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Theorem caucvgprlemloc 6754
Description: Lemma for caucvgpr 6761. The putative limit is located. (Contributed by Jim Kingdon, 27-Sep-2020.)
Hypotheses
Ref Expression
caucvgpr.f  |-  ( ph  ->  F : N. --> Q. )
caucvgpr.cau  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <Q  ( ( F `
 k )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  /\  ( F `  k ) 
<Q  ( ( F `  n )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  )
) ) ) )
caucvgpr.bnd  |-  ( ph  ->  A. j  e.  N.  A  <Q  ( F `  j ) )
caucvgpr.lim  |-  L  = 
<. { l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) } ,  { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u } >.
Assertion
Ref Expression
caucvgprlemloc  |-  ( ph  ->  A. s  e.  Q.  A. r  e.  Q.  (
s  <Q  r  ->  (
s  e.  ( 1st `  L )  \/  r  e.  ( 2nd `  L
) ) ) )
Distinct variable groups:    A, j    j, F, l    u, F    ph, j,
r, s    s, l    u, j, r
Allowed substitution hints:    ph( u, k, n, l)    A( u, k, n, s, r, l)    F( k, n, s, r)    L( u, j, k, n, s, r, l)

Proof of Theorem caucvgprlemloc
Dummy variables  f  g  h  m  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltexnqi 6488 . . . . 5  |-  ( s 
<Q  r  ->  E. y  e.  Q.  ( s  +Q  y )  =  r )
21adantl 262 . . . 4  |-  ( ( ( ph  /\  (
s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  ->  E. y  e.  Q.  ( s  +Q  y )  =  r )
3 subhalfnqq 6493 . . . . . 6  |-  ( y  e.  Q.  ->  E. x  e.  Q.  ( x  +Q  x )  <Q  y
)
43ad2antrl 459 . . . . 5  |-  ( ( ( ( ph  /\  ( s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  r ) )  ->  E. x  e.  Q.  ( x  +Q  x
)  <Q  y )
5 archrecnq 6742 . . . . . . 7  |-  ( x  e.  Q.  ->  E. m  e.  N.  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x )
65ad2antrl 459 . . . . . 6  |-  ( ( ( ( ( ph  /\  ( s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  r ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  ->  E. m  e.  N.  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x )
7 simprr 484 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  r  e. 
Q. ) )  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  r ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x
) )  ->  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x )
8 nnnq 6501 . . . . . . . . . . . . . . 15  |-  ( m  e.  N.  ->  [ <. m ,  1o >. ]  ~Q  e.  Q. )
9 recclnq 6471 . . . . . . . . . . . . . . 15  |-  ( [
<. m ,  1o >. ]  ~Q  e.  Q.  ->  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  e.  Q. )
108, 9syl 14 . . . . . . . . . . . . . 14  |-  ( m  e.  N.  ->  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  e.  Q. )
1110ad2antrl 459 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  r  e. 
Q. ) )  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  r ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x
) )  ->  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  e.  Q. )
12 simplrl 487 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  r  e. 
Q. ) )  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  r ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x
) )  ->  x  e.  Q. )
13 lt2addnq 6483 . . . . . . . . . . . . 13  |-  ( ( ( ( *Q `  [ <. m ,  1o >. ]  ~Q  )  e. 
Q.  /\  x  e.  Q. )  /\  (
( *Q `  [ <. m ,  1o >. ]  ~Q  )  e.  Q.  /\  x  e.  Q. )
)  ->  ( (
( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x
)  ->  ( ( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  <Q 
( x  +Q  x
) ) )
1411, 12, 11, 12, 13syl22anc 1136 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  r  e. 
Q. ) )  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  r ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x
) )  ->  (
( ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x )  ->  (
( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  ( x  +Q  x ) ) )
157, 7, 14mp2and 409 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  r  e. 
Q. ) )  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  r ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x
) )  ->  (
( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  ( x  +Q  x ) )
16 simplrr 488 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  r  e. 
Q. ) )  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  r ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x
) )  ->  (
x  +Q  x ) 
<Q  y )
17 ltsonq 6477 . . . . . . . . . . . 12  |-  <Q  Or  Q.
18 ltrelnq 6444 . . . . . . . . . . . 12  |-  <Q  C_  ( Q.  X.  Q. )
1917, 18sotri 4707 . . . . . . . . . . 11  |-  ( ( ( ( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  <Q 
( x  +Q  x
)  /\  ( x  +Q  x )  <Q  y
)  ->  ( ( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  <Q 
y )
2015, 16, 19syl2anc 391 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  r  e. 
Q. ) )  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  r ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x
) )  ->  (
( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  y )
21 simplrl 487 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  ->  s  e.  Q. )
2221ad3antrrr 461 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  r  e. 
Q. ) )  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  r ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x
) )  ->  s  e.  Q. )
23 ltanqi 6481 . . . . . . . . . 10  |-  ( ( ( ( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  <Q 
y  /\  s  e.  Q. )  ->  ( s  +Q  ( ( *Q
`  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) ) 
<Q  ( s  +Q  y
) )
2420, 22, 23syl2anc 391 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  r  e. 
Q. ) )  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  r ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x
) )  ->  (
s  +Q  ( ( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) )  <Q  ( s  +Q  y ) )
25 simprr 484 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  r ) )  ->  ( s  +Q  y )  =  r )
2625ad2antrr 457 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  r  e. 
Q. ) )  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  r ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x
) )  ->  (
s  +Q  y )  =  r )
2724, 26breqtrd 3785 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  r  e. 
Q. ) )  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  r ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x
) )  ->  (
s  +Q  ( ( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) )  <Q  r )
28 addclnq 6454 . . . . . . . . . . 11  |-  ( ( ( *Q `  [ <. m ,  1o >. ]  ~Q  )  e.  Q.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( ( *Q
`  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  e. 
Q. )
2911, 11, 28syl2anc 391 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  r  e. 
Q. ) )  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  r ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x
) )  ->  (
( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  e.  Q. )
30 addclnq 6454 . . . . . . . . . 10  |-  ( ( s  e.  Q.  /\  ( ( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  e. 
Q. )  ->  (
s  +Q  ( ( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) )  e.  Q. )
3122, 29, 30syl2anc 391 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  r  e. 
Q. ) )  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  r ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x
) )  ->  (
s  +Q  ( ( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) )  e.  Q. )
32 simplrr 488 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  ->  r  e.  Q. )
3332ad3antrrr 461 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  r  e. 
Q. ) )  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  r ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x
) )  ->  r  e.  Q. )
34 caucvgpr.f . . . . . . . . . . . 12  |-  ( ph  ->  F : N. --> Q. )
3534ad5antr 465 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  r  e. 
Q. ) )  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  r ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x
) )  ->  F : N. --> Q. )
36 simprl 483 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  r  e. 
Q. ) )  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  r ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x
) )  ->  m  e.  N. )
3735, 36ffvelrnd 5290 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  r  e. 
Q. ) )  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  r ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x
) )  ->  ( F `  m )  e.  Q. )
38 addclnq 6454 . . . . . . . . . 10  |-  ( ( ( F `  m
)  e.  Q.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  e.  Q. )
3937, 11, 38syl2anc 391 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  r  e. 
Q. ) )  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  r ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x
) )  ->  (
( F `  m
)  +Q  ( *Q
`  [ <. m ,  1o >. ]  ~Q  )
)  e.  Q. )
40 sowlin 4054 . . . . . . . . . 10  |-  ( ( 
<Q  Or  Q.  /\  (
( s  +Q  (
( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
) )  e.  Q.  /\  r  e.  Q.  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  e.  Q. )
)  ->  ( (
s  +Q  ( ( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) )  <Q  r  ->  ( ( s  +Q  (
( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
) )  <Q  (
( F `  m
)  +Q  ( *Q
`  [ <. m ,  1o >. ]  ~Q  )
)  \/  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) 
<Q  r ) ) )
4117, 40mpan 400 . . . . . . . . 9  |-  ( ( ( s  +Q  (
( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
) )  e.  Q.  /\  r  e.  Q.  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  e.  Q. )  ->  ( ( s  +Q  ( ( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) ) 
<Q  r  ->  ( ( s  +Q  ( ( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) )  <Q  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  \/  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  r ) ) )
4231, 33, 39, 41syl3anc 1135 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  r  e. 
Q. ) )  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  r ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x
) )  ->  (
( s  +Q  (
( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
) )  <Q  r  ->  ( ( s  +Q  ( ( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) ) 
<Q  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  \/  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) 
<Q  r ) ) )
4327, 42mpd 13 . . . . . . 7  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  r  e. 
Q. ) )  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  r ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x
) )  ->  (
( s  +Q  (
( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
) )  <Q  (
( F `  m
)  +Q  ( *Q
`  [ <. m ,  1o >. ]  ~Q  )
)  \/  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) 
<Q  r ) )
4422adantr 261 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ph  /\  (
s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  r ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  /\  (
m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x ) )  /\  ( s  +Q  ( ( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) ) 
<Q  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
) )  ->  s  e.  Q. )
45 simplrl 487 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ph  /\  (
s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  r ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  /\  (
m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x ) )  /\  ( s  +Q  ( ( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) ) 
<Q  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
) )  ->  m  e.  N. )
46 simpr 103 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ph  /\  (
s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  r ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  /\  (
m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x ) )  /\  ( s  +Q  ( ( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) ) 
<Q  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
) )  ->  (
s  +Q  ( ( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) )  <Q  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) )
4711adantr 261 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ph  /\  (
s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  r ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  /\  (
m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x ) )  /\  ( s  +Q  ( ( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) ) 
<Q  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
) )  ->  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  e.  Q. )
48 addassnqg 6461 . . . . . . . . . . . . . . 15  |-  ( ( s  e.  Q.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  e.  Q.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  e.  Q. )  ->  (
( s  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  +Q  ( *Q
`  [ <. m ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( ( *Q
`  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) ) )
4944, 47, 47, 48syl3anc 1135 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ph  /\  (
s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  r ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  /\  (
m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x ) )  /\  ( s  +Q  ( ( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) ) 
<Q  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
) )  ->  (
( s  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  +Q  ( *Q
`  [ <. m ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( ( *Q
`  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) ) )
5049breq1d 3771 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ph  /\  (
s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  r ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  /\  (
m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x ) )  /\  ( s  +Q  ( ( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) ) 
<Q  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
) )  ->  (
( ( s  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <->  ( s  +Q  ( ( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) ) 
<Q  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
) ) )
5146, 50mpbird 156 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ph  /\  (
s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  r ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  /\  (
m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x ) )  /\  ( s  +Q  ( ( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) ) 
<Q  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
) )  ->  (
( s  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  +Q  ( *Q
`  [ <. m ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) )
52 ltanqg 6479 . . . . . . . . . . . . . 14  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
5352adantl 262 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ph  /\  ( s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  r ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  /\  (
m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x ) )  /\  ( s  +Q  ( ( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) ) 
<Q  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
) )  /\  (
f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. ) )  -> 
( f  <Q  g  <->  ( h  +Q  f ) 
<Q  ( h  +Q  g
) ) )
54 addclnq 6454 . . . . . . . . . . . . . 14  |-  ( ( s  e.  Q.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( s  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  e.  Q. )
5544, 47, 54syl2anc 391 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ph  /\  (
s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  r ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  /\  (
m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x ) )  /\  ( s  +Q  ( ( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) ) 
<Q  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
) )  ->  (
s  +Q  ( *Q
`  [ <. m ,  1o >. ]  ~Q  )
)  e.  Q. )
5637adantr 261 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ph  /\  (
s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  r ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  /\  (
m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x ) )  /\  ( s  +Q  ( ( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) ) 
<Q  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
) )  ->  ( F `  m )  e.  Q. )
57 addcomnqg 6460 . . . . . . . . . . . . . 14  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
5857adantl 262 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ph  /\  ( s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  r ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  /\  (
m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x ) )  /\  ( s  +Q  ( ( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) ) 
<Q  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
) )  /\  (
f  e.  Q.  /\  g  e.  Q. )
)  ->  ( f  +Q  g )  =  ( g  +Q  f ) )
5953, 55, 56, 47, 58caovord2d 5657 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ph  /\  (
s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  r ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  /\  (
m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x ) )  /\  ( s  +Q  ( ( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) ) 
<Q  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
) )  ->  (
( s  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  ( F `  m )  <->  ( (
s  +Q  ( *Q
`  [ <. m ,  1o >. ]  ~Q  )
)  +Q  ( *Q
`  [ <. m ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) ) )
6051, 59mpbird 156 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ph  /\  (
s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  r ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  /\  (
m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x ) )  /\  ( s  +Q  ( ( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) ) 
<Q  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
) )  ->  (
s  +Q  ( *Q
`  [ <. m ,  1o >. ]  ~Q  )
)  <Q  ( F `  m ) )
61 opeq1 3546 . . . . . . . . . . . . . . . 16  |-  ( j  =  m  ->  <. j ,  1o >.  =  <. m ,  1o >. )
6261eceq1d 6129 . . . . . . . . . . . . . . 15  |-  ( j  =  m  ->  [ <. j ,  1o >. ]  ~Q  =  [ <. m ,  1o >. ]  ~Q  )
6362fveq2d 5169 . . . . . . . . . . . . . 14  |-  ( j  =  m  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )
6463oveq2d 5515 . . . . . . . . . . . . 13  |-  ( j  =  m  ->  (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) )
65 fveq2 5165 . . . . . . . . . . . . 13  |-  ( j  =  m  ->  ( F `  j )  =  ( F `  m ) )
6664, 65breq12d 3774 . . . . . . . . . . . 12  |-  ( j  =  m  ->  (
( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( s  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  <Q 
( F `  m
) ) )
6766rspcev 2653 . . . . . . . . . . 11  |-  ( ( m  e.  N.  /\  ( s  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  ( F `  m ) )  ->  E. j  e.  N.  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) )
6845, 60, 67syl2anc 391 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ph  /\  (
s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  r ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  /\  (
m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x ) )  /\  ( s  +Q  ( ( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) ) 
<Q  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
) )  ->  E. j  e.  N.  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) )
69 oveq1 5506 . . . . . . . . . . . . 13  |-  ( l  =  s  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) )
7069breq1d 3771 . . . . . . . . . . . 12  |-  ( l  =  s  ->  (
( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
7170rexbidv 2324 . . . . . . . . . . 11  |-  ( l  =  s  ->  ( E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  E. j  e.  N.  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
72 caucvgpr.lim . . . . . . . . . . . . 13  |-  L  = 
<. { l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) } ,  { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u } >.
7372fveq2i 5168 . . . . . . . . . . . 12  |-  ( 1st `  L )  =  ( 1st `  <. { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) } ,  { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u } >. )
74 nqex 6442 . . . . . . . . . . . . . 14  |-  Q.  e.  _V
7574rabex 3898 . . . . . . . . . . . . 13  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  _V
7674rabex 3898 . . . . . . . . . . . . 13  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  e.  _V
7775, 76op1st 5760 . . . . . . . . . . . 12  |-  ( 1st `  <. { l  e. 
Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) } ,  {
u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u } >. )  =  { l  e. 
Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) }
7873, 77eqtri 2060 . . . . . . . . . . 11  |-  ( 1st `  L )  =  {
l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }
7971, 78elrab2 2697 . . . . . . . . . 10  |-  ( s  e.  ( 1st `  L
)  <->  ( s  e. 
Q.  /\  E. j  e.  N.  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
8044, 68, 79sylanbrc 394 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ph  /\  (
s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  r ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  /\  (
m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x ) )  /\  ( s  +Q  ( ( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) ) 
<Q  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
) )  ->  s  e.  ( 1st `  L
) )
8180ex 108 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  r  e. 
Q. ) )  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  r ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x
) )  ->  (
( s  +Q  (
( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
) )  <Q  (
( F `  m
)  +Q  ( *Q
`  [ <. m ,  1o >. ]  ~Q  )
)  ->  s  e.  ( 1st `  L ) ) )
8233adantr 261 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ph  /\  (
s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  r ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  /\  (
m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x ) )  /\  ( ( F `
 m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  <Q 
r )  ->  r  e.  Q. )
8365, 63oveq12d 5517 . . . . . . . . . . . . 13  |-  ( j  =  m  ->  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) )
8483breq1d 3771 . . . . . . . . . . . 12  |-  ( j  =  m  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  r  <->  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  <Q 
r ) )
8584rspcev 2653 . . . . . . . . . . 11  |-  ( ( m  e.  N.  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  r )  ->  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  r )
8636, 85sylan 267 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ph  /\  (
s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  r ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  /\  (
m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x ) )  /\  ( ( F `
 m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  <Q 
r )  ->  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
r )
87 breq2 3765 . . . . . . . . . . . 12  |-  ( u  =  r  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
r ) )
8887rexbidv 2324 . . . . . . . . . . 11  |-  ( u  =  r  ->  ( E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
r ) )
8972fveq2i 5168 . . . . . . . . . . . 12  |-  ( 2nd `  L )  =  ( 2nd `  <. { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) } ,  { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u } >. )
9075, 76op2nd 5761 . . . . . . . . . . . 12  |-  ( 2nd `  <. { l  e. 
Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) } ,  {
u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u } >. )  =  { u  e. 
Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }
9189, 90eqtri 2060 . . . . . . . . . . 11  |-  ( 2nd `  L )  =  {
u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u }
9288, 91elrab2 2697 . . . . . . . . . 10  |-  ( r  e.  ( 2nd `  L
)  <->  ( r  e. 
Q.  /\  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
r ) )
9382, 86, 92sylanbrc 394 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ph  /\  (
s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  r ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  /\  (
m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x ) )  /\  ( ( F `
 m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  <Q 
r )  ->  r  e.  ( 2nd `  L
) )
9493ex 108 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  r  e. 
Q. ) )  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  r ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x
) )  ->  (
( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  r  ->  r  e.  ( 2nd `  L
) ) )
9581, 94orim12d 700 . . . . . . 7  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  r  e. 
Q. ) )  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  r ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x
) )  ->  (
( ( s  +Q  ( ( *Q `  [ <. m ,  1o >. ]  ~Q  )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) ) 
<Q  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  \/  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) 
<Q  r )  ->  (
s  e.  ( 1st `  L )  \/  r  e.  ( 2nd `  L
) ) ) )
9643, 95mpd 13 . . . . . 6  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  r  e. 
Q. ) )  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  r ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( m  e.  N.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x
) )  ->  (
s  e.  ( 1st `  L )  \/  r  e.  ( 2nd `  L
) ) )
976, 96rexlimddv 2434 . . . . 5  |-  ( ( ( ( ( ph  /\  ( s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  r ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  ->  (
s  e.  ( 1st `  L )  \/  r  e.  ( 2nd `  L
) ) )
984, 97rexlimddv 2434 . . . 4  |-  ( ( ( ( ph  /\  ( s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  r ) )  ->  ( s  e.  ( 1st `  L
)  \/  r  e.  ( 2nd `  L
) ) )
992, 98rexlimddv 2434 . . 3  |-  ( ( ( ph  /\  (
s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  ->  ( s  e.  ( 1st `  L
)  \/  r  e.  ( 2nd `  L
) ) )
10099ex 108 . 2  |-  ( (
ph  /\  ( s  e.  Q.  /\  r  e. 
Q. ) )  -> 
( s  <Q  r  ->  ( s  e.  ( 1st `  L )  \/  r  e.  ( 2nd `  L ) ) ) )
101100ralrimivva 2398 1  |-  ( ph  ->  A. s  e.  Q.  A. r  e.  Q.  (
s  <Q  r  ->  (
s  e.  ( 1st `  L )  \/  r  e.  ( 2nd `  L
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    \/ wo 629    /\ w3a 885    = wceq 1243    e. wcel 1393   A.wral 2303   E.wrex 2304   {crab 2307   <.cop 3375   class class class wbr 3761    Or wor 4029   -->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    +Q cplq 6361   *Qcrq 6363    <Q cltq 6364
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-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
This theorem is referenced by:  caucvgprlemcl  6755
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