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Theorem caucvgprlemopl 7378
Description: Lemma for caucvgpr 7391. 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 5713 . . . . . . 7  |-  ( l  =  s  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) )
21breq1d 3885 . . . . . 6  |-  ( l  =  s  ->  (
( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
32rexbidv 2397 . . . . 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 5356 . . . . . 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 7072 . . . . . . . 8  |-  Q.  e.  _V
76rabex 4012 . . . . . . 7  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  _V
86rabex 4012 . . . . . . 7  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  e.  _V
97, 8op1st 5975 . . . . . 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 2120 . . . . 5  |-  ( 1st `  L )  =  {
l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }
113, 10elrab2 2796 . . . 4  |-  ( s  e.  ( 1st `  L
)  <->  ( s  e. 
Q.  /\  E. j  e.  N.  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
1211simprbi 271 . . 3  |-  ( s  e.  ( 1st `  L
)  ->  E. j  e.  N.  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) )
1312adantl 273 . 2  |-  ( (
ph  /\  s  e.  ( 1st `  L ) )  ->  E. j  e.  N.  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) )
14 simprr 502 . . . 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 7124 . . . 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 505 . . . . . . . . 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 7131 . . . . . . . . 9  |-  ( j  e.  N.  ->  [ <. j ,  1o >. ]  ~Q  e.  Q. )
19 recclnq 7101 . . . . . . . . 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 270 . . . . . . . . 9  |-  ( s  e.  ( 1st `  L
)  ->  s  e.  Q. )
2221ad3antlr 480 . . . . . . . 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 7116 . . . . . . . 8  |-  ( ( ( *Q `  [ <. j ,  1o >. ]  ~Q  )  e.  Q.  /\  s  e.  Q. )  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  (
( *Q `  [ <. j ,  1o >. ]  ~Q  )  +Q  s
) )
2420, 22, 23syl2anc 406 . . . . . . 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 7090 . . . . . . . 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 406 . . . . . . 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 3899 . . . . . 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 509 . . . . . 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 7107 . . . . . . 7  |-  <Q  Or  Q.
30 ltrelnq 7074 . . . . . . 7  |-  <Q  C_  ( Q.  X.  Q. )
3129, 30sotri 4870 . . . . . 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 406 . . . . 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 501 . . . . . 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 7117 . . . . . 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 406 . . . . 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 475 . . . . . . . . . . 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 475 . . . . . . . . . . 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 7090 . . . . . . . . . . 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 406 . . . . . . . . . 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 475 . . . . . . . . . 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 3897 . . . . . . . . 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 3901 . . . . . . . 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 500 . . . . . . . . 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 7109 . . . . . . . . 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 1184 . . . . . . . 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 475 . . . . . . . . 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 510 . . . . . . . . . . 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 475 . . . . . . . . . 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 7090 . . . . . . . . . . . . 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 406 . . . . . . . . . . . 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 2134 . . . . . . . . . . 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 3885 . . . . . . . . . 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 2440 . . . . . . . . 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 406 . . . . . . . 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 5713 . . . . . . . . . . 11  |-  ( l  =  r  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( r  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) )
6059breq1d 3885 . . . . . . . . . 10  |-  ( l  =  r  ->  (
( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( r  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
6160rexbidv 2397 . . . . . . . . 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 2796 . . . . . . . 8  |-  ( r  e.  ( 1st `  L
)  <->  ( r  e. 
Q.  /\  E. j  e.  N.  ( r  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
6345, 58, 62sylanbrc 411 . . . . . . 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 302 . . . . . 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 2493 . . . 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 2513 . 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 2513 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 1299    e. wcel 1448   A.wral 2375   E.wrex 2376   {crab 2379   <.cop 3477   class class class wbr 3875   -->wf 5055   ` cfv 5059  (class class class)co 5706   1stc1st 5967   1oc1o 6236   [cec 6357   N.cnpi 6981    <N clti 6984    ~Q ceq 6988   Q.cnq 6989    +Q cplq 6991   *Qcrq 6993    <Q cltq 6994
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-eprel 4149  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-irdg 6197  df-1o 6243  df-oadd 6247  df-omul 6248  df-er 6359  df-ec 6361  df-qs 6365  df-ni 7013  df-pli 7014  df-mi 7015  df-lti 7016  df-plpq 7053  df-mpq 7054  df-enq 7056  df-nqqs 7057  df-plqqs 7058  df-mqqs 7059  df-1nqqs 7060  df-rq 7061  df-ltnqqs 7062
This theorem is referenced by:  caucvgprlemrnd  7382
  Copyright terms: Public domain W3C validator