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Theorem caucvgprlemopl 7659
Description: Lemma for caucvgpr 7672. 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 5876 . . . . . . 7  |-  ( l  =  s  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) )
21breq1d 4010 . . . . . 6  |-  ( l  =  s  ->  (
( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
32rexbidv 2478 . . . . 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 5514 . . . . . 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 7353 . . . . . . . 8  |-  Q.  e.  _V
76rabex 4144 . . . . . . 7  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  _V
86rabex 4144 . . . . . . 7  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  e.  _V
97, 8op1st 6141 . . . . . 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 2198 . . . . 5  |-  ( 1st `  L )  =  {
l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }
113, 10elrab2 2896 . . . 4  |-  ( s  e.  ( 1st `  L
)  <->  ( s  e. 
Q.  /\  E. j  e.  N.  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
1211simprbi 275 . . 3  |-  ( s  e.  ( 1st `  L
)  ->  E. j  e.  N.  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) )
1312adantl 277 . 2  |-  ( (
ph  /\  s  e.  ( 1st `  L ) )  ->  E. j  e.  N.  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) )
14 simprr 531 . . . 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 7405 . . . 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 122 . . 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 535 . . . . . . . . 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 7412 . . . . . . . . 9  |-  ( j  e.  N.  ->  [ <. j ,  1o >. ]  ~Q  e.  Q. )
19 recclnq 7382 . . . . . . . . 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 274 . . . . . . . . 9  |-  ( s  e.  ( 1st `  L
)  ->  s  e.  Q. )
2221ad3antlr 493 . . . . . . . 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 7397 . . . . . . . 8  |-  ( ( ( *Q `  [ <. j ,  1o >. ]  ~Q  )  e.  Q.  /\  s  e.  Q. )  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q  (
( *Q `  [ <. j ,  1o >. ]  ~Q  )  +Q  s
) )
2420, 22, 23syl2anc 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
) ) ) )  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  <Q 
( ( *Q `  [ <. j ,  1o >. ]  ~Q  )  +Q  s ) )
25 addcomnqg 7371 . . . . . . . 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 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
) ) ) )  ->  ( ( *Q
`  [ <. j ,  1o >. ]  ~Q  )  +Q  s )  =  ( s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
) )
2724, 26breqtrd 4026 . . . . . 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 539 . . . . . 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 7388 . . . . . . 7  |-  <Q  Or  Q.
30 ltrelnq 7355 . . . . . . 7  |-  <Q  C_  ( Q.  X.  Q. )
3129, 30sotri 5020 . . . . . 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 411 . . . . 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 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
) ) ) )  ->  t  e.  Q. )
34 ltexnqq 7398 . . . . . 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 411 . . . . 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 147 . . . 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 488 . . . . . . . . . . 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 488 . . . . . . . . . . 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 7371 . . . . . . . . . . 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 411 . . . . . . . . . 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 488 . . . . . . . . . 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 4024 . . . . . . . . 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 110 . . . . . . . . 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 4028 . . . . . . . 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 528 . . . . . . . . 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 7390 . . . . . . . . 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 1238 . . . . . . . 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 167 . . . . . . 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 488 . . . . . . . . 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 540 . . . . . . . . . . 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 488 . . . . . . . . . 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 7371 . . . . . . . . . . . . 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 411 . . . . . . . . . . . 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 2212 . . . . . . . . . . 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 4010 . . . . . . . . . 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 167 . . . . . . . . 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 2526 . . . . . . . . 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 411 . . . . . . . 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 5876 . . . . . . . . . . 11  |-  ( l  =  r  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( r  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) )
6059breq1d 4010 . . . . . . . . . 10  |-  ( l  =  r  ->  (
( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( r  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
6160rexbidv 2478 . . . . . . . . 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 2896 . . . . . . . 8  |-  ( r  e.  ( 1st `  L
)  <->  ( r  e. 
Q.  /\  E. j  e.  N.  ( r  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
6345, 58, 62sylanbrc 417 . . . . . . 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 306 . . . . . 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 115 . . . . 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 2579 . . . 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 2599 . 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 2599 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 104    <-> wb 105    = wceq 1353    e. wcel 2148   A.wral 2455   E.wrex 2456   {crab 2459   <.cop 3594   class class class wbr 4000   -->wf 5208   ` cfv 5212  (class class class)co 5869   1stc1st 6133   1oc1o 6404   [cec 6527   N.cnpi 7262    <N clti 7265    ~Q ceq 7269   Q.cnq 7270    +Q cplq 7272   *Qcrq 7274    <Q cltq 7275
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343
This theorem is referenced by:  caucvgprlemrnd  7663
  Copyright terms: Public domain W3C validator