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Theorem nntopi 7026
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 2065 . . 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 2344 . 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 2065 . . 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 2344 . 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 2065 . . 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 2344 . 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 2065 . . 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 2344 . 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 6471 . . 3  |-  1o  e.  N.
11 eqid 2056 . . 3  |-  1  =  1
12 opeq1 3577 . . . . . . . . . . . . . . . . 17  |-  ( z  =  1o  ->  <. z ,  1o >.  =  <. 1o ,  1o >. )
1312eceq1d 6173 . . . . . . . . . . . . . . . 16  |-  ( z  =  1o  ->  [ <. z ,  1o >. ]  ~Q  =  [ <. 1o ,  1o >. ]  ~Q  )
14 df-1nqqs 6507 . . . . . . . . . . . . . . . 16  |-  1Q  =  [ <. 1o ,  1o >. ]  ~Q
1513, 14syl6eqr 2106 . . . . . . . . . . . . . . 15  |-  ( z  =  1o  ->  [ <. z ,  1o >. ]  ~Q  =  1Q )
1615breq2d 3804 . . . . . . . . . . . . . 14  |-  ( z  =  1o  ->  (
l  <Q  [ <. z ,  1o >. ]  ~Q  <->  l  <Q  1Q ) )
1716abbidv 2171 . . . . . . . . . . . . 13  |-  ( z  =  1o  ->  { l  |  l  <Q  [ <. z ,  1o >. ]  ~Q  }  =  { l  |  l  <Q  1Q } )
1815breq1d 3802 . . . . . . . . . . . . . 14  |-  ( z  =  1o  ->  ( [ <. z ,  1o >. ]  ~Q  <Q  u  <->  1Q 
<Q  u ) )
1918abbidv 2171 . . . . . . . . . . . . 13  |-  ( z  =  1o  ->  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u }  =  { u  |  1Q  <Q  u }
)
2017, 19opeq12d 3585 . . . . . . . . . . . 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 6623 . . . . . . . . . . . 12  |-  1P  =  <. { l  |  l 
<Q  1Q } ,  {
u  |  1Q  <Q  u } >.
2220, 21syl6eqr 2106 . . . . . . . . . . 11  |-  ( z  =  1o  ->  <. { l  |  l  <Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  =  1P )
2322oveq1d 5555 . . . . . . . . . 10  |-  ( z  =  1o  ->  ( <. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P )  =  ( 1P  +P.  1P ) )
2423opeq1d 3583 . . . . . . . . 9  |-  ( z  =  1o  ->  <. ( <. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >.  =  <. ( 1P  +P.  1P ) ,  1P >. )
2524eceq1d 6173 . . . . . . . 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 6875 . . . . . . . 8  |-  1R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
2725, 26syl6eqr 2106 . . . . . . 7  |-  ( z  =  1o  ->  [ <. (
<. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  =  1R )
2827opeq1d 3583 . . . . . 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 6955 . . . . . 6  |-  1  =  <. 1R ,  0R >.
3028, 29syl6eqr 2106 . . . . 5  |-  ( z  =  1o  ->  <. [ <. (
<. { l  |  l 
<Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  1 )
3130eqeq1d 2064 . . . 4  |-  ( z  =  1o  ->  ( <. [ <. ( <. { l  |  l  <Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  1  <->  1  =  1 ) )
3231rspcev 2673 . . 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 410 . 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 490 . . . . . . 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 6483 . . . . . . 7  |-  ( ( z  e.  N.  /\  1o  e.  N. )  -> 
( z  +N  1o )  e.  N. )
3634, 10, 35sylancl 398 . . . . . 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 6981 . . . . . . . 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 107 . . . . . . . 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 5555 . . . . . . 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 2090 . . . . . 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 3577 . . . . . . . . . . . . . . . 16  |-  ( v  =  ( z  +N  1o )  ->  <. v ,  1o >.  =  <. ( z  +N  1o ) ,  1o >. )
4342eceq1d 6173 . . . . . . . . . . . . . . 15  |-  ( v  =  ( z  +N  1o )  ->  [ <. v ,  1o >. ]  ~Q  =  [ <. ( z  +N  1o ) ,  1o >. ]  ~Q  )
4443breq2d 3804 . . . . . . . . . . . . . 14  |-  ( v  =  ( z  +N  1o )  ->  (
l  <Q  [ <. v ,  1o >. ]  ~Q  <->  l  <Q  [
<. ( z  +N  1o ) ,  1o >. ]  ~Q  ) )
4544abbidv 2171 . . . . . . . . . . . . 13  |-  ( v  =  ( z  +N  1o )  ->  { l  |  l  <Q  [ <. v ,  1o >. ]  ~Q  }  =  { l  |  l  <Q  [ <. (
z  +N  1o ) ,  1o >. ]  ~Q  } )
4643breq1d 3802 . . . . . . . . . . . . . 14  |-  ( v  =  ( z  +N  1o )  ->  ( [ <. v ,  1o >. ]  ~Q  <Q  u  <->  [
<. ( z  +N  1o ) ,  1o >. ]  ~Q  <Q  u ) )
4746abbidv 2171 . . . . . . . . . . . . 13  |-  ( v  =  ( z  +N  1o )  ->  { u  |  [ <. v ,  1o >. ]  ~Q  <Q  u }  =  { u  |  [ <. ( z  +N  1o ) ,  1o >. ]  ~Q  <Q  u } )
4845, 47opeq12d 3585 . . . . . . . . . . . 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 5555 . . . . . . . . . . 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 3583 . . . . . . . . . 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 6173 . . . . . . . . 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 3583 . . . . . . . 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 2064 . . . . . . 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 2673 . . . . . 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 397 . . . . 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 112 . . . 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 2450 . . 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 3577 . . . . . . . . . . . . 13  |-  ( v  =  z  ->  <. v ,  1o >.  =  <. z ,  1o >. )
5958eceq1d 6173 . . . . . . . . . . . 12  |-  ( v  =  z  ->  [ <. v ,  1o >. ]  ~Q  =  [ <. z ,  1o >. ]  ~Q  )
6059breq2d 3804 . . . . . . . . . . 11  |-  ( v  =  z  ->  (
l  <Q  [ <. v ,  1o >. ]  ~Q  <->  l  <Q  [
<. z ,  1o >. ]  ~Q  ) )
6160abbidv 2171 . . . . . . . . . 10  |-  ( v  =  z  ->  { l  |  l  <Q  [ <. v ,  1o >. ]  ~Q  }  =  { l  |  l  <Q  [ <. z ,  1o >. ]  ~Q  }
)
6259breq1d 3802 . . . . . . . . . . 11  |-  ( v  =  z  ->  ( [ <. v ,  1o >. ]  ~Q  <Q  u  <->  [
<. z ,  1o >. ]  ~Q  <Q  u )
)
6362abbidv 2171 . . . . . . . . . 10  |-  ( v  =  z  ->  { u  |  [ <. v ,  1o >. ]  ~Q  <Q  u }  =  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } )
6461, 63opeq12d 3585 . . . . . . . . 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 5555 . . . . . . . 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 3583 . . . . . . 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 6173 . . . . . 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 3583 . . . . 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 2064 . . . 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 2551 . . 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 154 . 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 7025 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 101    = wceq 1259    e. wcel 1409   {cab 2042   A.wral 2323   E.wrex 2324   <.cop 3406   |^|cint 3643   class class class wbr 3792  (class class class)co 5540   1oc1o 6025   [cec 6135   N.cnpi 6428    +N cpli 6429    ~Q ceq 6435   1Qc1q 6437    <Q cltq 6441   1Pc1p 6448    +P. cpp 6449    ~R cer 6452   0Rc0r 6454   1Rc1r 6455   1c1 6948    + caddc 6950
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-i1p 6623  df-iplp 6624  df-enr 6869  df-nr 6870  df-plr 6871  df-0r 6874  df-1r 6875  df-c 6953  df-1 6955  df-r 6957  df-add 6958
This theorem is referenced by:  axcaucvglemres  7031
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