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Theorem nntopi 7895
Description: Mapping from  NN to  N.. (Contributed by Jim Kingdon, 13-Jul-2021.)
Hypothesis
Ref Expression
nntopi.n  |-  N  = 
|^| { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1 )  e.  x ) }
Assertion
Ref Expression
nntopi  |-  ( A  e.  N  ->  E. z  e.  N.  <. [ <. ( <. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  A )
Distinct variable groups:    x, y    z, A    z, N, y, x   
u, l, z, y, x
Allowed substitution hints:    A( x, y, u, l)    N( u, l)

Proof of Theorem nntopi
Dummy variables  w  k  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nntopi.n . 2  |-  N  = 
|^| { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1 )  e.  x ) }
2 eqeq2 2187 . . 3  |-  ( w  =  1  ->  ( <. [ <. ( <. { l  |  l  <Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  w  <->  <. [ <. (
<. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  1 ) )
32rexbidv 2478 . 2  |-  ( w  =  1  ->  ( E. z  e.  N.  <. [ <. ( <. { l  |  l  <Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  w  <->  E. z  e.  N.  <. [ <. ( <. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  1 ) )
4 eqeq2 2187 . . 3  |-  ( w  =  k  ->  ( <. [ <. ( <. { l  |  l  <Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  w  <->  <. [ <. (
<. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  k ) )
54rexbidv 2478 . 2  |-  ( w  =  k  ->  ( E. z  e.  N.  <. [ <. ( <. { l  |  l  <Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  w  <->  E. z  e.  N.  <. [ <. ( <. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  k ) )
6 eqeq2 2187 . . 3  |-  ( w  =  ( k  +  1 )  ->  ( <. [ <. ( <. { l  |  l  <Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  w  <->  <. [ <. (
<. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  ( k  +  1 ) ) )
76rexbidv 2478 . 2  |-  ( w  =  ( k  +  1 )  ->  ( E. z  e.  N.  <. [ <. ( <. { l  |  l  <Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  w  <->  E. z  e.  N.  <. [ <. ( <. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  ( k  +  1 ) ) )
8 eqeq2 2187 . . 3  |-  ( w  =  A  ->  ( <. [ <. ( <. { l  |  l  <Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  w  <->  <. [ <. (
<. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  A ) )
98rexbidv 2478 . 2  |-  ( w  =  A  ->  ( E. z  e.  N.  <. [ <. ( <. { l  |  l  <Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  w  <->  E. z  e.  N.  <. [ <. ( <. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  A ) )
10 1pi 7316 . . 3  |-  1o  e.  N.
11 eqid 2177 . . 3  |-  1  =  1
12 opeq1 3780 . . . . . . . . . . . . . . . . 17  |-  ( z  =  1o  ->  <. z ,  1o >.  =  <. 1o ,  1o >. )
1312eceq1d 6573 . . . . . . . . . . . . . . . 16  |-  ( z  =  1o  ->  [ <. z ,  1o >. ]  ~Q  =  [ <. 1o ,  1o >. ]  ~Q  )
14 df-1nqqs 7352 . . . . . . . . . . . . . . . 16  |-  1Q  =  [ <. 1o ,  1o >. ]  ~Q
1513, 14eqtr4di 2228 . . . . . . . . . . . . . . 15  |-  ( z  =  1o  ->  [ <. z ,  1o >. ]  ~Q  =  1Q )
1615breq2d 4017 . . . . . . . . . . . . . 14  |-  ( z  =  1o  ->  (
l  <Q  [ <. z ,  1o >. ]  ~Q  <->  l  <Q  1Q ) )
1716abbidv 2295 . . . . . . . . . . . . 13  |-  ( z  =  1o  ->  { l  |  l  <Q  [ <. z ,  1o >. ]  ~Q  }  =  { l  |  l  <Q  1Q } )
1815breq1d 4015 . . . . . . . . . . . . . 14  |-  ( z  =  1o  ->  ( [ <. z ,  1o >. ]  ~Q  <Q  u  <->  1Q 
<Q  u ) )
1918abbidv 2295 . . . . . . . . . . . . 13  |-  ( z  =  1o  ->  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u }  =  { u  |  1Q  <Q  u }
)
2017, 19opeq12d 3788 . . . . . . . . . . . 12  |-  ( z  =  1o  ->  <. { l  |  l  <Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  =  <. { l  |  l  <Q  1Q } ,  { u  |  1Q  <Q  u } >. )
21 df-i1p 7468 . . . . . . . . . . . 12  |-  1P  =  <. { l  |  l 
<Q  1Q } ,  {
u  |  1Q  <Q  u } >.
2220, 21eqtr4di 2228 . . . . . . . . . . 11  |-  ( z  =  1o  ->  <. { l  |  l  <Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  =  1P )
2322oveq1d 5892 . . . . . . . . . 10  |-  ( z  =  1o  ->  ( <. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  =  ( 1P  +P.  1P ) )
2423opeq1d 3786 . . . . . . . . 9  |-  ( z  =  1o  ->  <. ( <. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >.  =  <. ( 1P  +P.  1P ) ,  1P >. )
2524eceq1d 6573 . . . . . . . 8  |-  ( z  =  1o  ->  [ <. (
<. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  )
26 df-1r 7733 . . . . . . . 8  |-  1R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
2725, 26eqtr4di 2228 . . . . . . 7  |-  ( z  =  1o  ->  [ <. (
<. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  =  1R )
2827opeq1d 3786 . . . . . 6  |-  ( z  =  1o  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  <. 1R ,  0R >. )
29 df-1 7821 . . . . . 6  |-  1  =  <. 1R ,  0R >.
3028, 29eqtr4di 2228 . . . . 5  |-  ( z  =  1o  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  1 )
3130eqeq1d 2186 . . . 4  |-  ( z  =  1o  ->  ( <. [ <. ( <. { l  |  l  <Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  1  <->  1  =  1 ) )
3231rspcev 2843 . . 3  |-  ( ( 1o  e.  N.  /\  1  =  1 )  ->  E. z  e.  N.  <. [ <. ( <. { l  |  l  <Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  1 )
3310, 11, 32mp2an 426 . 2  |-  E. z  e.  N.  <. [ <. ( <. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  1
34 simplr 528 . . . . . . 7  |-  ( ( ( k  e.  N  /\  z  e.  N. )  /\  <. [ <. ( <. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  k )  ->  z  e.  N. )
35 addclpi 7328 . . . . . . 7  |-  ( ( z  e.  N.  /\  1o  e.  N. )  -> 
( z  +N  1o )  e.  N. )
3634, 10, 35sylancl 413 . . . . . 6  |-  ( ( ( k  e.  N  /\  z  e.  N. )  /\  <. [ <. ( <. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  k )  ->  ( z  +N  1o )  e.  N. )
37 pitonnlem2 7848 . . . . . . . 8  |-  ( z  e.  N.  ->  ( <. [ <. ( <. { l  |  l  <Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  +  1 )  =  <. [ <. ( <. { l  |  l 
<Q  [ <. ( z  +N  1o ) ,  1o >. ]  ~Q  } ,  { u  |  [ <. ( z  +N  1o ) ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )
3834, 37syl 14 . . . . . . 7  |-  ( ( ( k  e.  N  /\  z  e.  N. )  /\  <. [ <. ( <. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  k )  ->  ( <. [ <. (
<. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  +  1 )  =  <. [ <. ( <. { l  |  l 
<Q  [ <. ( z  +N  1o ) ,  1o >. ]  ~Q  } ,  { u  |  [ <. ( z  +N  1o ) ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )
39 simpr 110 . . . . . . . 8  |-  ( ( ( k  e.  N  /\  z  e.  N. )  /\  <. [ <. ( <. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  k )  ->  <. [ <. ( <. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  k )
4039oveq1d 5892 . . . . . . 7  |-  ( ( ( k  e.  N  /\  z  e.  N. )  /\  <. [ <. ( <. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  k )  ->  ( <. [ <. (
<. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  +  1 )  =  ( k  +  1 ) )
4138, 40eqtr3d 2212 . . . . . 6  |-  ( ( ( k  e.  N  /\  z  e.  N. )  /\  <. [ <. ( <. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  k )  ->  <. [ <. ( <. { l  |  l 
<Q  [ <. ( z  +N  1o ) ,  1o >. ]  ~Q  } ,  { u  |  [ <. ( z  +N  1o ) ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  (
k  +  1 ) )
42 opeq1 3780 . . . . . . . . . . . . . . . 16  |-  ( v  =  ( z  +N  1o )  ->  <. v ,  1o >.  =  <. ( z  +N  1o ) ,  1o >. )
4342eceq1d 6573 . . . . . . . . . . . . . . 15  |-  ( v  =  ( z  +N  1o )  ->  [ <. v ,  1o >. ]  ~Q  =  [ <. ( z  +N  1o ) ,  1o >. ]  ~Q  )
4443breq2d 4017 . . . . . . . . . . . . . 14  |-  ( v  =  ( z  +N  1o )  ->  (
l  <Q  [ <. v ,  1o >. ]  ~Q  <->  l  <Q  [
<. ( z  +N  1o ) ,  1o >. ]  ~Q  ) )
4544abbidv 2295 . . . . . . . . . . . . 13  |-  ( v  =  ( z  +N  1o )  ->  { l  |  l  <Q  [ <. v ,  1o >. ]  ~Q  }  =  { l  |  l  <Q  [ <. (
z  +N  1o ) ,  1o >. ]  ~Q  } )
4643breq1d 4015 . . . . . . . . . . . . . 14  |-  ( v  =  ( z  +N  1o )  ->  ( [ <. v ,  1o >. ]  ~Q  <Q  u  <->  [
<. ( z  +N  1o ) ,  1o >. ]  ~Q  <Q  u ) )
4746abbidv 2295 . . . . . . . . . . . . 13  |-  ( v  =  ( z  +N  1o )  ->  { u  |  [ <. v ,  1o >. ]  ~Q  <Q  u }  =  { u  |  [ <. ( z  +N  1o ) ,  1o >. ]  ~Q  <Q  u } )
4845, 47opeq12d 3788 . . . . . . . . . . . 12  |-  ( v  =  ( z  +N  1o )  ->  <. { l  |  l  <Q  [ <. v ,  1o >. ]  ~Q  } ,  { u  |  [ <. v ,  1o >. ]  ~Q  <Q  u } >.  =  <. { l  |  l  <Q  [ <. ( z  +N  1o ) ,  1o >. ]  ~Q  } ,  { u  |  [ <. ( z  +N  1o ) ,  1o >. ]  ~Q  <Q  u } >. )
4948oveq1d 5892 . . . . . . . . . . 11  |-  ( v  =  ( z  +N  1o )  ->  ( <. { l  |  l 
<Q  [ <. v ,  1o >. ]  ~Q  } ,  { u  |  [ <. v ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  =  ( <. { l  |  l  <Q  [ <. (
z  +N  1o ) ,  1o >. ]  ~Q  } ,  { u  |  [ <. ( z  +N  1o ) ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )
)
5049opeq1d 3786 . . . . . . . . . 10  |-  ( v  =  ( z  +N  1o )  ->  <. ( <. { l  |  l 
<Q  [ <. v ,  1o >. ]  ~Q  } ,  { u  |  [ <. v ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >.  =  <. (
<. { l  |  l 
<Q  [ <. ( z  +N  1o ) ,  1o >. ]  ~Q  } ,  { u  |  [ <. ( z  +N  1o ) ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. )
5150eceq1d 6573 . . . . . . . . 9  |-  ( v  =  ( z  +N  1o )  ->  [ <. (
<. { l  |  l 
<Q  [ <. v ,  1o >. ]  ~Q  } ,  { u  |  [ <. v ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  =  [ <. ( <. { l  |  l  <Q  [ <. ( z  +N  1o ) ,  1o >. ]  ~Q  } ,  { u  |  [ <. ( z  +N  1o ) ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )
5251opeq1d 3786 . . . . . . . 8  |-  ( v  =  ( z  +N  1o )  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. v ,  1o >. ]  ~Q  } ,  { u  |  [ <. v ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  <. [ <. (
<. { l  |  l 
<Q  [ <. ( z  +N  1o ) ,  1o >. ]  ~Q  } ,  { u  |  [ <. ( z  +N  1o ) ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )
5352eqeq1d 2186 . . . . . . 7  |-  ( v  =  ( z  +N  1o )  ->  ( <. [ <. ( <. { l  |  l  <Q  [ <. v ,  1o >. ]  ~Q  } ,  { u  |  [ <. v ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  ( k  +  1 )  <->  <. [ <. (
<. { l  |  l 
<Q  [ <. ( z  +N  1o ) ,  1o >. ]  ~Q  } ,  { u  |  [ <. ( z  +N  1o ) ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  (
k  +  1 ) ) )
5453rspcev 2843 . . . . . 6  |-  ( ( ( z  +N  1o )  e.  N.  /\  <. [
<. ( <. { l  |  l  <Q  [ <. (
z  +N  1o ) ,  1o >. ]  ~Q  } ,  { u  |  [ <. ( z  +N  1o ) ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  ( k  +  1 ) )  ->  E. v  e.  N.  <. [ <. ( <. { l  |  l  <Q  [ <. v ,  1o >. ]  ~Q  } ,  { u  |  [ <. v ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  ( k  +  1 ) )
5536, 41, 54syl2anc 411 . . . . 5  |-  ( ( ( k  e.  N  /\  z  e.  N. )  /\  <. [ <. ( <. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  k )  ->  E. v  e.  N.  <. [ <. ( <. { l  |  l  <Q  [ <. v ,  1o >. ]  ~Q  } ,  { u  |  [ <. v ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  ( k  +  1 ) )
5655ex 115 . . . 4  |-  ( ( k  e.  N  /\  z  e.  N. )  ->  ( <. [ <. ( <. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  k  ->  E. v  e.  N.  <. [ <. ( <. { l  |  l  <Q  [ <. v ,  1o >. ]  ~Q  } ,  { u  |  [ <. v ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  ( k  +  1 ) ) )
5756rexlimdva 2594 . . 3  |-  ( k  e.  N  ->  ( E. z  e.  N.  <. [ <. ( <. { l  |  l  <Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  k  ->  E. v  e.  N.  <. [ <. ( <. { l  |  l  <Q  [ <. v ,  1o >. ]  ~Q  } ,  { u  |  [ <. v ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  ( k  +  1 ) ) )
58 opeq1 3780 . . . . . . . . . . . . 13  |-  ( v  =  z  ->  <. v ,  1o >.  =  <. z ,  1o >. )
5958eceq1d 6573 . . . . . . . . . . . 12  |-  ( v  =  z  ->  [ <. v ,  1o >. ]  ~Q  =  [ <. z ,  1o >. ]  ~Q  )
6059breq2d 4017 . . . . . . . . . . 11  |-  ( v  =  z  ->  (
l  <Q  [ <. v ,  1o >. ]  ~Q  <->  l  <Q  [
<. z ,  1o >. ]  ~Q  ) )
6160abbidv 2295 . . . . . . . . . 10  |-  ( v  =  z  ->  { l  |  l  <Q  [ <. v ,  1o >. ]  ~Q  }  =  { l  |  l  <Q  [ <. z ,  1o >. ]  ~Q  }
)
6259breq1d 4015 . . . . . . . . . . 11  |-  ( v  =  z  ->  ( [ <. v ,  1o >. ]  ~Q  <Q  u  <->  [
<. z ,  1o >. ]  ~Q  <Q  u )
)
6362abbidv 2295 . . . . . . . . . 10  |-  ( v  =  z  ->  { u  |  [ <. v ,  1o >. ]  ~Q  <Q  u }  =  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } )
6461, 63opeq12d 3788 . . . . . . . . 9  |-  ( v  =  z  ->  <. { l  |  l  <Q  [ <. v ,  1o >. ]  ~Q  } ,  { u  |  [ <. v ,  1o >. ]  ~Q  <Q  u } >.  =  <. { l  |  l  <Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >. )
6564oveq1d 5892 . . . . . . . 8  |-  ( v  =  z  ->  ( <. { l  |  l 
<Q  [ <. v ,  1o >. ]  ~Q  } ,  { u  |  [ <. v ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  =  ( <. { l  |  l  <Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) )
6665opeq1d 3786 . . . . . . 7  |-  ( v  =  z  ->  <. ( <. { l  |  l 
<Q  [ <. v ,  1o >. ]  ~Q  } ,  { u  |  [ <. v ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >.  =  <. (
<. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. )
6766eceq1d 6573 . . . . . 6  |-  ( v  =  z  ->  [ <. (
<. { l  |  l 
<Q  [ <. v ,  1o >. ]  ~Q  } ,  { u  |  [ <. v ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  =  [ <. ( <. { l  |  l  <Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )
6867opeq1d 3786 . . . . 5  |-  ( v  =  z  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. v ,  1o >. ]  ~Q  } ,  { u  |  [ <. v ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  <. [ <. (
<. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )
6968eqeq1d 2186 . . . 4  |-  ( v  =  z  ->  ( <. [ <. ( <. { l  |  l  <Q  [ <. v ,  1o >. ]  ~Q  } ,  { u  |  [ <. v ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  ( k  +  1 )  <->  <. [ <. (
<. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  ( k  +  1 ) ) )
7069cbvrexv 2706 . . 3  |-  ( E. v  e.  N.  <. [
<. ( <. { l  |  l  <Q  [ <. v ,  1o >. ]  ~Q  } ,  { u  |  [ <. v ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  ( k  +  1 )  <->  E. z  e.  N.  <. [ <. ( <. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  ( k  +  1 ) )
7157, 70imbitrdi 161 . 2  |-  ( k  e.  N  ->  ( E. z  e.  N.  <. [ <. ( <. { l  |  l  <Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  k  ->  E. z  e.  N.  <. [ <. ( <. { l  |  l  <Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  ( k  +  1 ) ) )
721, 3, 5, 7, 9, 33, 71nnindnn 7894 1  |-  ( A  e.  N  ->  E. z  e.  N.  <. [ <. ( <. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   {cab 2163   A.wral 2455   E.wrex 2456   <.cop 3597   |^|cint 3846   class class class wbr 4005  (class class class)co 5877   1oc1o 6412   [cec 6535   N.cnpi 7273    +N cpli 7274    ~Q ceq 7280   1Qc1q 7282    <Q cltq 7286   1Pc1p 7293    +P. cpp 7294    ~R cer 7297   0Rc0r 7299   1Rc1r 7300   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-r 7823  df-add 7824
This theorem is referenced by:  axcaucvglemres  7900
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