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Theorem caucvgprprlemmu 7244
Description: Lemma for caucvgprpr 7261. 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 6864 . . . . 5  |-  1o  e.  N.
32a1i 9 . . . 4  |-  ( ph  ->  1o  e.  N. )
41, 3ffvelrnd 5429 . . 3  |-  ( ph  ->  ( F `  1o )  e.  P. )
5 prop 7024 . . 3  |-  ( ( F `  1o )  e.  P.  ->  <. ( 1st `  ( F `  1o ) ) ,  ( 2nd `  ( F `
 1o ) )
>.  e.  P. )
6 prmu 7027 . . 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 498 . . . 4  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  ->  x  e.  Q. )
9 1nq 6915 . . . 4  |-  1Q  e.  Q.
10 addclnq 6924 . . . 4  |-  ( ( x  e.  Q.  /\  1Q  e.  Q. )  -> 
( x  +Q  1Q )  e.  Q. )
118, 9, 10sylancl 404 . . 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 499 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  ->  x  e.  ( 2nd `  ( F `  1o ) ) )
144adantr 270 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  -> 
( F `  1o )  e.  P. )
15 nqpru 7101 . . . . . . . . 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 403 . . . . . . . 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 145 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  -> 
( F `  1o )  <P  <. { p  |  p  <Q  x } ,  { q  |  x 
<Q  q } >. )
18 ltaprg 7168 . . . . . . . . 9  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
f  <P  g  <->  ( h  +P.  f )  <P  (
h  +P.  g )
) )
1918adantl 271 . . . . . . . 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 7096 . . . . . . . . 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 7096 . . . . . . . . 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 7127 . . . . . . . . 9  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
2524adantl 271 . . . . . . . 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 5806 . . . . . . 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 145 . . . . . 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 6900 . . . . . . . . . . . . 13  |-  1Q  =  [ <. 1o ,  1o >. ]  ~Q
2928fveq2i 5302 . . . . . . . . . . . 12  |-  ( *Q
`  1Q )  =  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )
30 rec1nq 6944 . . . . . . . . . . . 12  |-  ( *Q
`  1Q )  =  1Q
3129, 30eqtr3i 2110 . . . . . . . . . . 11  |-  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  )  =  1Q
3231breq2i 3851 . . . . . . . . . 10  |-  ( p 
<Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <->  p  <Q  1Q )
3332abbii 2203 . . . . . . . . 9  |-  { p  |  p  <Q  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  ) }  =  { p  |  p  <Q  1Q }
3431breq1i 3850 . . . . . . . . . 10  |-  ( ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  q  <->  1Q  <Q  q )
3534abbii 2203 . . . . . . . . 9  |-  { q  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q 
q }  =  {
q  |  1Q  <Q  q }
3633, 35opeq12i 3625 . . . . . . . 8  |-  <. { p  |  p  <Q  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q 
q } >.  =  <. { p  |  p  <Q  1Q } ,  { q  |  1Q  <Q  q } >.
3736oveq2i 5655 . . . . . . 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 7110 . . . . . . 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 404 . . . . . 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 3873 . . . . 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 5299 . . . . . . . 8  |-  ( r  =  1o  ->  ( F `  r )  =  ( F `  1o ) )
43 opeq1 3620 . . . . . . . . . . . . 13  |-  ( r  =  1o  ->  <. r ,  1o >.  =  <. 1o ,  1o >. )
4443eceq1d 6318 . . . . . . . . . . . 12  |-  ( r  =  1o  ->  [ <. r ,  1o >. ]  ~Q  =  [ <. 1o ,  1o >. ]  ~Q  )
4544fveq2d 5303 . . . . . . . . . . 11  |-  ( r  =  1o  ->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) )
4645breq2d 3855 . . . . . . . . . 10  |-  ( r  =  1o  ->  (
p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <->  p  <Q  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  ) ) )
4746abbidv 2205 . . . . . . . . 9  |-  ( r  =  1o  ->  { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) }  =  { p  |  p  <Q  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  ) } )
4845breq1d 3853 . . . . . . . . . 10  |-  ( r  =  1o  ->  (
( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. 1o ,  1o >. ]  ~Q  )  <Q  q ) )
4948abbidv 2205 . . . . . . . . 9  |-  ( r  =  1o  ->  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q }  =  {
q  |  ( *Q
`  [ <. 1o ,  1o >. ]  ~Q  )  <Q  q } )
5047, 49opeq12d 3628 . . . . . . . 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 5662 . . . . . . 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 3853 . . . . . 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 2722 . . . . 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 403 . . . 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 3847 . . . . . . . . 9  |-  ( u  =  ( x  +Q  1Q )  ->  ( p 
<Q  u  <->  p  <Q  ( x  +Q  1Q ) ) )
5655abbidv 2205 . . . . . . . 8  |-  ( u  =  ( x  +Q  1Q )  ->  { p  |  p  <Q  u }  =  { p  |  p 
<Q  ( x  +Q  1Q ) } )
57 breq1 3846 . . . . . . . . 9  |-  ( u  =  ( x  +Q  1Q )  ->  ( u 
<Q  q  <->  ( x  +Q  1Q )  <Q  q ) )
5857abbidv 2205 . . . . . . . 8  |-  ( u  =  ( x  +Q  1Q )  ->  { q  |  u  <Q  q }  =  { q  |  ( x  +Q  1Q )  <Q  q } )
5956, 58opeq12d 3628 . . . . . . 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 3855 . . . . . 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 2381 . . . . 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 5302 . . . . . 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 6912 . . . . . . . 8  |-  Q.  e.  _V
6564rabex 3981 . . . . . . 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 3981 . . . . . . 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 5910 . . . . . 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 2108 . . . . 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 2774 . . . 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 408 . . 3  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  -> 
( x  +Q  1Q )  e.  ( 2nd `  L ) )
71 eleq1 2150 . . . 4  |-  ( t  =  ( x  +Q  1Q )  ->  ( t  e.  ( 2nd `  L
)  <->  ( x  +Q  1Q )  e.  ( 2nd `  L ) ) )
7271rspcev 2722 . . 3  |-  ( ( ( x  +Q  1Q )  e.  Q.  /\  (
x  +Q  1Q )  e.  ( 2nd `  L
) )  ->  E. t  e.  Q.  t  e.  ( 2nd `  L ) )
7311, 70, 72syl2anc 403 . 2  |-  ( (
ph  /\  ( x  e.  Q.  /\  x  e.  ( 2nd `  ( F `  1o )
) ) )  ->  E. t  e.  Q.  t  e.  ( 2nd `  L ) )
747, 73rexlimddv 2493 1  |-  ( ph  ->  E. t  e.  Q.  t  e.  ( 2nd `  L ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438   {cab 2074   A.wral 2359   E.wrex 2360   {crab 2363   <.cop 3447   class class class wbr 3843   -->wf 5006   ` cfv 5010  (class class class)co 5644   1stc1st 5901   2ndc2nd 5902   1oc1o 6166   [cec 6280   N.cnpi 6821    <N clti 6824    ~Q ceq 6828   Q.cnq 6829   1Qc1q 6830    +Q cplq 6831   *Qcrq 6833    <Q cltq 6834   P.cnp 6840    +P. cpp 6842    <P cltp 6844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3952  ax-sep 3955  ax-nul 3963  ax-pow 4007  ax-pr 4034  ax-un 4258  ax-setind 4351  ax-iinf 4401
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-int 3687  df-iun 3730  df-br 3844  df-opab 3898  df-mpt 3899  df-tr 3935  df-eprel 4114  df-id 4118  df-po 4121  df-iso 4122  df-iord 4191  df-on 4193  df-suc 4196  df-iom 4404  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-ima 4449  df-iota 4975  df-fun 5012  df-fn 5013  df-f 5014  df-f1 5015  df-fo 5016  df-f1o 5017  df-fv 5018  df-ov 5647  df-oprab 5648  df-mpt2 5649  df-1st 5903  df-2nd 5904  df-recs 6062  df-irdg 6127  df-1o 6173  df-2o 6174  df-oadd 6177  df-omul 6178  df-er 6282  df-ec 6284  df-qs 6288  df-ni 6853  df-pli 6854  df-mi 6855  df-lti 6856  df-plpq 6893  df-mpq 6894  df-enq 6896  df-nqqs 6897  df-plqqs 6898  df-mqqs 6899  df-1nqqs 6900  df-rq 6901  df-ltnqqs 6902  df-enq0 6973  df-nq0 6974  df-0nq0 6975  df-plq0 6976  df-mq0 6977  df-inp 7015  df-iplp 7017  df-iltp 7019
This theorem is referenced by:  caucvgprprlemm  7245
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