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Theorem caucvgprlemloc 7213
Description: Lemma for caucvgpr 7220. 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 6947 . . . . 5  |-  ( s 
<Q  r  ->  E. y  e.  Q.  ( s  +Q  y )  =  r )
21adantl 271 . . . 4  |-  ( ( ( ph  /\  (
s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  ->  E. y  e.  Q.  ( s  +Q  y )  =  r )
3 subhalfnqq 6952 . . . . . 6  |-  ( y  e.  Q.  ->  E. x  e.  Q.  ( x  +Q  x )  <Q  y
)
43ad2antrl 474 . . . . 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 7201 . . . . . . 7  |-  ( x  e.  Q.  ->  E. m  e.  N.  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x )
65ad2antrl 474 . . . . . 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 499 . . . . . . . . . . . 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 6960 . . . . . . . . . . . . . . 15  |-  ( m  e.  N.  ->  [ <. m ,  1o >. ]  ~Q  e.  Q. )
9 recclnq 6930 . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . 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 502 . . . . . . . . . . . . 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 6942 . . . . . . . . . . . . 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 1175 . . . . . . . . . . . 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 424 . . . . . . . . . . 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 503 . . . . . . . . . . 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 6936 . . . . . . . . . . . 12  |-  <Q  Or  Q.
18 ltrelnq 6903 . . . . . . . . . . . 12  |-  <Q  C_  ( Q.  X.  Q. )
1917, 18sotri 4814 . . . . . . . . . . 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 403 . . . . . . . . . 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 502 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  ->  s  e.  Q. )
2221ad3antrrr 476 . . . . . . . . . 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 6940 . . . . . . . . . 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 403 . . . . . . . . 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 499 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  r ) )  ->  ( s  +Q  y )  =  r )
2625ad2antrr 472 . . . . . . . . 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 3861 . . . . . . . 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 6913 . . . . . . . . . . 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 403 . . . . . . . . . 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 6913 . . . . . . . . . 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 403 . . . . . . . . 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 503 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  ->  r  e.  Q. )
3332ad3antrrr 476 . . . . . . . . 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 480 . . . . . . . . . . 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 498 . . . . . . . . . . 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 5419 . . . . . . . . . 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 6913 . . . . . . . . . 10  |-  ( ( ( F `  m
)  e.  Q.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  e.  Q. )
3937, 11, 38syl2anc 403 . . . . . . . . 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 4138 . . . . . . . . . 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 415 . . . . . . . . 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 1174 . . . . . . . 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 270 . . . . . . . . . 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 502 . . . . . . . . . . 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 108 . . . . . . . . . . . . 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 270 . . . . . . . . . . . . . . 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 6920 . . . . . . . . . . . . . . 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 1174 . . . . . . . . . . . . . 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 3847 . . . . . . . . . . . . 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 165 . . . . . . . . . . . 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 6938 . . . . . . . . . . . . . 14  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
5352adantl 271 . . . . . . . . . . . . 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 6913 . . . . . . . . . . . . . 14  |-  ( ( s  e.  Q.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( s  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  e.  Q. )
5544, 47, 54syl2anc 403 . . . . . . . . . . . . 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 270 . . . . . . . . . . . . 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 6919 . . . . . . . . . . . . . 14  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
5857adantl 271 . . . . . . . . . . . . 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 5796 . . . . . . . . . . . 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 165 . . . . . . . . . . 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 3617 . . . . . . . . . . . . . . . 16  |-  ( j  =  m  ->  <. j ,  1o >.  =  <. m ,  1o >. )
6261eceq1d 6308 . . . . . . . . . . . . . . 15  |-  ( j  =  m  ->  [ <. j ,  1o >. ]  ~Q  =  [ <. m ,  1o >. ]  ~Q  )
6362fveq2d 5293 . . . . . . . . . . . . . 14  |-  ( j  =  m  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )
6463oveq2d 5650 . . . . . . . . . . . . 13  |-  ( j  =  m  ->  (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) )
65 fveq2 5289 . . . . . . . . . . . . 13  |-  ( j  =  m  ->  ( F `  j )  =  ( F `  m ) )
6664, 65breq12d 3850 . . . . . . . . . . . 12  |-  ( j  =  m  ->  (
( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( s  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  <Q 
( F `  m
) ) )
6766rspcev 2722 . . . . . . . . . . 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 403 . . . . . . . . . 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 5641 . . . . . . . . . . . . 13  |-  ( l  =  s  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) )
7069breq1d 3847 . . . . . . . . . . . 12  |-  ( l  =  s  ->  (
( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
7170rexbidv 2381 . . . . . . . . . . 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 5292 . . . . . . . . . . . 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 6901 . . . . . . . . . . . . . 14  |-  Q.  e.  _V
7574rabex 3975 . . . . . . . . . . . . 13  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  _V
7674rabex 3975 . . . . . . . . . . . . 13  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  e.  _V
7775, 76op1st 5899 . . . . . . . . . . . 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 2108 . . . . . . . . . . 11  |-  ( 1st `  L )  =  {
l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }
7971, 78elrab2 2772 . . . . . . . . . 10  |-  ( s  e.  ( 1st `  L
)  <->  ( s  e. 
Q.  /\  E. j  e.  N.  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
8044, 68, 79sylanbrc 408 . . . . . . . . 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 113 . . . . . . . 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 270 . . . . . . . . . 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 5652 . . . . . . . . . . . . 13  |-  ( j  =  m  ->  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) )
8483breq1d 3847 . . . . . . . . . . . 12  |-  ( j  =  m  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  r  <->  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  <Q 
r ) )
8584rspcev 2722 . . . . . . . . . . 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 277 . . . . . . . . . 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 3841 . . . . . . . . . . . 12  |-  ( u  =  r  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
r ) )
8887rexbidv 2381 . . . . . . . . . . 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 5292 . . . . . . . . . . . 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 5900 . . . . . . . . . . . 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 2108 . . . . . . . . . . 11  |-  ( 2nd `  L )  =  {
u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u }
9288, 91elrab2 2772 . . . . . . . . . 10  |-  ( r  e.  ( 2nd `  L
)  <->  ( r  e. 
Q.  /\  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
r ) )
9382, 86, 92sylanbrc 408 . . . . . . . . 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 113 . . . . . . . 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 735 . . . . . . 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 2493 . . . . 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 2493 . . . 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 2493 . . 3  |-  ( ( ( ph  /\  (
s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  ->  ( s  e.  ( 1st `  L
)  \/  r  e.  ( 2nd `  L
) ) )
10099ex 113 . 2  |-  ( (
ph  /\  ( s  e.  Q.  /\  r  e. 
Q. ) )  -> 
( s  <Q  r  ->  ( s  e.  ( 1st `  L )  \/  r  e.  ( 2nd `  L ) ) ) )
101100ralrimivva 2455 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 102    <-> wb 103    \/ wo 664    /\ w3a 924    = wceq 1289    e. wcel 1438   A.wral 2359   E.wrex 2360   {crab 2363   <.cop 3444   class class class wbr 3837    Or wor 4113   -->wf 4998   ` cfv 5002  (class class class)co 5634   1stc1st 5891   2ndc2nd 5892   1oc1o 6156   [cec 6270   N.cnpi 6810    <N clti 6813    ~Q ceq 6817   Q.cnq 6818    +Q cplq 6820   *Qcrq 6822    <Q cltq 6823
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 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
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 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-eprel 4107  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-1o 6163  df-oadd 6167  df-omul 6168  df-er 6272  df-ec 6274  df-qs 6278  df-ni 6842  df-pli 6843  df-mi 6844  df-lti 6845  df-plpq 6882  df-mpq 6883  df-enq 6885  df-nqqs 6886  df-plqqs 6887  df-mqqs 6888  df-1nqqs 6889  df-rq 6890  df-ltnqqs 6891
This theorem is referenced by:  caucvgprlemcl  7214
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