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Theorem caucvgprprlemmu 6774
Description: Lemma for caucvgprpr 6791. 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 6394 . . . . 5  |-  1o  e.  N.
32a1i 9 . . . 4  |-  ( ph  ->  1o  e.  N. )
41, 3ffvelrnd 5290 . . 3  |-  ( ph  ->  ( F `  1o )  e.  P. )
5 prop 6554 . . 3  |-  ( ( F `  1o )  e.  P.  ->  <. ( 1st `  ( F `  1o ) ) ,  ( 2nd `  ( F `
 1o ) )
>.  e.  P. )
6 prmu 6557 . . 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 483 . . . 4  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  ->  x  e.  Q. )
9 1nq 6445 . . . 4  |-  1Q  e.  Q.
10 addclnq 6454 . . . 4  |-  ( ( x  e.  Q.  /\  1Q  e.  Q. )  -> 
( x  +Q  1Q )  e.  Q. )
118, 9, 10sylancl 392 . . 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 484 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  ->  x  e.  ( 2nd `  ( F `  1o ) ) )
144adantr 261 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  -> 
( F `  1o )  e.  P. )
15 nqpru 6631 . . . . . . . . 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 391 . . . . . . . 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 135 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  -> 
( F `  1o )  <P  <. { p  |  p  <Q  x } ,  { q  |  x 
<Q  q } >. )
18 ltaprg 6698 . . . . . . . . 9  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
f  <P  g  <->  ( h  +P.  f )  <P  (
h  +P.  g )
) )
1918adantl 262 . . . . . . . 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 6626 . . . . . . . . 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 6626 . . . . . . . . 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 6657 . . . . . . . . 9  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
2524adantl 262 . . . . . . . 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 5657 . . . . . . 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 135 . . . . . 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 6430 . . . . . . . . . . . . 13  |-  1Q  =  [ <. 1o ,  1o >. ]  ~Q
2928fveq2i 5168 . . . . . . . . . . . 12  |-  ( *Q
`  1Q )  =  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )
30 rec1nq 6474 . . . . . . . . . . . 12  |-  ( *Q
`  1Q )  =  1Q
3129, 30eqtr3i 2062 . . . . . . . . . . 11  |-  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  )  =  1Q
3231breq2i 3769 . . . . . . . . . 10  |-  ( p 
<Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <->  p  <Q  1Q )
3332abbii 2153 . . . . . . . . 9  |-  { p  |  p  <Q  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  ) }  =  { p  |  p  <Q  1Q }
3431breq1i 3768 . . . . . . . . . 10  |-  ( ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  q  <->  1Q  <Q  q )
3534abbii 2153 . . . . . . . . 9  |-  { q  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q 
q }  =  {
q  |  1Q  <Q  q }
3633, 35opeq12i 3551 . . . . . . . 8  |-  <. { p  |  p  <Q  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q 
q } >.  =  <. { p  |  p  <Q  1Q } ,  { q  |  1Q  <Q  q } >.
3736oveq2i 5510 . . . . . . 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 6640 . . . . . . 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 392 . . . . . 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 3791 . . . . 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 5165 . . . . . . . 8  |-  ( r  =  1o  ->  ( F `  r )  =  ( F `  1o ) )
43 opeq1 3546 . . . . . . . . . . . . 13  |-  ( r  =  1o  ->  <. r ,  1o >.  =  <. 1o ,  1o >. )
4443eceq1d 6129 . . . . . . . . . . . 12  |-  ( r  =  1o  ->  [ <. r ,  1o >. ]  ~Q  =  [ <. 1o ,  1o >. ]  ~Q  )
4544fveq2d 5169 . . . . . . . . . . 11  |-  ( r  =  1o  ->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) )
4645breq2d 3773 . . . . . . . . . 10  |-  ( r  =  1o  ->  (
p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <->  p  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) ) )
4746abbidv 2155 . . . . . . . . 9  |-  ( r  =  1o  ->  { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) }  =  { p  |  p  <Q  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  ) } )
4845breq1d 3771 . . . . . . . . . 10  |-  ( r  =  1o  ->  (
( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  q ) )
4948abbidv 2155 . . . . . . . . 9  |-  ( r  =  1o  ->  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q }  =  {
q  |  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  )  <Q  q } )
5047, 49opeq12d 3554 . . . . . . . 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 5517 . . . . . . 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 3771 . . . . . 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 2653 . . . . 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 391 . . . 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 3765 . . . . . . . . 9  |-  ( u  =  ( x  +Q  1Q )  ->  ( p 
<Q  u  <->  p  <Q  ( x  +Q  1Q ) ) )
5655abbidv 2155 . . . . . . . 8  |-  ( u  =  ( x  +Q  1Q )  ->  { p  |  p  <Q  u }  =  { p  |  p 
<Q  ( x  +Q  1Q ) } )
57 breq1 3764 . . . . . . . . 9  |-  ( u  =  ( x  +Q  1Q )  ->  ( u 
<Q  q  <->  ( x  +Q  1Q )  <Q  q ) )
5857abbidv 2155 . . . . . . . 8  |-  ( u  =  ( x  +Q  1Q )  ->  { q  |  u  <Q  q }  =  { q  |  ( x  +Q  1Q )  <Q  q } )
5956, 58opeq12d 3554 . . . . . . 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 3773 . . . . . 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 2324 . . . . 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 5168 . . . . . 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 6442 . . . . . . . 8  |-  Q.  e.  _V
6564rabex 3898 . . . . . . 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 3898 . . . . . . 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 5761 . . . . . 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 2060 . . . . 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 2697 . . . 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 394 . . 3  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  -> 
( x  +Q  1Q )  e.  ( 2nd `  L ) )
71 eleq1 2100 . . . 4  |-  ( t  =  ( x  +Q  1Q )  ->  ( t  e.  ( 2nd `  L
)  <->  ( x  +Q  1Q )  e.  ( 2nd `  L ) ) )
7271rspcev 2653 . . 3  |-  ( ( ( x  +Q  1Q )  e.  Q.  /\  (
x  +Q  1Q )  e.  ( 2nd `  L
) )  ->  E. t  e.  Q.  t  e.  ( 2nd `  L ) )
7311, 70, 72syl2anc 391 . 2  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  ->  E. t  e.  Q.  t  e.  ( 2nd `  L ) )
747, 73rexlimddv 2434 1  |-  ( ph  ->  E. t  e.  Q.  t  e.  ( 2nd `  L ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    /\ w3a 885    = wceq 1243    e. wcel 1393   {cab 2026   A.wral 2303   E.wrex 2304   {crab 2307   <.cop 3375   class class class wbr 3761   -->wf 4885   ` cfv 4889  (class class class)co 5499   1stc1st 5752   2ndc2nd 5753   1oc1o 5981   [cec 6091   N.cnpi 6351    <N clti 6354    ~Q ceq 6358   Q.cnq 6359   1Qc1q 6360    +Q cplq 6361   *Qcrq 6363    <Q cltq 6364   P.cnp 6370    +P. cpp 6372    <P cltp 6374
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-iplp 6547  df-iltp 6549
This theorem is referenced by:  caucvgprprlemm  6775
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