14  |-  ( w  =  ( k  +N  1o )  ->  [ <. w ,  1o >. ]  ~Q  =  [ <. ( k  +N  1o ) ,  1o >. ]  ~Q  )
2928breq2d 4017 . . . . . . . . . . . . 13  |-  ( w  =  ( k  +N  1o )  ->  (
l  <Q  [ <. w ,  1o >. ]  ~Q  <->  l  <Q  [
<. ( k  +N  1o ) ,  1o >. ]  ~Q  ) )
3029abbidv 2295 . . . . . . . . . . . 12  |-  ( w  =  ( k  +N  1o )  ->  { l  |  l  <Q  [ <. w ,  1o >. ]  ~Q  }  =  { l  |  l  <Q  [ <. (
k  +N  1o ) ,  1o >. ]  ~Q  } )
3128breq1d 4015 . . . . . . . . . . . . 13  |-  ( w  =  ( k  +N  1o )  ->  ( [ <. w ,  1o >. ]  ~Q  <Q  u  <->  [
<. ( k  +N  1o ) ,  1o >. ]  ~Q  <Q  u ) )
3231abbidv 2295 . . . . . . . . . . . 12  |-  ( w  =  ( k  +N  1o )  ->  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u }  =  { u  |  [ <. ( k  +N  1o ) ,  1o >. ]  ~Q  <Q  u } )
3330, 32opeq12d 3788 . . . . . . . . . . 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 5892 . . . . . . . . . 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 3786 . . . . . . . . 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 6573 . . . . . . . 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 3786 . . . . . . 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 2246 . . . . . 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 230 . . . . 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 3780 . . . . . . . . . . . . . . 15  |-  ( w  =  N  ->  <. w ,  1o >.  =  <. N ,  1o >. )
4140eceq1d 6573 . . . . . . . . . . . . . 14  |-  ( w  =  N  ->  [ <. w ,  1o >. ]  ~Q  =  [ <. N ,  1o >. ]  ~Q  )
4241breq2d 4017 . . . . . . . . . . . . 13  |-  ( w  =  N  ->  (
l  <Q  [ <. w ,  1o >. ]  ~Q  <->  l  <Q  [
<. N ,  1o >. ]  ~Q  ) )
4342abbidv 2295 . . . . . . . . . . . 12  |-  ( w  =  N  ->  { l  |  l  <Q  [ <. w ,  1o >. ]  ~Q  }  =  { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  }
)
4441breq1d 4015 . . . . . . . . . . . . 13  |-  ( w  =  N  ->  ( [ <. w ,  1o >. ]  ~Q  <Q  u  <->  [
<. N ,  1o >. ]  ~Q  <Q  u )
)
4544abbidv 2295 . . . . . . . . . . . 12  |-  ( w  =  N  ->  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u }  =  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } )
4643, 45opeq12d 3788 . . . . . . . . . . 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 5892 . . . . . . . . . 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 3786 . . . . . . . . 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 6573 . . . . . . . 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 3786 . . . . . . 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 2246 . . . . . 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 230 . . . . 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 7846 . . . . . . . 8  |-  <. [ <. (
<. { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  1
5453eleq1i 2243 . . . . . . 7  |-  ( <. [ <. ( <. { l  |  l  <Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z  <->  1  e.  z )
5554biimpri 133 . . . . . 6  |-  ( 1  e.  z  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  z )
5655adantr 276 . . . . 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 5884 . . . . . . . . . . 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 2246 . . . . . . . . . 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 2840 . . . . . . . . 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 491 . . . . . . . 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 7848 . . . . . . . . . 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 2246 . . . . . . . . 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 276 . . . . . . . 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 149 . . . . . . 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 115 . . . . . 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 7343 . . . 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 1874 . . 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 2241 . . . . 5  |-  ( x  =  z  ->  (
1  e.  x  <->  1  e.  z ) )
70 eleq2 2241 . . . . . 6  |-  ( x  =  z  ->  (
( y  +  1 )  e.  x  <->  ( y  +  1 )  e.  z ) )
7170raleqbi1dv 2681 . . . . 5  |-  ( x  =  z  ->  ( A. y  e.  x  ( y  +  1 )  e.  x  <->  A. y  e.  z  ( y  +  1 )  e.  z ) )
7269, 71anbi12d 473 . . . 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 2899 . . 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 134 . 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 7554 . . . . . . 7  |-  ( N  e.  N.  ->  <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  e.  P. )
76 1pr 7555 . . . . . . 7  |-  1P  e.  P.
77 addclpr 7538 . . . . . . 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 413 . . . . . 6  |-  ( N  e.  N.  ->  ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  e. 
P. )
79 opelxpi 4660 . . . . . 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 413 . . . . 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 7738 . . . . . 6  |-  ~R  e.  _V
8281ecelqsi 6591 . . . . 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 7751 . . . 4  |-  0R  e.  R.
85 opelxpi 4660 . . . 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 413 . . 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 3854 . . 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 167 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 104    <-> wb 105   A.wal 1351    = wceq 1353    e. wcel 2148   {cab 2163   A.wral 2455   <.cop 3597   |^|cint 3846   class class class wbr 4005    X. cxp 4626  (class class class)co 5877   1oc1o 6412   [cec 6535   /.cqs 6536   N.cnpi 7273    +N cpli 7274    ~Q ceq 7280    <Q cltq 7286   P.cnp 7292   1Pc1p 7293    +P. cpp 7294    ~R cer 7297   R.cnr 7298   0Rc0r 7299   1c1 7814    + caddc 7816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-2o 6420  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-pli 7306  df-mi 7307  df-lti 7308  df-plpq 7345  df-mpq 7346  df-enq 7348  df-nqqs 7349  df-plqqs 7350  df-mqqs 7351  df-1nqqs 7352  df-rq 7353  df-ltnqqs 7354  df-enq0 7425  df-nq0 7426  df-0nq0 7427  df-plq0 7428  df-mq0 7429  df-inp 7467  df-i1p 7468  df-iplp 7469  df-enr 7727  df-nr 7728  df-plr 7729  df-0r 7732  df-1r 7733  df-c 7819  df-1 7821  df-add 7824
This theorem is referenced by:  axarch  7892  axcaucvglemcl  7896  axcaucvglemval  7898  axcaucvglemcau  7899  axcaucvglemres  7900
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