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Theorem caucvgprlemopl 7501
Description: Lemma for caucvgpr 7514. 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 5789 . . . . . . 7  |-  ( l  =  s  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) )
21breq1d 3947 . . . . . 6  |-  ( l  =  s  ->  (
( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
32rexbidv 2439 . . . . 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 5432 . . . . . 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 7195 . . . . . . . 8  |-  Q.  e.  _V
76rabex 4080 . . . . . . 7  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  _V
86rabex 4080 . . . . . . 7  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  e.  _V
97, 8op1st 6052 . . . . . 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 2161 . . . . 5  |-  ( 1st `  L )  =  {
l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }
113, 10elrab2 2847 . . . 4  |-  ( s  e.  ( 1st `  L
)  <->  ( s  e. 
Q.  /\  E. j  e.  N.  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
1211simprbi 273 . . 3  |-  ( s  e.  ( 1st `  L
)  ->  E. j  e.  N.  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) )
1312adantl 275 . 2  |-  ( (
ph  /\  s  e.  ( 1st `  L ) )  ->  E. j  e.  N.  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) )
14 simprr 522 . . . 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 7247 . . . 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 121 . . 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 525 . . . . . . . . 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 7254 . . . . . . . . 9  |-  ( j  e.  N.  ->  [ <. j ,  1o >. ]  ~Q  e.  Q. )
19 recclnq 7224 . . . . . . . . 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 272 . . . . . . . . 9  |-  ( s  e.  ( 1st `  L
)  ->  s  e.  Q. )
2221ad3antlr 485 . . . . . . . 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 7239 . . . . . . . 8  |-  ( ( ( *Q `  [ <. j ,  1o >. ]  ~Q  )  e.  Q.  /\  s  e.  Q. )  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  (
( *Q `  [ <. j ,  1o >. ]  ~Q  )  +Q  s
) )
2420, 22, 23syl2anc 409 . . . . . . 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 7213 . . . . . . . 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 409 . . . . . . 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 3962 . . . . . 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 529 . . . . . 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 7230 . . . . . . 7  |-  <Q  Or  Q.
30 ltrelnq 7197 . . . . . . 7  |-  <Q  C_  ( Q.  X.  Q. )
3129, 30sotri 4942 . . . . . 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 409 . . . . 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 521 . . . . . 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 7240 . . . . . 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 409 . . . . 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 146 . . . 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 480 . . . . . . . . . . 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 480 . . . . . . . . . . 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 7213 . . . . . . . . . . 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 409 . . . . . . . . . 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 480 . . . . . . . . . 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 3960 . . . . . . . . 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 109 . . . . . . . . 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 3964 . . . . . . . 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 520 . . . . . . . . 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 7232 . . . . . . . . 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 1217 . . . . . . . 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 166 . . . . . . 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 480 . . . . . . . . 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 530 . . . . . . . . . . 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 480 . . . . . . . . . 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 7213 . . . . . . . . . . . . 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 409 . . . . . . . . . . . 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 2175 . . . . . . . . . . 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 3947 . . . . . . . . . 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 166 . . . . . . . . 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 2484 . . . . . . . . 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 409 . . . . . . . 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 5789 . . . . . . . . . . 11  |-  ( l  =  r  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( r  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) )
6059breq1d 3947 . . . . . . . . . 10  |-  ( l  =  r  ->  (
( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( r  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
6160rexbidv 2439 . . . . . . . . 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 2847 . . . . . . . 8  |-  ( r  e.  ( 1st `  L
)  <->  ( r  e. 
Q.  /\  E. j  e.  N.  ( r  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
6345, 58, 62sylanbrc 414 . . . . . . 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 304 . . . . . 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 114 . . . . 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 2537 . . . 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 2557 . 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 2557 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 103    <-> wb 104    = wceq 1332    e. wcel 1481   A.wral 2417   E.wrex 2418   {crab 2421   <.cop 3535   class class class wbr 3937   -->wf 5127   ` cfv 5131  (class class class)co 5782   1stc1st 6044   1oc1o 6314   [cec 6435   N.cnpi 7104    <N clti 7107    ~Q ceq 7111   Q.cnq 7112    +Q cplq 7114   *Qcrq 7116    <Q cltq 7117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-eprel 4219  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-1o 6321  df-oadd 6325  df-omul 6326  df-er 6437  df-ec 6439  df-qs 6443  df-ni 7136  df-pli 7137  df-mi 7138  df-lti 7139  df-plpq 7176  df-mpq 7177  df-enq 7179  df-nqqs 7180  df-plqqs 7181  df-mqqs 7182  df-1nqqs 7183  df-rq 7184  df-ltnqqs 7185
This theorem is referenced by:  caucvgprlemrnd  7505
  Copyright terms: Public domain W3C validator