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Theorem caucvgprprlemmu 7351
Description: Lemma for caucvgprpr 7368. 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 6971 . . . . 5  |-  1o  e.  N.
32a1i 9 . . . 4  |-  ( ph  ->  1o  e.  N. )
41, 3ffvelrnd 5474 . . 3  |-  ( ph  ->  ( F `  1o )  e.  P. )
5 prop 7131 . . 3  |-  ( ( F `  1o )  e.  P.  ->  <. ( 1st `  ( F `  1o ) ) ,  ( 2nd `  ( F `
 1o ) )
>.  e.  P. )
6 prmu 7134 . . 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 499 . . . 4  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  ->  x  e.  Q. )
9 1nq 7022 . . . 4  |-  1Q  e.  Q.
10 addclnq 7031 . . . 4  |-  ( ( x  e.  Q.  /\  1Q  e.  Q. )  -> 
( x  +Q  1Q )  e.  Q. )
118, 9, 10sylancl 405 . . 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 500 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  ->  x  e.  ( 2nd `  ( F `  1o ) ) )
144adantr 271 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  -> 
( F `  1o )  e.  P. )
15 nqpru 7208 . . . . . . . . 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 404 . . . . . . . 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 146 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  -> 
( F `  1o )  <P  <. { p  |  p  <Q  x } ,  { q  |  x 
<Q  q } >. )
18 ltaprg 7275 . . . . . . . . 9  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
f  <P  g  <->  ( h  +P.  f )  <P  (
h  +P.  g )
) )
1918adantl 272 . . . . . . . 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 7203 . . . . . . . . 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 7203 . . . . . . . . 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 7234 . . . . . . . . 9  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
2524adantl 272 . . . . . . . 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 5852 . . . . . . 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 146 . . . . . 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 7007 . . . . . . . . . . . . 13  |-  1Q  =  [ <. 1o ,  1o >. ]  ~Q
2928fveq2i 5343 . . . . . . . . . . . 12  |-  ( *Q
`  1Q )  =  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )
30 rec1nq 7051 . . . . . . . . . . . 12  |-  ( *Q
`  1Q )  =  1Q
3129, 30eqtr3i 2117 . . . . . . . . . . 11  |-  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  )  =  1Q
3231breq2i 3875 . . . . . . . . . 10  |-  ( p 
<Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <->  p  <Q  1Q )
3332abbii 2210 . . . . . . . . 9  |-  { p  |  p  <Q  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  ) }  =  { p  |  p  <Q  1Q }
3431breq1i 3874 . . . . . . . . . 10  |-  ( ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  q  <->  1Q  <Q  q )
3534abbii 2210 . . . . . . . . 9  |-  { q  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q 
q }  =  {
q  |  1Q  <Q  q }
3633, 35opeq12i 3649 . . . . . . . 8  |-  <. { p  |  p  <Q  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q 
q } >.  =  <. { p  |  p  <Q  1Q } ,  { q  |  1Q  <Q  q } >.
3736oveq2i 5701 . . . . . . 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 7217 . . . . . . 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 405 . . . . . 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 3897 . . . . 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 5340 . . . . . . . 8  |-  ( r  =  1o  ->  ( F `  r )  =  ( F `  1o ) )
43 opeq1 3644 . . . . . . . . . . . . 13  |-  ( r  =  1o  ->  <. r ,  1o >.  =  <. 1o ,  1o >. )
4443eceq1d 6368 . . . . . . . . . . . 12  |-  ( r  =  1o  ->  [ <. r ,  1o >. ]  ~Q  =  [ <. 1o ,  1o >. ]  ~Q  )
4544fveq2d 5344 . . . . . . . . . . 11  |-  ( r  =  1o  ->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) )
4645breq2d 3879 . . . . . . . . . 10  |-  ( r  =  1o  ->  (
p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <->  p  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) ) )
4746abbidv 2212 . . . . . . . . 9  |-  ( r  =  1o  ->  { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) }  =  { p  |  p  <Q  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  ) } )
4845breq1d 3877 . . . . . . . . . 10  |-  ( r  =  1o  ->  (
( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  q ) )
4948abbidv 2212 . . . . . . . . 9  |-  ( r  =  1o  ->  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q }  =  {
q  |  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  )  <Q  q } )
5047, 49opeq12d 3652 . . . . . . . 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 5708 . . . . . . 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 3877 . . . . . 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 2736 . . . . 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 404 . . . 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 3871 . . . . . . . . 9  |-  ( u  =  ( x  +Q  1Q )  ->  ( p 
<Q  u  <->  p  <Q  ( x  +Q  1Q ) ) )
5655abbidv 2212 . . . . . . . 8  |-  ( u  =  ( x  +Q  1Q )  ->  { p  |  p  <Q  u }  =  { p  |  p 
<Q  ( x  +Q  1Q ) } )
57 breq1 3870 . . . . . . . . 9  |-  ( u  =  ( x  +Q  1Q )  ->  ( u 
<Q  q  <->  ( x  +Q  1Q )  <Q  q ) )
5857abbidv 2212 . . . . . . . 8  |-  ( u  =  ( x  +Q  1Q )  ->  { q  |  u  <Q  q }  =  { q  |  ( x  +Q  1Q )  <Q  q } )
5956, 58opeq12d 3652 . . . . . . 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 3879 . . . . . 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 2392 . . . . 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 5343 . . . . . 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 7019 . . . . . . . 8  |-  Q.  e.  _V
6564rabex 4004 . . . . . . 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 4004 . . . . . . 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 5956 . . . . . 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 2115 . . . . 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 2788 . . . 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 409 . . 3  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  -> 
( x  +Q  1Q )  e.  ( 2nd `  L ) )
71 eleq1 2157 . . . 4  |-  ( t  =  ( x  +Q  1Q )  ->  ( t  e.  ( 2nd `  L
)  <->  ( x  +Q  1Q )  e.  ( 2nd `  L ) ) )
7271rspcev 2736 . . 3  |-  ( ( ( x  +Q  1Q )  e.  Q.  /\  (
x  +Q  1Q )  e.  ( 2nd `  L
) )  ->  E. t  e.  Q.  t  e.  ( 2nd `  L ) )
7311, 70, 72syl2anc 404 . 2  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  ->  E. t  e.  Q.  t  e.  ( 2nd `  L ) )
747, 73rexlimddv 2507 1  |-  ( ph  ->  E. t  e.  Q.  t  e.  ( 2nd `  L ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 927    = wceq 1296    e. wcel 1445   {cab 2081   A.wral 2370   E.wrex 2371   {crab 2374   <.cop 3469   class class class wbr 3867   -->wf 5045   ` cfv 5049  (class class class)co 5690   1stc1st 5947   2ndc2nd 5948   1oc1o 6212   [cec 6330   N.cnpi 6928    <N clti 6931    ~Q ceq 6935   Q.cnq 6936   1Qc1q 6937    +Q cplq 6938   *Qcrq 6940    <Q cltq 6941   P.cnp 6947    +P. cpp 6949    <P cltp 6951
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431
This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-eprel 4140  df-id 4144  df-po 4147  df-iso 4148  df-iord 4217  df-on 4219  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-irdg 6173  df-1o 6219  df-2o 6220  df-oadd 6223  df-omul 6224  df-er 6332  df-ec 6334  df-qs 6338  df-ni 6960  df-pli 6961  df-mi 6962  df-lti 6963  df-plpq 7000  df-mpq 7001  df-enq 7003  df-nqqs 7004  df-plqqs 7005  df-mqqs 7006  df-1nqqs 7007  df-rq 7008  df-ltnqqs 7009  df-enq0 7080  df-nq0 7081  df-0nq0 7082  df-plq0 7083  df-mq0 7084  df-inp 7122  df-iplp 7124  df-iltp 7126
This theorem is referenced by:  caucvgprprlemm  7352
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