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Theorem pitonn 7535
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 3652 . . . . . . . . . . . . . . 15  |-  ( w  =  1o  ->  <. w ,  1o >.  =  <. 1o ,  1o >. )
21eceq1d 6395 . . . . . . . . . . . . . 14  |-  ( w  =  1o  ->  [ <. w ,  1o >. ]  ~Q  =  [ <. 1o ,  1o >. ]  ~Q  )
32breq2d 3887 . . . . . . . . . . . . 13  |-  ( w  =  1o  ->  (
l  <Q  [ <. w ,  1o >. ]  ~Q  <->  l  <Q  [
<. 1o ,  1o >. ]  ~Q  ) )
43abbidv 2217 . . . . . . . . . . . 12  |-  ( w  =  1o  ->  { l  |  l  <Q  [ <. w ,  1o >. ]  ~Q  }  =  { l  |  l  <Q  [ <. 1o ,  1o >. ]  ~Q  }
)
52breq1d 3885 . . . . . . . . . . . . 13  |-  ( w  =  1o  ->  ( [ <. w ,  1o >. ]  ~Q  <Q  u  <->  [
<. 1o ,  1o >. ]  ~Q  <Q  u )
)
65abbidv 2217 . . . . . . . . . . . 12  |-  ( w  =  1o  ->  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u }  =  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } )
74, 6opeq12d 3660 . . . . . . . . . . 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 5721 . . . . . . . . . 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 3658 . . . . . . . . 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 6395 . . . . . . . 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 3658 . . . . . . 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 2168 . . . . . 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 3652 . . . . . . . . . . . . . . 15  |-  ( w  =  k  ->  <. w ,  1o >.  =  <. k ,  1o >. )
1514eceq1d 6395 . . . . . . . . . . . . . 14  |-  ( w  =  k  ->  [ <. w ,  1o >. ]  ~Q  =  [ <. k ,  1o >. ]  ~Q  )
1615breq2d 3887 . . . . . . . . . . . . 13  |-  ( w  =  k  ->  (
l  <Q  [ <. w ,  1o >. ]  ~Q  <->  l  <Q  [
<. k ,  1o >. ]  ~Q  ) )
1716abbidv 2217 . . . . . . . . . . . 12  |-  ( w  =  k  ->  { l  |  l  <Q  [ <. w ,  1o >. ]  ~Q  }  =  { l  |  l  <Q  [ <. k ,  1o >. ]  ~Q  }
)
1815breq1d 3885 . . . . . . . . . . . . 13  |-  ( w  =  k  ->  ( [ <. w ,  1o >. ]  ~Q  <Q  u  <->  [
<. k ,  1o >. ]  ~Q  <Q  u )
)
1918abbidv 2217 . . . . . . . . . . . 12  |-  ( w  =  k  ->  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u }  =  { u  |  [ <. k ,  1o >. ]  ~Q  <Q  u } )
2017, 19opeq12d 3660 . . . . . . . . . . 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 5721 . . . . . . . . . 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 3658 . . . . . . . . 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 6395 . . . . . . . 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 3658 . . . . . . 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 2168 . . . . . 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 3652 . . . . . . . . . . . . . . 15  |-  ( w  =  ( k  +N  1o )  ->  <. w ,  1o >.  =  <. ( k  +N  1o ) ,  1o >. )
2827eceq1d 6395 . . . . . . . . . . . . . 14  |-  ( w  =  ( k  +N  1o )  ->  [ <. w ,  1o >. ]  ~Q  =  [ <. ( k  +N  1o ) ,  1o >. ]  ~Q  )
2928breq2d 3887 . . . . . . . . . . . . 13  |-  ( w  =  ( k  +N  1o )  ->  (
l  <Q  [ <. w ,  1o >. ]  ~Q  <->  l  <Q  [
<. ( k  +N  1o ) ,  1o >. ]  ~Q  ) )
3029abbidv 2217 . . . . . . . . . . . 12  |-  ( w  =  ( k  +N  1o )  ->  { l  |  l  <Q  [ <. w ,  1o >. ]  ~Q  }  =  { l  |  l  <Q  [ <. (
k  +N  1o ) ,  1o >. ]  ~Q  } )
3128breq1d 3885 . . . . . . . . . . . . 13  |-  ( w  =  ( k  +N  1o )  ->  ( [ <. w ,  1o >. ]  ~Q  <Q  u  <->  [
<. ( k  +N  1o ) ,  1o >. ]  ~Q  <Q  u ) )
3231abbidv 2217 . . . . . . . . . . . 12  |-  ( w  =  ( k  +N  1o )  ->  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u }  =  { u  |  [ <. ( k  +N  1o ) ,  1o >. ]  ~Q  <Q  u } )
3330, 32opeq12d 3660 . . . . . . . . . . 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 5721 . . . . . . . . . 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 3658 . . . . . . . . 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 6395 . . . . . . . 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 3658 . . . . . . 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 2168 . . . . . 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 3652 . . . . . . . . . . . . . . 15  |-  ( w  =  N  ->  <. w ,  1o >.  =  <. N ,  1o >. )
4140eceq1d 6395 . . . . . . . . . . . . . 14  |-  ( w  =  N  ->  [ <. w ,  1o >. ]  ~Q  =  [ <. N ,  1o >. ]  ~Q  )
4241breq2d 3887 . . . . . . . . . . . . 13  |-  ( w  =  N  ->  (
l  <Q  [ <. w ,  1o >. ]  ~Q  <->  l  <Q  [
<. N ,  1o >. ]  ~Q  ) )
4342abbidv 2217 . . . . . . . . . . . 12  |-  ( w  =  N  ->  { l  |  l  <Q  [ <. w ,  1o >. ]  ~Q  }  =  { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  }
)
4441breq1d 3885 . . . . . . . . . . . . 13  |-  ( w  =  N  ->  ( [ <. w ,  1o >. ]  ~Q  <Q  u  <->  [
<. N ,  1o >. ]  ~Q  <Q  u )
)
4544abbidv 2217 . . . . . . . . . . . 12  |-  ( w  =  N  ->  { u  |  [ <. w ,  1o >. ]  ~Q  <Q  u }  =  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } )
4643, 45opeq12d 3660 . . . . . . . . . . 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 5721 . . . . . . . . . 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 3658 . . . . . . . . 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 6395 . . . . . . . 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 3658 . . . . . . 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 2168 . . . . . 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 7532 . . . . . . . 8  |-  <. [ <. (
<. { l  |  l 
<Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  1
5453eleq1i 2165 . . . . . . 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 272 . . . . 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 5713 . . . . . . . . . . 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 2168 . . . . . . . . . 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 2741 . . . . . . . . 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 478 . . . . . . . 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 7534 . . . . . . . . . 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 2168 . . . . . . . . 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 272 . . . . . . . 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 7051 . . . 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 1813 . . 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 2163 . . . . 5  |-  ( x  =  z  ->  (
1  e.  x  <->  1  e.  z ) )
70 eleq2 2163 . . . . . 6  |-  ( x  =  z  ->  (
( y  +  1 )  e.  x  <->  ( y  +  1 )  e.  z ) )
7170raleqbi1dv 2592 . . . . 5  |-  ( x  =  z  ->  ( A. y  e.  x  ( y  +  1 )  e.  x  <->  A. y  e.  z  ( y  +  1 )  e.  z ) )
7269, 71anbi12d 460 . . . 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 2797 . . 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 7262 . . . . . . 7  |-  ( N  e.  N.  ->  <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  e.  P. )
76 1pr 7263 . . . . . . 7  |-  1P  e.  P.
77 addclpr 7246 . . . . . . 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 407 . . . . . 6  |-  ( N  e.  N.  ->  ( <. { l  |  l 
<Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  e. 
P. )
79 opelxpi 4509 . . . . . 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 407 . . . . 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 7433 . . . . . 6  |-  ~R  e.  _V
8281ecelqsi 6413 . . . . 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 7446 . . . 4  |-  0R  e.  R.
85 opelxpi 4509 . . . 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 407 . . 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 3726 . . 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 1297    = wceq 1299    e. wcel 1448   {cab 2086   A.wral 2375   <.cop 3477   |^|cint 3718   class class class wbr 3875    X. cxp 4475  (class class class)co 5706   1oc1o 6236   [cec 6357   /.cqs 6358   N.cnpi 6981    +N cpli 6982    ~Q ceq 6988    <Q cltq 6994   P.cnp 7000   1Pc1p 7001    +P. cpp 7002    ~R cer 7005   R.cnr 7006   0Rc0r 7007   1c1 7501    + caddc 7503
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-eprel 4149  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-irdg 6197  df-1o 6243  df-2o 6244  df-oadd 6247  df-omul 6248  df-er 6359  df-ec 6361  df-qs 6365  df-ni 7013  df-pli 7014  df-mi 7015  df-lti 7016  df-plpq 7053  df-mpq 7054  df-enq 7056  df-nqqs 7057  df-plqqs 7058  df-mqqs 7059  df-1nqqs 7060  df-rq 7061  df-ltnqqs 7062  df-enq0 7133  df-nq0 7134  df-0nq0 7135  df-plq0 7136  df-mq0 7137  df-inp 7175  df-i1p 7176  df-iplp 7177  df-enr 7422  df-nr 7423  df-plr 7424  df-0r 7427  df-1r 7428  df-c 7506  df-1 7508  df-add 7511
This theorem is referenced by:  axarch  7576  axcaucvglemcl  7580  axcaucvglemval  7582  axcaucvglemcau  7583  axcaucvglemres  7584
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