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Theorem caucvgprlemloc 7704
Description: Lemma for caucvgpr 7711. 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 7438 . . . . 5  |-  ( s 
<Q  r  ->  E. y  e.  Q.  ( s  +Q  y )  =  r )
21adantl 277 . . . 4  |-  ( ( ( ph  /\  (
s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  ->  E. y  e.  Q.  ( s  +Q  y )  =  r )
3 subhalfnqq 7443 . . . . . 6  |-  ( y  e.  Q.  ->  E. x  e.  Q.  ( x  +Q  x )  <Q  y
)
43ad2antrl 490 . . . . 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 7692 . . . . . . 7  |-  ( x  e.  Q.  ->  E. m  e.  N.  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  <Q  x )
65ad2antrl 490 . . . . . 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 531 . . . . . . . . . . . 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 7451 . . . . . . . . . . . . . . 15  |-  ( m  e.  N.  ->  [ <. m ,  1o >. ]  ~Q  e.  Q. )
9 recclnq 7421 . . . . . . . . . . . . . . 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 490 . . . . . . . . . . . . 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 535 . . . . . . . . . . . . 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 7433 . . . . . . . . . . . . 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 1250 . . . . . . . . . . . 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 433 . . . . . . . . . . 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 536 . . . . . . . . . . 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 7427 . . . . . . . . . . . 12  |-  <Q  Or  Q.
18 ltrelnq 7394 . . . . . . . . . . . 12  |-  <Q  C_  ( Q.  X.  Q. )
1917, 18sotri 5042 . . . . . . . . . . 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 411 . . . . . . . . . 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 535 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  ->  s  e.  Q. )
2221ad3antrrr 492 . . . . . . . . . 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 7431 . . . . . . . . . 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 411 . . . . . . . . 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 531 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  r ) )  ->  ( s  +Q  y )  =  r )
2625ad2antrr 488 . . . . . . . . 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 4044 . . . . . . . 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 7404 . . . . . . . . . . 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 411 . . . . . . . . . 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 7404 . . . . . . . . . 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 411 . . . . . . . . 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 536 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  ->  r  e.  Q. )
3332ad3antrrr 492 . . . . . . . . 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 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
) )  ->  F : N. --> Q. )
36 simprl 529 . . . . . . . . . . 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, 36ffvelcdmd 5673 . . . . . . . . . 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 7404 . . . . . . . . . 10  |-  ( ( ( F `  m
)  e.  Q.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  e.  Q. )
3937, 11, 38syl2anc 411 . . . . . . . . 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 4338 . . . . . . . . . 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 424 . . . . . . . . 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 1249 . . . . . . . 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 276 . . . . . . . . . 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 535 . . . . . . . . . . 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 110 . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . . 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 7411 . . . . . . . . . . . . . . 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 1249 . . . . . . . . . . . . . 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 4028 . . . . . . . . . . . . 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 167 . . . . . . . . . . . 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 7429 . . . . . . . . . . . . . 14  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
5352adantl 277 . . . . . . . . . . . . 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 7404 . . . . . . . . . . . . . 14  |-  ( ( s  e.  Q.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( s  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  e.  Q. )
5544, 47, 54syl2anc 411 . . . . . . . . . . . . 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 276 . . . . . . . . . . . . 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 7410 . . . . . . . . . . . . . 14  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
5857adantl 277 . . . . . . . . . . . . 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 6066 . . . . . . . . . . . 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 167 . . . . . . . . . . 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 3793 . . . . . . . . . . . . . . . 16  |-  ( j  =  m  ->  <. j ,  1o >.  =  <. m ,  1o >. )
6261eceq1d 6595 . . . . . . . . . . . . . . 15  |-  ( j  =  m  ->  [ <. j ,  1o >. ]  ~Q  =  [ <. m ,  1o >. ]  ~Q  )
6362fveq2d 5538 . . . . . . . . . . . . . 14  |-  ( j  =  m  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )
6463oveq2d 5912 . . . . . . . . . . . . 13  |-  ( j  =  m  ->  (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) )
65 fveq2 5534 . . . . . . . . . . . . 13  |-  ( j  =  m  ->  ( F `  j )  =  ( F `  m ) )
6664, 65breq12d 4031 . . . . . . . . . . . 12  |-  ( j  =  m  ->  (
( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( s  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  <Q 
( F `  m
) ) )
6766rspcev 2856 . . . . . . . . . . 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 411 . . . . . . . . . 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 5903 . . . . . . . . . . . . 13  |-  ( l  =  s  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) )
7069breq1d 4028 . . . . . . . . . . . 12  |-  ( l  =  s  ->  (
( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
7170rexbidv 2491 . . . . . . . . . . 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 5537 . . . . . . . . . . . 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 7392 . . . . . . . . . . . . . 14  |-  Q.  e.  _V
7574rabex 4162 . . . . . . . . . . . . 13  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  _V
7674rabex 4162 . . . . . . . . . . . . 13  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  e.  _V
7775, 76op1st 6171 . . . . . . . . . . . 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 2210 . . . . . . . . . . 11  |-  ( 1st `  L )  =  {
l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }
7971, 78elrab2 2911 . . . . . . . . . 10  |-  ( s  e.  ( 1st `  L
)  <->  ( s  e. 
Q.  /\  E. j  e.  N.  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
8044, 68, 79sylanbrc 417 . . . . . . . . 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 115 . . . . . . . 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 276 . . . . . . . . . 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 5914 . . . . . . . . . . . . 13  |-  ( j  =  m  ->  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) )
8483breq1d 4028 . . . . . . . . . . . 12  |-  ( j  =  m  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  r  <->  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  <Q 
r ) )
8584rspcev 2856 . . . . . . . . . . 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 283 . . . . . . . . . 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 4022 . . . . . . . . . . . 12  |-  ( u  =  r  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
r ) )
8887rexbidv 2491 . . . . . . . . . . 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 5537 . . . . . . . . . . . 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 6172 . . . . . . . . . . . 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 2210 . . . . . . . . . . 11  |-  ( 2nd `  L )  =  {
u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u }
9288, 91elrab2 2911 . . . . . . . . . 10  |-  ( r  e.  ( 2nd `  L
)  <->  ( r  e. 
Q.  /\  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
r ) )
9382, 86, 92sylanbrc 417 . . . . . . . . 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 115 . . . . . . . 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 787 . . . . . . 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 2612 . . . . 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 2612 . . . 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 2612 . . 3  |-  ( ( ( ph  /\  (
s  e.  Q.  /\  r  e.  Q. )
)  /\  s  <Q  r )  ->  ( s  e.  ( 1st `  L
)  \/  r  e.  ( 2nd `  L
) ) )
10099ex 115 . 2  |-  ( (
ph  /\  ( s  e.  Q.  /\  r  e. 
Q. ) )  -> 
( s  <Q  r  ->  ( s  e.  ( 1st `  L )  \/  r  e.  ( 2nd `  L ) ) ) )
101100ralrimivva 2572 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 104    <-> wb 105    \/ wo 709    /\ w3a 980    = wceq 1364    e. wcel 2160   A.wral 2468   E.wrex 2469   {crab 2472   <.cop 3610   class class class wbr 4018    Or wor 4313   -->wf 5231   ` cfv 5235  (class class class)co 5896   1stc1st 6163   2ndc2nd 6164   1oc1o 6434   [cec 6557   N.cnpi 7301    <N clti 7304    ~Q ceq 7308   Q.cnq 7309    +Q cplq 7311   *Qcrq 7313    <Q cltq 7314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-irdg 6395  df-1o 6441  df-oadd 6445  df-omul 6446  df-er 6559  df-ec 6561  df-qs 6565  df-ni 7333  df-pli 7334  df-mi 7335  df-lti 7336  df-plpq 7373  df-mpq 7374  df-enq 7376  df-nqqs 7377  df-plqqs 7378  df-mqqs 7379  df-1nqqs 7380  df-rq 7381  df-ltnqqs 7382
This theorem is referenced by:  caucvgprlemcl  7705
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