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Theorem caucvgprlemopl 6910
Description: Lemma for caucvgpr 6923. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 20-Oct-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
caucvgprlemopl  |-  ( (
ph  /\  s  e.  ( 1st `  L ) )  ->  E. r  e.  Q.  ( s  <Q 
r  /\  r  e.  ( 1st `  L ) ) )
Distinct variable groups:    A, j    F, l, r, s    u, F   
j, L, r, s   
j, l, s    ph, j,
r, s    u, j,
r, s
Allowed substitution hints:    ph( u, k, n, l)    A( u, k, n, s, r, l)    F( j, k, n)    L( u, k, n, l)

Proof of Theorem caucvgprlemopl
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 oveq1 5544 . . . . . . 7  |-  ( l  =  s  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) )
21breq1d 3797 . . . . . 6  |-  ( l  =  s  ->  (
( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
32rexbidv 2370 . . . . 5  |-  ( 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
) ) )
4 caucvgpr.lim . . . . . . 7  |-  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 } >.
54fveq2i 5206 . . . . . 6  |-  ( 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 } >. )
6 nqex 6604 . . . . . . . 8  |-  Q.  e.  _V
76rabex 3924 . . . . . . 7  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  _V
86rabex 3924 . . . . . . 7  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  e.  _V
97, 8op1st 5798 . . . . . 6  |-  ( 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
) }
105, 9eqtri 2102 . . . . 5  |-  ( 1st `  L )  =  {
l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }
113, 10elrab2 2752 . . . 4  |-  ( s  e.  ( 1st `  L
)  <->  ( s  e. 
Q.  /\  E. j  e.  N.  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
1211simprbi 269 . . 3  |-  ( s  e.  ( 1st `  L
)  ->  E. j  e.  N.  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) )
1312adantl 271 . 2  |-  ( (
ph  /\  s  e.  ( 1st `  L ) )  ->  E. j  e.  N.  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) )
14 simprr 499 . . . 4  |-  ( ( ( ph  /\  s  e.  ( 1st `  L
) )  /\  (
j  e.  N.  /\  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  ->  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) )
15 ltbtwnnqq 6656 . . . 4  |-  ( ( s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  E. t  e.  Q.  ( ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) 
<Q  t  /\  t  <Q  ( F `  j
) ) )
1614, 15sylib 120 . . 3  |-  ( ( ( ph  /\  s  e.  ( 1st `  L
) )  /\  (
j  e.  N.  /\  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  ->  E. t  e.  Q.  ( ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
t  /\  t  <Q  ( F `  j ) ) )
17 simplrl 502 . . . . . . . . 9  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( j  e.  N.  /\  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  t  /\  t  <Q  ( F `  j
) ) ) )  ->  j  e.  N. )
18 nnnq 6663 . . . . . . . . 9  |-  ( j  e.  N.  ->  [ <. j ,  1o >. ]  ~Q  e.  Q. )
19 recclnq 6633 . . . . . . . . 9  |-  ( [
<. j ,  1o >. ]  ~Q  e.  Q.  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  e.  Q. )
2017, 18, 193syl 17 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( j  e.  N.  /\  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  t  /\  t  <Q  ( F `  j
) ) ) )  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  e. 
Q. )
2111simplbi 268 . . . . . . . . 9  |-  ( s  e.  ( 1st `  L
)  ->  s  e.  Q. )
2221ad3antlr 477 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( j  e.  N.  /\  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  t  /\  t  <Q  ( F `  j
) ) ) )  ->  s  e.  Q. )
23 ltaddnq 6648 . . . . . . . 8  |-  ( ( ( *Q `  [ <. j ,  1o >. ]  ~Q  )  e.  Q.  /\  s  e.  Q. )  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  (
( *Q `  [ <. j ,  1o >. ]  ~Q  )  +Q  s
) )
2420, 22, 23syl2anc 403 . . . . . . 7  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( j  e.  N.  /\  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  t  /\  t  <Q  ( F `  j
) ) ) )  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q 
( ( *Q `  [ <. j ,  1o >. ]  ~Q  )  +Q  s ) )
25 addcomnqg 6622 . . . . . . . 8  |-  ( ( ( *Q `  [ <. j ,  1o >. ]  ~Q  )  e.  Q.  /\  s  e.  Q. )  ->  ( ( *Q `  [ <. j ,  1o >. ]  ~Q  )  +Q  s )  =  ( s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
) )
2620, 22, 25syl2anc 403 . . . . . . 7  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( j  e.  N.  /\  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  t  /\  t  <Q  ( F `  j
) ) ) )  ->  ( ( *Q
`  [ <. j ,  1o >. ]  ~Q  )  +Q  s )  =  ( s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
) )
2724, 26breqtrd 3811 . . . . . 6  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( j  e.  N.  /\  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  t  /\  t  <Q  ( F `  j
) ) ) )  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q 
( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
) )
28 simprrl 506 . . . . . 6  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( j  e.  N.  /\  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  t  /\  t  <Q  ( F `  j
) ) ) )  ->  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
t )
29 ltsonq 6639 . . . . . . 7  |-  <Q  Or  Q.
30 ltrelnq 6606 . . . . . . 7  |-  <Q  C_  ( Q.  X.  Q. )
3129, 30sotri 4744 . . . . . 6  |-  ( ( ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  /\  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
t )  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  t )
3227, 28, 31syl2anc 403 . . . . 5  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( j  e.  N.  /\  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  t  /\  t  <Q  ( F `  j
) ) ) )  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q 
t )
33 simprl 498 . . . . . 6  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( j  e.  N.  /\  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  t  /\  t  <Q  ( F `  j
) ) ) )  ->  t  e.  Q. )
34 ltexnqq 6649 . . . . . 6  |-  ( ( ( *Q `  [ <. j ,  1o >. ]  ~Q  )  e.  Q.  /\  t  e.  Q. )  ->  ( ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q 
t  <->  E. r  e.  Q.  ( ( *Q `  [ <. j ,  1o >. ]  ~Q  )  +Q  r )  =  t ) )
3520, 33, 34syl2anc 403 . . . . 5  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( j  e.  N.  /\  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  t  /\  t  <Q  ( F `  j
) ) ) )  ->  ( ( *Q
`  [ <. j ,  1o >. ]  ~Q  )  <Q  t  <->  E. r  e.  Q.  ( ( *Q `  [ <. j ,  1o >. ]  ~Q  )  +Q  r )  =  t ) )
3632, 35mpbid 145 . . . 4  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( j  e.  N.  /\  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  t  /\  t  <Q  ( F `  j
) ) ) )  ->  E. r  e.  Q.  ( ( *Q `  [ <. j ,  1o >. ]  ~Q  )  +Q  r )  =  t )
3722ad2antrr 472 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( j  e.  N.  /\  (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  t  /\  t  <Q  ( F `  j
) ) ) )  /\  r  e.  Q. )  /\  ( ( *Q
`  [ <. j ,  1o >. ]  ~Q  )  +Q  r )  =  t )  ->  s  e.  Q. )
3820ad2antrr 472 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( j  e.  N.  /\  (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  t  /\  t  <Q  ( F `  j
) ) ) )  /\  r  e.  Q. )  /\  ( ( *Q
`  [ <. j ,  1o >. ]  ~Q  )  +Q  r )  =  t )  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  e. 
Q. )
39 addcomnqg 6622 . . . . . . . . . . 11  |-  ( ( s  e.  Q.  /\  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  =  ( ( *Q `  [ <. j ,  1o >. ]  ~Q  )  +Q  s ) )
4037, 38, 39syl2anc 403 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( j  e.  N.  /\  (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  t  /\  t  <Q  ( F `  j
) ) ) )  /\  r  e.  Q. )  /\  ( ( *Q
`  [ <. j ,  1o >. ]  ~Q  )  +Q  r )  =  t )  ->  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  =  ( ( *Q `  [ <. j ,  1o >. ]  ~Q  )  +Q  s ) )
4128ad2antrr 472 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( j  e.  N.  /\  (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  t  /\  t  <Q  ( F `  j
) ) ) )  /\  r  e.  Q. )  /\  ( ( *Q
`  [ <. j ,  1o >. ]  ~Q  )  +Q  r )  =  t )  ->  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
t )
4240, 41eqbrtrrd 3809 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( j  e.  N.  /\  (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  t  /\  t  <Q  ( F `  j
) ) ) )  /\  r  e.  Q. )  /\  ( ( *Q
`  [ <. j ,  1o >. ]  ~Q  )  +Q  r )  =  t )  ->  ( ( *Q `  [ <. j ,  1o >. ]  ~Q  )  +Q  s )  <Q  t
)
43 simpr 108 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( j  e.  N.  /\  (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  t  /\  t  <Q  ( F `  j
) ) ) )  /\  r  e.  Q. )  /\  ( ( *Q
`  [ <. j ,  1o >. ]  ~Q  )  +Q  r )  =  t )  ->  ( ( *Q `  [ <. j ,  1o >. ]  ~Q  )  +Q  r )  =  t )
4442, 43breqtrrd 3813 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( j  e.  N.  /\  (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  t  /\  t  <Q  ( F `  j
) ) ) )  /\  r  e.  Q. )  /\  ( ( *Q
`  [ <. j ,  1o >. ]  ~Q  )  +Q  r )  =  t )  ->  ( ( *Q `  [ <. j ,  1o >. ]  ~Q  )  +Q  s )  <Q  (
( *Q `  [ <. j ,  1o >. ]  ~Q  )  +Q  r
) )
45 simplr 497 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( j  e.  N.  /\  (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  t  /\  t  <Q  ( F `  j
) ) ) )  /\  r  e.  Q. )  /\  ( ( *Q
`  [ <. j ,  1o >. ]  ~Q  )  +Q  r )  =  t )  ->  r  e.  Q. )
46 ltanqg 6641 . . . . . . . . 9  |-  ( ( s  e.  Q.  /\  r  e.  Q.  /\  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  e.  Q. )  ->  (
s  <Q  r  <->  ( ( *Q `  [ <. j ,  1o >. ]  ~Q  )  +Q  s )  <Q  (
( *Q `  [ <. j ,  1o >. ]  ~Q  )  +Q  r
) ) )
4737, 45, 38, 46syl3anc 1170 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( j  e.  N.  /\  (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  t  /\  t  <Q  ( F `  j
) ) ) )  /\  r  e.  Q. )  /\  ( ( *Q
`  [ <. j ,  1o >. ]  ~Q  )  +Q  r )  =  t )  ->  ( s  <Q  r  <->  ( ( *Q
`  [ <. j ,  1o >. ]  ~Q  )  +Q  s )  <Q  (
( *Q `  [ <. j ,  1o >. ]  ~Q  )  +Q  r
) ) )
4844, 47mpbird 165 . . . . . . 7  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( j  e.  N.  /\  (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  t  /\  t  <Q  ( F `  j
) ) ) )  /\  r  e.  Q. )  /\  ( ( *Q
`  [ <. j ,  1o >. ]  ~Q  )  +Q  r )  =  t )  ->  s  <Q  r )
4917ad2antrr 472 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( j  e.  N.  /\  (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  t  /\  t  <Q  ( F `  j
) ) ) )  /\  r  e.  Q. )  /\  ( ( *Q
`  [ <. j ,  1o >. ]  ~Q  )  +Q  r )  =  t )  ->  j  e.  N. )
50 simprrr 507 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( j  e.  N.  /\  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  t  /\  t  <Q  ( F `  j
) ) ) )  ->  t  <Q  ( F `  j )
)
5150ad2antrr 472 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( j  e.  N.  /\  (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  t  /\  t  <Q  ( F `  j
) ) ) )  /\  r  e.  Q. )  /\  ( ( *Q
`  [ <. j ,  1o >. ]  ~Q  )  +Q  r )  =  t )  ->  t  <Q  ( F `  j ) )
52 addcomnqg 6622 . . . . . . . . . . . . 13  |-  ( ( ( *Q `  [ <. j ,  1o >. ]  ~Q  )  e.  Q.  /\  r  e.  Q. )  ->  ( ( *Q `  [ <. j ,  1o >. ]  ~Q  )  +Q  r )  =  ( r  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
) )
5338, 45, 52syl2anc 403 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( j  e.  N.  /\  (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  t  /\  t  <Q  ( F `  j
) ) ) )  /\  r  e.  Q. )  /\  ( ( *Q
`  [ <. j ,  1o >. ]  ~Q  )  +Q  r )  =  t )  ->  ( ( *Q `  [ <. j ,  1o >. ]  ~Q  )  +Q  r )  =  ( r  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
) )
5453, 43eqtr3d 2116 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( j  e.  N.  /\  (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  t  /\  t  <Q  ( F `  j
) ) ) )  /\  r  e.  Q. )  /\  ( ( *Q
`  [ <. j ,  1o >. ]  ~Q  )  +Q  r )  =  t )  ->  ( r  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  =  t )
5554breq1d 3797 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( j  e.  N.  /\  (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  t  /\  t  <Q  ( F `  j
) ) ) )  /\  r  e.  Q. )  /\  ( ( *Q
`  [ <. j ,  1o >. ]  ~Q  )  +Q  r )  =  t )  ->  ( (
r  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  t  <Q  ( F `  j ) ) )
5651, 55mpbird 165 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( j  e.  N.  /\  (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  t  /\  t  <Q  ( F `  j
) ) ) )  /\  r  e.  Q. )  /\  ( ( *Q
`  [ <. j ,  1o >. ]  ~Q  )  +Q  r )  =  t )  ->  ( r  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) )
57 rspe 2413 . . . . . . . . 9  |-  ( ( j  e.  N.  /\  ( r  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) )  ->  E. j  e.  N.  ( r  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) )
5849, 56, 57syl2anc 403 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( j  e.  N.  /\  (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  t  /\  t  <Q  ( F `  j
) ) ) )  /\  r  e.  Q. )  /\  ( ( *Q
`  [ <. j ,  1o >. ]  ~Q  )  +Q  r )  =  t )  ->  E. j  e.  N.  ( r  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) )
59 oveq1 5544 . . . . . . . . . . 11  |-  ( l  =  r  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( r  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) )
6059breq1d 3797 . . . . . . . . . 10  |-  ( l  =  r  ->  (
( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( r  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
6160rexbidv 2370 . . . . . . . . 9  |-  ( l  =  r  ->  ( E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  E. j  e.  N.  ( r  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
6261, 10elrab2 2752 . . . . . . . 8  |-  ( r  e.  ( 1st `  L
)  <->  ( r  e. 
Q.  /\  E. j  e.  N.  ( r  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
6345, 58, 62sylanbrc 408 . . . . . . 7  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( j  e.  N.  /\  (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  t  /\  t  <Q  ( F `  j
) ) ) )  /\  r  e.  Q. )  /\  ( ( *Q
`  [ <. j ,  1o >. ]  ~Q  )  +Q  r )  =  t )  ->  r  e.  ( 1st `  L ) )
6448, 63jca 300 . . . . . 6  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( j  e.  N.  /\  (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  t  /\  t  <Q  ( F `  j
) ) ) )  /\  r  e.  Q. )  /\  ( ( *Q
`  [ <. j ,  1o >. ]  ~Q  )  +Q  r )  =  t )  ->  ( s  <Q  r  /\  r  e.  ( 1st `  L
) ) )
6564ex 113 . . . . 5  |-  ( ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( j  e.  N.  /\  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  t  /\  t  <Q  ( F `  j
) ) ) )  /\  r  e.  Q. )  ->  ( ( ( *Q `  [ <. j ,  1o >. ]  ~Q  )  +Q  r )  =  t  ->  ( s  <Q  r  /\  r  e.  ( 1st `  L
) ) ) )
6665reximdva 2464 . . . 4  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( j  e.  N.  /\  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  t  /\  t  <Q  ( F `  j
) ) ) )  ->  ( E. r  e.  Q.  ( ( *Q
`  [ <. j ,  1o >. ]  ~Q  )  +Q  r )  =  t  ->  E. r  e.  Q.  ( s  <Q  r  /\  r  e.  ( 1st `  L ) ) ) )
6736, 66mpd 13 . . 3  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( j  e.  N.  /\  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  t  /\  t  <Q  ( F `  j
) ) ) )  ->  E. r  e.  Q.  ( s  <Q  r  /\  r  e.  ( 1st `  L ) ) )
6816, 67rexlimddv 2482 . 2  |-  ( ( ( ph  /\  s  e.  ( 1st `  L
) )  /\  (
j  e.  N.  /\  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) ) )  ->  E. r  e.  Q.  ( s  <Q  r  /\  r  e.  ( 1st `  L ) ) )
6913, 68rexlimddv 2482 1  |-  ( (
ph  /\  s  e.  ( 1st `  L ) )  ->  E. r  e.  Q.  ( s  <Q 
r  /\  r  e.  ( 1st `  L ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   A.wral 2349   E.wrex 2350   {crab 2353   <.cop 3403   class class class wbr 3787   -->wf 4922   ` cfv 4926  (class class class)co 5537   1stc1st 5790   1oc1o 6052   [cec 6163   N.cnpi 6513    <N clti 6516    ~Q ceq 6520   Q.cnq 6521    +Q cplq 6523   *Qcrq 6525    <Q cltq 6526
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3895  ax-sep 3898  ax-nul 3906  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-iinf 4331
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3253  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-int 3639  df-iun 3682  df-br 3788  df-opab 3842  df-mpt 3843  df-tr 3878  df-eprel 4046  df-id 4050  df-po 4053  df-iso 4054  df-iord 4123  df-on 4125  df-suc 4128  df-iom 4334  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-f1 4931  df-fo 4932  df-f1o 4933  df-fv 4934  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-irdg 6013  df-1o 6059  df-oadd 6063  df-omul 6064  df-er 6165  df-ec 6167  df-qs 6171  df-ni 6545  df-pli 6546  df-mi 6547  df-lti 6548  df-plpq 6585  df-mpq 6586  df-enq 6588  df-nqqs 6589  df-plqqs 6590  df-mqqs 6591  df-1nqqs 6592  df-rq 6593  df-ltnqqs 6594
This theorem is referenced by:  caucvgprlemrnd  6914
  Copyright terms: Public domain W3C validator