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Theorem nntopi 6949
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 2049 . . 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 2324 . 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 2049 . . 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 2324 . 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 2049 . . 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 2324 . 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 2049 . . 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 2324 . 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 6394 . . 3  |-  1o  e.  N.
11 eqid 2040 . . 3  |-  1  =  1
12 opeq1 3546 . . . . . . . . . . . . . . . . 17  |-  ( z  =  1o  ->  <. z ,  1o >.  =  <. 1o ,  1o >. )
1312eceq1d 6129 . . . . . . . . . . . . . . . 16  |-  ( z  =  1o  ->  [ <. z ,  1o >. ]  ~Q  =  [ <. 1o ,  1o >. ]  ~Q  )
14 df-1nqqs 6430 . . . . . . . . . . . . . . . 16  |-  1Q  =  [ <. 1o ,  1o >. ]  ~Q
1513, 14syl6eqr 2090 . . . . . . . . . . . . . . 15  |-  ( z  =  1o  ->  [ <. z ,  1o >. ]  ~Q  =  1Q )
1615breq2d 3773 . . . . . . . . . . . . . 14  |-  ( z  =  1o  ->  (
l  <Q  [ <. z ,  1o >. ]  ~Q  <->  l  <Q  1Q ) )
1716abbidv 2155 . . . . . . . . . . . . 13  |-  ( z  =  1o  ->  { l  |  l  <Q  [ <. z ,  1o >. ]  ~Q  }  =  { l  |  l  <Q  1Q } )
1815breq1d 3771 . . . . . . . . . . . . . 14  |-  ( z  =  1o  ->  ( [ <. z ,  1o >. ]  ~Q  <Q  u  <->  1Q 
<Q  u ) )
1918abbidv 2155 . . . . . . . . . . . . 13  |-  ( z  =  1o  ->  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u }  =  { u  |  1Q  <Q  u }
)
2017, 19opeq12d 3554 . . . . . . . . . . . 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 6546 . . . . . . . . . . . 12  |-  1P  =  <. { l  |  l 
<Q  1Q } ,  {
u  |  1Q  <Q  u } >.
2220, 21syl6eqr 2090 . . . . . . . . . . 11  |-  ( z  =  1o  ->  <. { l  |  l  <Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  =  1P )
2322oveq1d 5514 . . . . . . . . . 10  |-  ( z  =  1o  ->  ( <. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  =  ( 1P  +P.  1P ) )
2423opeq1d 3552 . . . . . . . . 9  |-  ( z  =  1o  ->  <. ( <. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >.  =  <. ( 1P  +P.  1P ) ,  1P >. )
2524eceq1d 6129 . . . . . . . 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 6798 . . . . . . . 8  |-  1R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
2725, 26syl6eqr 2090 . . . . . . 7  |-  ( z  =  1o  ->  [ <. (
<. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  =  1R )
2827opeq1d 3552 . . . . . 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 6878 . . . . . 6  |-  1  =  <. 1R ,  0R >.
3028, 29syl6eqr 2090 . . . . 5  |-  ( z  =  1o  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  1 )
3130eqeq1d 2048 . . . 4  |-  ( z  =  1o  ->  ( <. [ <. ( <. { l  |  l  <Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  1  <->  1  =  1 ) )
3231rspcev 2653 . . 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 402 . 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 482 . . . . . . 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 6406 . . . . . . 7  |-  ( ( z  e.  N.  /\  1o  e.  N. )  -> 
( z  +N  1o )  e.  N. )
3634, 10, 35sylancl 392 . . . . . 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 6904 . . . . . . . 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 103 . . . . . . . 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 5514 . . . . . . 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 2074 . . . . . 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 3546 . . . . . . . . . . . . . . . 16  |-  ( v  =  ( z  +N  1o )  ->  <. v ,  1o >.  =  <. ( z  +N  1o ) ,  1o >. )
4342eceq1d 6129 . . . . . . . . . . . . . . 15  |-  ( v  =  ( z  +N  1o )  ->  [ <. v ,  1o >. ]  ~Q  =  [ <. ( z  +N  1o ) ,  1o >. ]  ~Q  )
4443breq2d 3773 . . . . . . . . . . . . . 14  |-  ( v  =  ( z  +N  1o )  ->  (
l  <Q  [ <. v ,  1o >. ]  ~Q  <->  l  <Q  [
<. ( z  +N  1o ) ,  1o >. ]  ~Q  ) )
4544abbidv 2155 . . . . . . . . . . . . 13  |-  ( v  =  ( z  +N  1o )  ->  { l  |  l  <Q  [ <. v ,  1o >. ]  ~Q  }  =  { l  |  l  <Q  [ <. (
z  +N  1o ) ,  1o >. ]  ~Q  } )
4643breq1d 3771 . . . . . . . . . . . . . 14  |-  ( v  =  ( z  +N  1o )  ->  ( [ <. v ,  1o >. ]  ~Q  <Q  u  <->  [
<. ( z  +N  1o ) ,  1o >. ]  ~Q  <Q  u ) )
4746abbidv 2155 . . . . . . . . . . . . 13  |-  ( v  =  ( z  +N  1o )  ->  { u  |  [ <. v ,  1o >. ]  ~Q  <Q  u }  =  { u  |  [ <. ( z  +N  1o ) ,  1o >. ]  ~Q  <Q  u } )
4845, 47opeq12d 3554 . . . . . . . . . . . 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 5514 . . . . . . . . . . 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 3552 . . . . . . . . . 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 6129 . . . . . . . . 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 3552 . . . . . . . 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 2048 . . . . . . 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 2653 . . . . . 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 391 . . . . 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 108 . . . 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 2430 . . 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 3546 . . . . . . . . . . . . 13  |-  ( v  =  z  ->  <. v ,  1o >.  =  <. z ,  1o >. )
5958eceq1d 6129 . . . . . . . . . . . 12  |-  ( v  =  z  ->  [ <. v ,  1o >. ]  ~Q  =  [ <. z ,  1o >. ]  ~Q  )
6059breq2d 3773 . . . . . . . . . . 11  |-  ( v  =  z  ->  (
l  <Q  [ <. v ,  1o >. ]  ~Q  <->  l  <Q  [
<. z ,  1o >. ]  ~Q  ) )
6160abbidv 2155 . . . . . . . . . 10  |-  ( v  =  z  ->  { l  |  l  <Q  [ <. v ,  1o >. ]  ~Q  }  =  { l  |  l  <Q  [ <. z ,  1o >. ]  ~Q  }
)
6259breq1d 3771 . . . . . . . . . . 11  |-  ( v  =  z  ->  ( [ <. v ,  1o >. ]  ~Q  <Q  u  <->  [
<. z ,  1o >. ]  ~Q  <Q  u )
)
6362abbidv 2155 . . . . . . . . . 10  |-  ( v  =  z  ->  { u  |  [ <. v ,  1o >. ]  ~Q  <Q  u }  =  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } )
6461, 63opeq12d 3554 . . . . . . . . 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 5514 . . . . . . . 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 3552 . . . . . . 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 6129 . . . . . 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 3552 . . . . 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 2048 . . . 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 2531 . . 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, 70syl6ib 150 . 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 6948 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 97    = wceq 1243    e. wcel 1393   {cab 2026   A.wral 2303   E.wrex 2304   <.cop 3375   |^|cint 3612   class class class wbr 3761  (class class class)co 5499   1oc1o 5981   [cec 6091   N.cnpi 6351    +N cpli 6352    ~Q ceq 6358   1Qc1q 6360    <Q cltq 6364   1Pc1p 6371    +P. cpp 6372    ~R cer 6375   0Rc0r 6377   1Rc1r 6378   1c1 6871    + caddc 6873
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 4166  ax-setind 4256  ax-iinf 4298
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 4099  df-on 4101  df-suc 4104  df-iom 4301  df-xp 4338  df-rel 4339  df-cnv 4340  df-co 4341  df-dm 4342  df-rn 4343  df-res 4344  df-ima 4345  df-iota 4854  df-fun 4891  df-fn 4892  df-f 4893  df-f1 4894  df-fo 4895  df-f1o 4896  df-fv 4897  df-ov 5502  df-oprab 5503  df-mpt2 5504  df-1st 5754  df-2nd 5755  df-recs 5907  df-irdg 5944  df-1o 5988  df-2o 5989  df-oadd 5992  df-omul 5993  df-er 6093  df-ec 6095  df-qs 6099  df-ni 6383  df-pli 6384  df-mi 6385  df-lti 6386  df-plpq 6423  df-mpq 6424  df-enq 6426  df-nqqs 6427  df-plqqs 6428  df-mqqs 6429  df-1nqqs 6430  df-rq 6431  df-ltnqqs 6432  df-enq0 6503  df-nq0 6504  df-0nq0 6505  df-plq0 6506  df-mq0 6507  df-inp 6545  df-i1p 6546  df-iplp 6547  df-enr 6792  df-nr 6793  df-plr 6794  df-0r 6797  df-1r 6798  df-c 6876  df-1 6878  df-r 6880  df-add 6881
This theorem is referenced by:  axcaucvglemres  6954
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