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Theorem caucvgprprlemmu 7526
Description: Lemma for caucvgprpr 7543. 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 7146 . . . . 5  |-  1o  e.  N.
32a1i 9 . . . 4  |-  ( ph  ->  1o  e.  N. )
41, 3ffvelrnd 5563 . . 3  |-  ( ph  ->  ( F `  1o )  e.  P. )
5 prop 7306 . . 3  |-  ( ( F `  1o )  e.  P.  ->  <. ( 1st `  ( F `  1o ) ) ,  ( 2nd `  ( F `
 1o ) )
>.  e.  P. )
6 prmu 7309 . . 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 521 . . . 4  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  ->  x  e.  Q. )
9 1nq 7197 . . . 4  |-  1Q  e.  Q.
10 addclnq 7206 . . . 4  |-  ( ( x  e.  Q.  /\  1Q  e.  Q. )  -> 
( x  +Q  1Q )  e.  Q. )
118, 9, 10sylancl 410 . . 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 522 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  ->  x  e.  ( 2nd `  ( F `  1o ) ) )
144adantr 274 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  -> 
( F `  1o )  e.  P. )
15 nqpru 7383 . . . . . . . . 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 409 . . . . . . . 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 7450 . . . . . . . . 9  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
f  <P  g  <->  ( h  +P.  f )  <P  (
h  +P.  g )
) )
1918adantl 275 . . . . . . . 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 7378 . . . . . . . . 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 7378 . . . . . . . . 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 7409 . . . . . . . . 9  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
2524adantl 275 . . . . . . . 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 5947 . . . . . . 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 7182 . . . . . . . . . . . . 13  |-  1Q  =  [ <. 1o ,  1o >. ]  ~Q
2928fveq2i 5431 . . . . . . . . . . . 12  |-  ( *Q
`  1Q )  =  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )
30 rec1nq 7226 . . . . . . . . . . . 12  |-  ( *Q
`  1Q )  =  1Q
3129, 30eqtr3i 2163 . . . . . . . . . . 11  |-  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  )  =  1Q
3231breq2i 3944 . . . . . . . . . 10  |-  ( p 
<Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <->  p  <Q  1Q )
3332abbii 2256 . . . . . . . . 9  |-  { p  |  p  <Q  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  ) }  =  { p  |  p  <Q  1Q }
3431breq1i 3943 . . . . . . . . . 10  |-  ( ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  q  <->  1Q  <Q  q )
3534abbii 2256 . . . . . . . . 9  |-  { q  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q 
q }  =  {
q  |  1Q  <Q  q }
3633, 35opeq12i 3717 . . . . . . . 8  |-  <. { p  |  p  <Q  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q 
q } >.  =  <. { p  |  p  <Q  1Q } ,  { q  |  1Q  <Q  q } >.
3736oveq2i 5792 . . . . . . 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 7392 . . . . . . 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 410 . . . . . 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 3967 . . . . 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 5428 . . . . . . . 8  |-  ( r  =  1o  ->  ( F `  r )  =  ( F `  1o ) )
43 opeq1 3712 . . . . . . . . . . . . 13  |-  ( r  =  1o  ->  <. r ,  1o >.  =  <. 1o ,  1o >. )
4443eceq1d 6472 . . . . . . . . . . . 12  |-  ( r  =  1o  ->  [ <. r ,  1o >. ]  ~Q  =  [ <. 1o ,  1o >. ]  ~Q  )
4544fveq2d 5432 . . . . . . . . . . 11  |-  ( r  =  1o  ->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) )
4645breq2d 3948 . . . . . . . . . 10  |-  ( r  =  1o  ->  (
p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <->  p  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) ) )
4746abbidv 2258 . . . . . . . . 9  |-  ( r  =  1o  ->  { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) }  =  { p  |  p  <Q  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  ) } )
4845breq1d 3946 . . . . . . . . . 10  |-  ( r  =  1o  ->  (
( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  q ) )
4948abbidv 2258 . . . . . . . . 9  |-  ( r  =  1o  ->  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q }  =  {
q  |  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  )  <Q  q } )
5047, 49opeq12d 3720 . . . . . . . 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 5799 . . . . . . 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 3946 . . . . . 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 2792 . . . . 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 409 . . . 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 3940 . . . . . . . . 9  |-  ( u  =  ( x  +Q  1Q )  ->  ( p 
<Q  u  <->  p  <Q  ( x  +Q  1Q ) ) )
5655abbidv 2258 . . . . . . . 8  |-  ( u  =  ( x  +Q  1Q )  ->  { p  |  p  <Q  u }  =  { p  |  p 
<Q  ( x  +Q  1Q ) } )
57 breq1 3939 . . . . . . . . 9  |-  ( u  =  ( x  +Q  1Q )  ->  ( u 
<Q  q  <->  ( x  +Q  1Q )  <Q  q ) )
5857abbidv 2258 . . . . . . . 8  |-  ( u  =  ( x  +Q  1Q )  ->  { q  |  u  <Q  q }  =  { q  |  ( x  +Q  1Q )  <Q  q } )
5956, 58opeq12d 3720 . . . . . . 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 3948 . . . . . 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 2439 . . . . 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 5431 . . . . . 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 7194 . . . . . . . 8  |-  Q.  e.  _V
6564rabex 4079 . . . . . . 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 4079 . . . . . . 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 6052 . . . . . 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 2161 . . . . 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 2846 . . . 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 414 . . 3  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  -> 
( x  +Q  1Q )  e.  ( 2nd `  L ) )
71 eleq1 2203 . . . 4  |-  ( t  =  ( x  +Q  1Q )  ->  ( t  e.  ( 2nd `  L
)  <->  ( x  +Q  1Q )  e.  ( 2nd `  L ) ) )
7271rspcev 2792 . . 3  |-  ( ( ( x  +Q  1Q )  e.  Q.  /\  (
x  +Q  1Q )  e.  ( 2nd `  L
) )  ->  E. t  e.  Q.  t  e.  ( 2nd `  L ) )
7311, 70, 72syl2anc 409 . 2  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  ->  E. t  e.  Q.  t  e.  ( 2nd `  L ) )
747, 73rexlimddv 2557 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 963    = wceq 1332    e. wcel 1481   {cab 2126   A.wral 2417   E.wrex 2418   {crab 2421   <.cop 3534   class class class wbr 3936   -->wf 5126   ` cfv 5130  (class class class)co 5781   1stc1st 6043   2ndc2nd 6044   1oc1o 6313   [cec 6434   N.cnpi 7103    <N clti 7106    ~Q ceq 7110   Q.cnq 7111   1Qc1q 7112    +Q cplq 7113   *Qcrq 7115    <Q cltq 7116   P.cnp 7122    +P. cpp 7124    <P cltp 7126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-eprel 4218  df-id 4222  df-po 4225  df-iso 4226  df-iord 4295  df-on 4297  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-recs 6209  df-irdg 6274  df-1o 6320  df-2o 6321  df-oadd 6324  df-omul 6325  df-er 6436  df-ec 6438  df-qs 6442  df-ni 7135  df-pli 7136  df-mi 7137  df-lti 7138  df-plpq 7175  df-mpq 7176  df-enq 7178  df-nqqs 7179  df-plqqs 7180  df-mqqs 7181  df-1nqqs 7182  df-rq 7183  df-ltnqqs 7184  df-enq0 7255  df-nq0 7256  df-0nq0 7257  df-plq0 7258  df-mq0 7259  df-inp 7297  df-iplp 7299  df-iltp 7301
This theorem is referenced by:  caucvgprprlemm  7527
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