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Theorem caucvgprprlemmu 6851
Description: Lemma for caucvgprpr 6868. The upper cut of the putative limit is inhabited. (Contributed by Jim Kingdon, 29-Dec-2020.)
Hypotheses
Ref Expression
caucvgprpr.f  |-  ( ph  ->  F : N. --> P. )
caucvgprpr.cau  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <P  ( ( F `
 k )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( F `  k
)  <P  ( ( F `
 n )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )
) ) )
caucvgprpr.bnd  |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )
caucvgprpr.lim  |-  L  = 
<. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  r ) } ,  { u  e.  Q.  |  E. r  e.  N.  ( ( F `
 r )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  u } ,  { q  |  u 
<Q  q } >. } >.
Assertion
Ref Expression
caucvgprprlemmu  |-  ( ph  ->  E. t  e.  Q.  t  e.  ( 2nd `  L ) )
Distinct variable groups:    A, m    m, F    A, r, m    F, r, u    t, L    q, p, r, u
Allowed substitution hints:    ph( u, t, k, m, n, r, q, p, l)    A( u, t, k, n, q, p, l)    F( t, k, n, q, p, l)    L( u, k, m, n, r, q, p, l)

Proof of Theorem caucvgprprlemmu
Dummy variables  f  g  h  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 caucvgprpr.f . . . 4  |-  ( ph  ->  F : N. --> P. )
2 1pi 6471 . . . . 5  |-  1o  e.  N.
32a1i 9 . . . 4  |-  ( ph  ->  1o  e.  N. )
41, 3ffvelrnd 5331 . . 3  |-  ( ph  ->  ( F `  1o )  e.  P. )
5 prop 6631 . . 3  |-  ( ( F `  1o )  e.  P.  ->  <. ( 1st `  ( F `  1o ) ) ,  ( 2nd `  ( F `
 1o ) )
>.  e.  P. )
6 prmu 6634 . . 3  |-  ( <.
( 1st `  ( F `  1o )
) ,  ( 2nd `  ( F `  1o ) ) >.  e.  P.  ->  E. x  e.  Q.  x  e.  ( 2nd `  ( F `  1o ) ) )
74, 5, 63syl 17 . 2  |-  ( ph  ->  E. x  e.  Q.  x  e.  ( 2nd `  ( F `  1o ) ) )
8 simprl 491 . . . 4  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  ->  x  e.  Q. )
9 1nq 6522 . . . 4  |-  1Q  e.  Q.
10 addclnq 6531 . . . 4  |-  ( ( x  e.  Q.  /\  1Q  e.  Q. )  -> 
( x  +Q  1Q )  e.  Q. )
118, 9, 10sylancl 398 . . 3  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  -> 
( x  +Q  1Q )  e.  Q. )
122a1i 9 . . . . 5  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  ->  1o  e.  N. )
13 simprr 492 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  ->  x  e.  ( 2nd `  ( F `  1o ) ) )
144adantr 265 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  -> 
( F `  1o )  e.  P. )
15 nqpru 6708 . . . . . . . . 9  |-  ( ( x  e.  Q.  /\  ( F `  1o )  e.  P. )  -> 
( x  e.  ( 2nd `  ( F `
 1o ) )  <-> 
( F `  1o )  <P  <. { p  |  p  <Q  x } ,  { q  |  x 
<Q  q } >. )
)
168, 14, 15syl2anc 397 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  -> 
( x  e.  ( 2nd `  ( F `
 1o ) )  <-> 
( F `  1o )  <P  <. { p  |  p  <Q  x } ,  { q  |  x 
<Q  q } >. )
)
1713, 16mpbid 139 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  -> 
( F `  1o )  <P  <. { p  |  p  <Q  x } ,  { q  |  x 
<Q  q } >. )
18 ltaprg 6775 . . . . . . . . 9  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
f  <P  g  <->  ( h  +P.  f )  <P  (
h  +P.  g )
) )
1918adantl 266 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o ) ) ) )  /\  ( f  e. 
P.  /\  g  e.  P.  /\  h  e.  P. ) )  ->  (
f  <P  g  <->  ( h  +P.  f )  <P  (
h  +P.  g )
) )
20 nqprlu 6703 . . . . . . . . 9  |-  ( x  e.  Q.  ->  <. { p  |  p  <Q  x } ,  { q  |  x 
<Q  q } >.  e.  P. )
218, 20syl 14 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  ->  <. { p  |  p 
<Q  x } ,  {
q  |  x  <Q  q } >.  e.  P. )
22 nqprlu 6703 . . . . . . . . 9  |-  ( 1Q  e.  Q.  ->  <. { p  |  p  <Q  1Q } ,  { q  |  1Q  <Q  q } >.  e.  P. )
239, 22mp1i 10 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  ->  <. { p  |  p 
<Q  1Q } ,  {
q  |  1Q  <Q  q } >.  e.  P. )
24 addcomprg 6734 . . . . . . . . 9  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
2524adantl 266 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o ) ) ) )  /\  ( f  e. 
P.  /\  g  e.  P. ) )  ->  (
f  +P.  g )  =  ( g  +P.  f ) )
2619, 14, 21, 23, 25caovord2d 5698 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  -> 
( ( F `  1o )  <P  <. { p  |  p  <Q  x } ,  { q  |  x 
<Q  q } >.  <->  ( ( F `  1o )  +P.  <. { p  |  p  <Q  1Q } ,  { q  |  1Q  <Q  q } >. )  <P  ( <. { p  |  p  <Q  x } ,  { q  |  x 
<Q  q } >.  +P.  <. { p  |  p  <Q  1Q } ,  { q  |  1Q  <Q  q } >. ) ) )
2717, 26mpbid 139 . . . . . 6  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  -> 
( ( F `  1o )  +P.  <. { p  |  p  <Q  1Q } ,  { q  |  1Q  <Q  q } >. )  <P  ( <. { p  |  p  <Q  x } ,  { q  |  x 
<Q  q } >.  +P.  <. { p  |  p  <Q  1Q } ,  { q  |  1Q  <Q  q } >. ) )
28 df-1nqqs 6507 . . . . . . . . . . . . 13  |-  1Q  =  [ <. 1o ,  1o >. ]  ~Q
2928fveq2i 5209 . . . . . . . . . . . 12  |-  ( *Q
`  1Q )  =  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )
30 rec1nq 6551 . . . . . . . . . . . 12  |-  ( *Q
`  1Q )  =  1Q
3129, 30eqtr3i 2078 . . . . . . . . . . 11  |-  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  )  =  1Q
3231breq2i 3800 . . . . . . . . . 10  |-  ( p 
<Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <->  p  <Q  1Q )
3332abbii 2169 . . . . . . . . 9  |-  { p  |  p  <Q  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  ) }  =  { p  |  p  <Q  1Q }
3431breq1i 3799 . . . . . . . . . 10  |-  ( ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  q  <->  1Q  <Q  q )
3534abbii 2169 . . . . . . . . 9  |-  { q  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q 
q }  =  {
q  |  1Q  <Q  q }
3633, 35opeq12i 3582 . . . . . . . 8  |-  <. { p  |  p  <Q  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q 
q } >.  =  <. { p  |  p  <Q  1Q } ,  { q  |  1Q  <Q  q } >.
3736oveq2i 5551 . . . . . . 7  |-  ( ( F `  1o )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  q } >. )  =  ( ( F `  1o )  +P.  <. { p  |  p  <Q  1Q } ,  { q  |  1Q  <Q  q } >. )
3837a1i 9 . . . . . 6  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  -> 
( ( F `  1o )  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q 
q } >. )  =  ( ( F `
 1o )  +P. 
<. { p  |  p 
<Q  1Q } ,  {
q  |  1Q  <Q  q } >. ) )
39 addnqpr 6717 . . . . . . 7  |-  ( ( x  e.  Q.  /\  1Q  e.  Q. )  ->  <. { p  |  p 
<Q  ( x  +Q  1Q ) } ,  { q  |  ( x  +Q  1Q )  <Q  q }
>.  =  ( <. { p  |  p  <Q  x } ,  { q  |  x  <Q  q } >.  +P.  <. { p  |  p  <Q  1Q } ,  { q  |  1Q  <Q  q } >. )
)
408, 9, 39sylancl 398 . . . . . 6  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  ->  <. { p  |  p 
<Q  ( x  +Q  1Q ) } ,  { q  |  ( x  +Q  1Q )  <Q  q }
>.  =  ( <. { p  |  p  <Q  x } ,  { q  |  x  <Q  q } >.  +P.  <. { p  |  p  <Q  1Q } ,  { q  |  1Q  <Q  q } >. )
)
4127, 38, 403brtr4d 3822 . . . . 5  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  -> 
( ( F `  1o )  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  ( x  +Q  1Q ) } ,  { q  |  ( x  +Q  1Q )  <Q  q }
>. )
42 fveq2 5206 . . . . . . . 8  |-  ( r  =  1o  ->  ( F `  r )  =  ( F `  1o ) )
43 opeq1 3577 . . . . . . . . . . . . 13  |-  ( r  =  1o  ->  <. r ,  1o >.  =  <. 1o ,  1o >. )
4443eceq1d 6173 . . . . . . . . . . . 12  |-  ( r  =  1o  ->  [ <. r ,  1o >. ]  ~Q  =  [ <. 1o ,  1o >. ]  ~Q  )
4544fveq2d 5210 . . . . . . . . . . 11  |-  ( r  =  1o  ->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) )
4645breq2d 3804 . . . . . . . . . 10  |-  ( r  =  1o  ->  (
p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <->  p  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) ) )
4746abbidv 2171 . . . . . . . . 9  |-  ( r  =  1o  ->  { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) }  =  { p  |  p  <Q  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  ) } )
4845breq1d 3802 . . . . . . . . . 10  |-  ( r  =  1o  ->  (
( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  q ) )
4948abbidv 2171 . . . . . . . . 9  |-  ( r  =  1o  ->  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q }  =  {
q  |  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  )  <Q  q } )
5047, 49opeq12d 3585 . . . . . . . 8  |-  ( r  =  1o  ->  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >.  =  <. { p  |  p  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q 
q } >. )
5142, 50oveq12d 5558 . . . . . . 7  |-  ( r  =  1o  ->  (
( F `  r
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  =  ( ( F `
 1o )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  q } >. ) )
5251breq1d 3802 . . . . . 6  |-  ( r  =  1o  ->  (
( ( F `  r )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  ( x  +Q  1Q ) } ,  { q  |  ( x  +Q  1Q )  <Q  q }
>. 
<->  ( ( F `  1o )  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  ( x  +Q  1Q ) } ,  { q  |  ( x  +Q  1Q )  <Q  q }
>. ) )
5352rspcev 2673 . . . . 5  |-  ( ( 1o  e.  N.  /\  ( ( F `  1o )  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  ( x  +Q  1Q ) } ,  { q  |  ( x  +Q  1Q )  <Q  q }
>. )  ->  E. r  e.  N.  ( ( F `
 r )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  ( x  +Q  1Q ) } ,  { q  |  ( x  +Q  1Q ) 
<Q  q } >. )
5412, 41, 53syl2anc 397 . . . 4  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  ->  E. r  e.  N.  ( ( F `  r )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  ( x  +Q  1Q ) } ,  { q  |  ( x  +Q  1Q )  <Q  q }
>. )
55 breq2 3796 . . . . . . . . 9  |-  ( u  =  ( x  +Q  1Q )  ->  ( p 
<Q  u  <->  p  <Q  ( x  +Q  1Q ) ) )
5655abbidv 2171 . . . . . . . 8  |-  ( u  =  ( x  +Q  1Q )  ->  { p  |  p  <Q  u }  =  { p  |  p 
<Q  ( x  +Q  1Q ) } )
57 breq1 3795 . . . . . . . . 9  |-  ( u  =  ( x  +Q  1Q )  ->  ( u 
<Q  q  <->  ( x  +Q  1Q )  <Q  q ) )
5857abbidv 2171 . . . . . . . 8  |-  ( u  =  ( x  +Q  1Q )  ->  { q  |  u  <Q  q }  =  { q  |  ( x  +Q  1Q )  <Q  q } )
5956, 58opeq12d 3585 . . . . . . 7  |-  ( u  =  ( x  +Q  1Q )  ->  <. { p  |  p  <Q  u } ,  { q  |  u 
<Q  q } >.  =  <. { p  |  p  <Q  ( x  +Q  1Q ) } ,  { q  |  ( x  +Q  1Q )  <Q  q }
>. )
6059breq2d 3804 . . . . . 6  |-  ( u  =  ( x  +Q  1Q )  ->  ( ( ( F `  r
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  u } ,  {
q  |  u  <Q  q } >.  <->  ( ( F `
 r )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  ( x  +Q  1Q ) } ,  { q  |  ( x  +Q  1Q ) 
<Q  q } >. )
)
6160rexbidv 2344 . . . . 5  |-  ( u  =  ( x  +Q  1Q )  ->  ( E. r  e.  N.  (
( F `  r
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  u } ,  {
q  |  u  <Q  q } >.  <->  E. r  e.  N.  ( ( F `  r )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  ( x  +Q  1Q ) } ,  { q  |  ( x  +Q  1Q )  <Q  q }
>. ) )
62 caucvgprpr.lim . . . . . . 7  |-  L  = 
<. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  r ) } ,  { u  e.  Q.  |  E. r  e.  N.  ( ( F `
 r )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  u } ,  { q  |  u 
<Q  q } >. } >.
6362fveq2i 5209 . . . . . 6  |-  ( 2nd `  L )  =  ( 2nd `  <. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q } >.  <P  ( F `  r ) } ,  { u  e.  Q.  |  E. r  e.  N.  ( ( F `
 r )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  u } ,  { q  |  u 
<Q  q } >. } >. )
64 nqex 6519 . . . . . . . 8  |-  Q.  e.  _V
6564rabex 3929 . . . . . . 7  |-  { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q } >.  <P  ( F `  r ) }  e.  _V
6664rabex 3929 . . . . . . 7  |-  { u  e.  Q.  |  E. r  e.  N.  ( ( F `
 r )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  u } ,  { q  |  u 
<Q  q } >. }  e.  _V
6765, 66op2nd 5802 . . . . . 6  |-  ( 2nd `  <. { l  e. 
Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P 
( F `  r
) } ,  {
u  e.  Q.  |  E. r  e.  N.  ( ( F `  r )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  u } ,  {
q  |  u  <Q  q } >. } >. )  =  { u  e.  Q.  |  E. r  e.  N.  ( ( F `  r )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  u } ,  {
q  |  u  <Q  q } >. }
6863, 67eqtri 2076 . . . . 5  |-  ( 2nd `  L )  =  {
u  e.  Q.  |  E. r  e.  N.  ( ( F `  r )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  u } ,  {
q  |  u  <Q  q } >. }
6961, 68elrab2 2723 . . . 4  |-  ( ( x  +Q  1Q )  e.  ( 2nd `  L
)  <->  ( ( x  +Q  1Q )  e. 
Q.  /\  E. r  e.  N.  ( ( F `
 r )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  ( x  +Q  1Q ) } ,  { q  |  ( x  +Q  1Q ) 
<Q  q } >. )
)
7011, 54, 69sylanbrc 402 . . 3  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  -> 
( x  +Q  1Q )  e.  ( 2nd `  L ) )
71 eleq1 2116 . . . 4  |-  ( t  =  ( x  +Q  1Q )  ->  ( t  e.  ( 2nd `  L
)  <->  ( x  +Q  1Q )  e.  ( 2nd `  L ) ) )
7271rspcev 2673 . . 3  |-  ( ( ( x  +Q  1Q )  e.  Q.  /\  (
x  +Q  1Q )  e.  ( 2nd `  L
) )  ->  E. t  e.  Q.  t  e.  ( 2nd `  L ) )
7311, 70, 72syl2anc 397 . 2  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  ->  E. t  e.  Q.  t  e.  ( 2nd `  L ) )
747, 73rexlimddv 2454 1  |-  ( ph  ->  E. t  e.  Q.  t  e.  ( 2nd `  L ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    /\ w3a 896    = wceq 1259    e. wcel 1409   {cab 2042   A.wral 2323   E.wrex 2324   {crab 2327   <.cop 3406   class class class wbr 3792   -->wf 4926   ` cfv 4930  (class class class)co 5540   1stc1st 5793   2ndc2nd 5794   1oc1o 6025   [cec 6135   N.cnpi 6428    <N clti 6431    ~Q ceq 6435   Q.cnq 6436   1Qc1q 6437    +Q cplq 6438   *Qcrq 6440    <Q cltq 6441   P.cnp 6447    +P. cpp 6449    <P cltp 6451
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-iplp 6624  df-iltp 6626
This theorem is referenced by:  caucvgprprlemm  6852
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