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Theorem caucvgprlemloc 6831
Description: Lemma for caucvgpr 6838. 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 6565 . . . . 5  |-  ( s 
<Q  r  ->  E. y  e.  Q.  ( s  +Q  y )  =  r )
21adantl 266 . . . 4  |-  ( ( ( ph  /\  (
s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  ->  E. y  e.  Q.  ( s  +Q  y )  =  r )
3 subhalfnqq 6570 . . . . . 6  |-  ( y  e.  Q.  ->  E. x  e.  Q.  ( x  +Q  x )  <Q  y
)
43ad2antrl 467 . . . . 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 6819 . . . . . . 7  |-  ( x  e.  Q.  ->  E. m  e.  N.  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x )
65ad2antrl 467 . . . . . 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 492 . . . . . . . . . . . 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 6578 . . . . . . . . . . . . . . 15  |-  ( m  e.  N.  ->  [ <. m ,  1o >. ]  ~Q  e.  Q. )
9 recclnq 6548 . . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . 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 495 . . . . . . . . . . . . 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 6560 . . . . . . . . . . . . 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 1147 . . . . . . . . . . . 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 417 . . . . . . . . . . 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 496 . . . . . . . . . . 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 6554 . . . . . . . . . . . 12  |-  <Q  Or  Q.
18 ltrelnq 6521 . . . . . . . . . . . 12  |-  <Q  C_  ( Q.  X.  Q. )
1917, 18sotri 4748 . . . . . . . . . . 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 397 . . . . . . . . . 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 495 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  ->  s  e.  Q. )
2221ad3antrrr 469 . . . . . . . . . 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 6558 . . . . . . . . . 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 397 . . . . . . . . 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 492 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  r ) )  ->  ( s  +Q  y )  =  r )
2625ad2antrr 465 . . . . . . . . 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 3816 . . . . . . . 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 6531 . . . . . . . . . . 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 397 . . . . . . . . . 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 6531 . . . . . . . . . 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 397 . . . . . . . . 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 496 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  ->  r  e.  Q. )
3332ad3antrrr 469 . . . . . . . . 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 473 . . . . . . . . . . 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 491 . . . . . . . . . . 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 5331 . . . . . . . . . 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 6531 . . . . . . . . . 10  |-  ( ( ( F `  m
)  e.  Q.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  e.  Q. )
3937, 11, 38syl2anc 397 . . . . . . . . 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 4085 . . . . . . . . . 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 408 . . . . . . . . 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 1146 . . . . . . . 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 265 . . . . . . . . . 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 495 . . . . . . . . . . 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 107 . . . . . . . . . . . . 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 265 . . . . . . . . . . . . . . 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 6538 . . . . . . . . . . . . . . 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 1146 . . . . . . . . . . . . . 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 3802 . . . . . . . . . . . . 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 160 . . . . . . . . . . . 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 6556 . . . . . . . . . . . . . 14  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
5352adantl 266 . . . . . . . . . . . . 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 6531 . . . . . . . . . . . . . 14  |-  ( ( s  e.  Q.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( s  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  e.  Q. )
5544, 47, 54syl2anc 397 . . . . . . . . . . . . 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 265 . . . . . . . . . . . . 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 6537 . . . . . . . . . . . . . 14  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
5857adantl 266 . . . . . . . . . . . . 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 5698 . . . . . . . . . . . 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 160 . . . . . . . . . . 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 3577 . . . . . . . . . . . . . . . 16  |-  ( j  =  m  ->  <. j ,  1o >.  =  <. m ,  1o >. )
6261eceq1d 6173 . . . . . . . . . . . . . . 15  |-  ( j  =  m  ->  [ <. j ,  1o >. ]  ~Q  =  [ <. m ,  1o >. ]  ~Q  )
6362fveq2d 5210 . . . . . . . . . . . . . 14  |-  ( j  =  m  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )
6463oveq2d 5556 . . . . . . . . . . . . 13  |-  ( j  =  m  ->  (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) )
65 fveq2 5206 . . . . . . . . . . . . 13  |-  ( j  =  m  ->  ( F `  j )  =  ( F `  m ) )
6664, 65breq12d 3805 . . . . . . . . . . . 12  |-  ( j  =  m  ->  (
( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( s  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  <Q 
( F `  m
) ) )
6766rspcev 2673 . . . . . . . . . . 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 397 . . . . . . . . . 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 5547 . . . . . . . . . . . . 13  |-  ( l  =  s  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) )
7069breq1d 3802 . . . . . . . . . . . 12  |-  ( l  =  s  ->  (
( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
7170rexbidv 2344 . . . . . . . . . . 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 5209 . . . . . . . . . . . 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 6519 . . . . . . . . . . . . . 14  |-  Q.  e.  _V
7574rabex 3929 . . . . . . . . . . . . 13  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  _V
7674rabex 3929 . . . . . . . . . . . . 13  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  e.  _V
7775, 76op1st 5801 . . . . . . . . . . . 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 2076 . . . . . . . . . . 11  |-  ( 1st `  L )  =  {
l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }
7971, 78elrab2 2723 . . . . . . . . . 10  |-  ( s  e.  ( 1st `  L
)  <->  ( s  e. 
Q.  /\  E. j  e.  N.  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
8044, 68, 79sylanbrc 402 . . . . . . . . 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 112 . . . . . . . 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 265 . . . . . . . . . 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 5558 . . . . . . . . . . . . 13  |-  ( j  =  m  ->  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) )
8483breq1d 3802 . . . . . . . . . . . 12  |-  ( j  =  m  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  r  <->  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  <Q 
r ) )
8584rspcev 2673 . . . . . . . . . . 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 271 . . . . . . . . . 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 3796 . . . . . . . . . . . 12  |-  ( u  =  r  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
r ) )
8887rexbidv 2344 . . . . . . . . . . 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 5209 . . . . . . . . . . . 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 5802 . . . . . . . . . . . 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 2076 . . . . . . . . . . 11  |-  ( 2nd `  L )  =  {
u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u }
9288, 91elrab2 2723 . . . . . . . . . 10  |-  ( r  e.  ( 2nd `  L
)  <->  ( r  e. 
Q.  /\  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
r ) )
9382, 86, 92sylanbrc 402 . . . . . . . . 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 112 . . . . . . . 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 710 . . . . . . 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 2454 . . . . 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 2454 . . . 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 2454 . . 3  |-  ( ( ( ph  /\  (
s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  ->  ( s  e.  ( 1st `  L
)  \/  r  e.  ( 2nd `  L
) ) )
10099ex 112 . 2  |-  ( (
ph  /\  ( s  e.  Q.  /\  r  e. 
Q. ) )  -> 
( s  <Q  r  ->  ( s  e.  ( 1st `  L )  \/  r  e.  ( 2nd `  L ) ) ) )
101100ralrimivva 2418 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 101    <-> wb 102    \/ wo 639    /\ w3a 896    = wceq 1259    e. wcel 1409   A.wral 2323   E.wrex 2324   {crab 2327   <.cop 3406   class class class wbr 3792    Or wor 4060   -->wf 4926   ` cfv 4930  (class class class)co 5540   1stc1st 5793   2ndc2nd 5794   1oc1o 6025   [cec 6135   N.cnpi 6428    <N clti 6431    ~Q ceq 6435   Q.cnq 6436    +Q cplq 6438   *Qcrq 6440    <Q cltq 6441
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509
This theorem is referenced by:  caucvgprlemcl  6832
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