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Theorem caucvgprlemladdrl 6834
Description: Lemma for caucvgpr 6838. Adding  S after embedding in positive reals, or adding it as a rational. (Contributed by Jim Kingdon, 8-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 } >.
caucvgprlemladd.s  |-  ( ph  ->  S  e.  Q. )
Assertion
Ref Expression
caucvgprlemladdrl  |-  ( ph  ->  { l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 j )  +Q  S ) }  C_  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )
Distinct variable groups:    A, j    j, F, u, l    n, F, k    k, L, j    S, l, u, j    j,
k, S
Allowed substitution hints:    ph( u, j, k, n, l)    A( u, k, n, l)    S( n)    L( u, n, l)

Proof of Theorem caucvgprlemladdrl
Dummy variables  r  f  g  h  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opeq1 3577 . . . . . . . . 9  |-  ( j  =  a  ->  <. j ,  1o >.  =  <. a ,  1o >. )
21eceq1d 6173 . . . . . . . 8  |-  ( j  =  a  ->  [ <. j ,  1o >. ]  ~Q  =  [ <. a ,  1o >. ]  ~Q  )
32fveq2d 5210 . . . . . . 7  |-  ( j  =  a  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )
43oveq2d 5556 . . . . . 6  |-  ( j  =  a  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( l  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )
5 fveq2 5206 . . . . . . 7  |-  ( j  =  a  ->  ( F `  j )  =  ( F `  a ) )
65oveq1d 5555 . . . . . 6  |-  ( j  =  a  ->  (
( F `  j
)  +Q  S )  =  ( ( F `
 a )  +Q  S ) )
74, 6breq12d 3805 . . . . 5  |-  ( j  =  a  ->  (
( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 j )  +Q  S )  <->  ( l  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) ) )
87cbvrexv 2551 . . . 4  |-  ( E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 j )  +Q  S )  <->  E. a  e.  N.  ( l  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )
98a1i 9 . . 3  |-  ( l  e.  Q.  ->  ( E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 j )  +Q  S )  <->  E. a  e.  N.  ( l  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) ) )
109rabbiia 2564 . 2  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 j )  +Q  S ) }  =  { l  e.  Q.  |  E. a  e.  N.  ( l  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 a )  +Q  S ) }
11 oveq1 5547 . . . . . . 7  |-  ( l  =  r  ->  (
l  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
)  =  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )
1211breq1d 3802 . . . . . 6  |-  ( l  =  r  ->  (
( l  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 a )  +Q  S )  <->  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) ) )
1312rexbidv 2344 . . . . 5  |-  ( l  =  r  ->  ( E. a  e.  N.  ( l  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 a )  +Q  S )  <->  E. a  e.  N.  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) ) )
1413elrab 2721 . . . 4  |-  ( r  e.  { l  e. 
Q.  |  E. a  e.  N.  ( l  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) }  <->  ( r  e.  Q.  /\  E. a  e.  N.  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) ) )
15 caucvgpr.f . . . . . . . . . . . . . . 15  |-  ( ph  ->  F : N. --> Q. )
1615ad4antr 471 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  /\  b  e.  N. )  ->  F : N. --> Q. )
17 caucvgpr.cau . . . . . . . . . . . . . . 15  |-  ( 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  )
) ) ) )
1817ad4antr 471 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  /\  b  e.  N. )  ->  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  )
) ) ) )
19 simpr 107 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  /\  b  e.  N. )  ->  b  e.  N. )
20 simpllr 494 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  /\  b  e.  N. )  ->  a  e.  N. )
2116, 18, 19, 20caucvgprlemnbj 6823 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  /\  b  e.  N. )  ->  -.  ( ( ( F `
 b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( F `  a
) )
2215ad3antrrr 469 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  ->  F : N. --> Q. )
2322ffvelrnda 5330 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  /\  b  e.  N. )  ->  ( F `  b )  e.  Q. )
24 nnnq 6578 . . . . . . . . . . . . . . . . . 18  |-  ( b  e.  N.  ->  [ <. b ,  1o >. ]  ~Q  e.  Q. )
25 recclnq 6548 . . . . . . . . . . . . . . . . . 18  |-  ( [
<. b ,  1o >. ]  ~Q  e.  Q.  ->  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  e.  Q. )
2619, 24, 253syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  /\  b  e.  N. )  ->  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  e.  Q. )
27 addclnq 6531 . . . . . . . . . . . . . . . . 17  |-  ( ( ( F `  b
)  e.  Q.  /\  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( ( F `  b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  e.  Q. )
2823, 26, 27syl2anc 397 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  /\  b  e.  N. )  ->  (
( F `  b
)  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  e.  Q. )
29 nnnq 6578 . . . . . . . . . . . . . . . . 17  |-  ( a  e.  N.  ->  [ <. a ,  1o >. ]  ~Q  e.  Q. )
30 recclnq 6548 . . . . . . . . . . . . . . . . 17  |-  ( [
<. a ,  1o >. ]  ~Q  e.  Q.  ->  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  e.  Q. )
3120, 29, 303syl 17 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  /\  b  e.  N. )  ->  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  e.  Q. )
32 caucvgprlemladd.s . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  S  e.  Q. )
3332ad4antr 471 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  /\  b  e.  N. )  ->  S  e.  Q. )
34 addassnqg 6538 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( F `  b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  e.  Q.  /\  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  e.  Q.  /\  S  e.  Q. )  ->  (
( ( ( F `
 b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  +Q  S )  =  ( ( ( F `  b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  +Q  ( ( *Q `  [ <. a ,  1o >. ]  ~Q  )  +Q  S ) ) )
3528, 31, 33, 34syl3anc 1146 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  /\  b  e.  N. )  ->  (
( ( ( F `
 b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  +Q  S )  =  ( ( ( F `  b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  +Q  ( ( *Q `  [ <. a ,  1o >. ]  ~Q  )  +Q  S ) ) )
3635breq1d 3802 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  /\  b  e.  N. )  ->  (
( ( ( ( F `  b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  +Q  S )  <Q 
( ( F `  a )  +Q  S
)  <->  ( ( ( F `  b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  +Q  ( ( *Q
`  [ <. a ,  1o >. ]  ~Q  )  +Q  S ) )  <Q 
( ( F `  a )  +Q  S
) ) )
37 ltanqg 6556 . . . . . . . . . . . . . . . 16  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
3837adantl 266 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  (
r  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 a )  +Q  S ) )  /\  b  e.  N. )  /\  ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )
)  ->  ( f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
39 addclnq 6531 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( F `  b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  e.  Q.  /\  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( ( ( F `
 b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  e. 
Q. )
4028, 31, 39syl2anc 397 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  /\  b  e.  N. )  ->  (
( ( F `  b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
)  e.  Q. )
4116, 20ffvelrnd 5331 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  /\  b  e.  N. )  ->  ( F `  a )  e.  Q. )
42 addcomnqg 6537 . . . . . . . . . . . . . . . 16  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
4342adantl 266 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  (
r  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 a )  +Q  S ) )  /\  b  e.  N. )  /\  ( f  e.  Q.  /\  g  e.  Q. )
)  ->  ( f  +Q  g )  =  ( g  +Q  f ) )
4438, 40, 41, 33, 43caovord2d 5698 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  /\  b  e.  N. )  ->  (
( ( ( F `
 b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( F `  a
)  <->  ( ( ( ( F `  b
)  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
)  +Q  S ) 
<Q  ( ( F `  a )  +Q  S
) ) )
45 addcomnqg 6537 . . . . . . . . . . . . . . . . 17  |-  ( ( S  e.  Q.  /\  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  =  ( ( *Q `  [ <. a ,  1o >. ]  ~Q  )  +Q  S ) )
4633, 31, 45syl2anc 397 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  /\  b  e.  N. )  ->  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  =  ( ( *Q
`  [ <. a ,  1o >. ]  ~Q  )  +Q  S ) )
4746oveq2d 5556 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  /\  b  e.  N. )  ->  (
( ( F `  b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  +Q  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )  =  ( ( ( F `  b
)  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  +Q  ( ( *Q `  [ <. a ,  1o >. ]  ~Q  )  +Q  S ) ) )
4847breq1d 3802 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  /\  b  e.  N. )  ->  (
( ( ( F `
 b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  +Q  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
) )  <Q  (
( F `  a
)  +Q  S )  <-> 
( ( ( F `
 b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  +Q  ( ( *Q `  [ <. a ,  1o >. ]  ~Q  )  +Q  S ) )  <Q 
( ( F `  a )  +Q  S
) ) )
4936, 44, 483bitr4rd 214 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  /\  b  e.  N. )  ->  (
( ( ( F `
 b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  +Q  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
) )  <Q  (
( F `  a
)  +Q  S )  <-> 
( ( ( F `
 b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( F `  a
) ) )
5021, 49mtbird 608 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  /\  b  e.  N. )  ->  -.  ( ( ( F `
 b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  +Q  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
) )  <Q  (
( F `  a
)  +Q  S ) )
5150nrexdv 2429 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  ->  -.  E. b  e.  N.  (
( ( F `  b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  +Q  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )  <Q  ( ( F `  a )  +Q  S ) )
5251intnand 851 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  ->  -.  ( ( ( F `
 a )  +Q  S )  e.  Q.  /\ 
E. b  e.  N.  ( ( ( F `
 b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  +Q  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
) )  <Q  (
( F `  a
)  +Q  S ) ) )
5317ad3antrrr 469 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  ->  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  )
) ) ) )
54 caucvgpr.bnd . . . . . . . . . . . . . . 15  |-  ( ph  ->  A. j  e.  N.  A  <Q  ( F `  j ) )
55 fveq2 5206 . . . . . . . . . . . . . . . . 17  |-  ( j  =  b  ->  ( F `  j )  =  ( F `  b ) )
5655breq2d 3804 . . . . . . . . . . . . . . . 16  |-  ( j  =  b  ->  ( A  <Q  ( F `  j )  <->  A  <Q  ( F `  b ) ) )
5756cbvralv 2550 . . . . . . . . . . . . . . 15  |-  ( A. j  e.  N.  A  <Q  ( F `  j
)  <->  A. b  e.  N.  A  <Q  ( F `  b ) )
5854, 57sylib 131 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. b  e.  N.  A  <Q  ( F `  b ) )
5958ad3antrrr 469 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  ->  A. b  e.  N.  A  <Q  ( F `  b )
)
60 caucvgpr.lim . . . . . . . . . . . . . 14  |-  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 } >.
61 opeq1 3577 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( j  =  b  ->  <. j ,  1o >.  =  <. b ,  1o >. )
6261eceq1d 6173 . . . . . . . . . . . . . . . . . . . . 21  |-  ( j  =  b  ->  [ <. j ,  1o >. ]  ~Q  =  [ <. b ,  1o >. ]  ~Q  )
6362fveq2d 5210 . . . . . . . . . . . . . . . . . . . 20  |-  ( j  =  b  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )
6463oveq2d 5556 . . . . . . . . . . . . . . . . . . 19  |-  ( j  =  b  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( l  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) )
6564, 55breq12d 3805 . . . . . . . . . . . . . . . . . 18  |-  ( j  =  b  ->  (
( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( l  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q 
( F `  b
) ) )
6665cbvrexv 2551 . . . . . . . . . . . . . . . . 17  |-  ( E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  E. b  e.  N.  ( l  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q 
( F `  b
) )
6766a1i 9 . . . . . . . . . . . . . . . 16  |-  ( l  e.  Q.  ->  ( E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  E. b  e.  N.  ( l  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q 
( F `  b
) ) )
6867rabbiia 2564 . . . . . . . . . . . . . . 15  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  =  { l  e.  Q.  |  E. b  e.  N.  ( l  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  <Q  ( F `  b ) }
6955, 63oveq12d 5558 . . . . . . . . . . . . . . . . . . 19  |-  ( j  =  b  ->  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( ( F `  b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) )
7069breq1d 3802 . . . . . . . . . . . . . . . . . 18  |-  ( j  =  b  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  ( ( F `  b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q  u ) )
7170cbvrexv 2551 . . . . . . . . . . . . . . . . 17  |-  ( E. j  e.  N.  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  E. b  e.  N.  ( ( F `
 b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q  u )
7271a1i 9 . . . . . . . . . . . . . . . 16  |-  ( u  e.  Q.  ->  ( E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  E. b  e.  N.  ( ( F `
 b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q  u ) )
7372rabbiia 2564 . . . . . . . . . . . . . . 15  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  =  {
u  e.  Q.  |  E. b  e.  N.  ( ( F `  b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  <Q  u }
7468, 73opeq12i 3582 . . . . . . . . . . . . . 14  |-  <. { 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. b  e.  N.  ( l  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  <Q  ( F `  b ) } ,  { u  e.  Q.  |  E. b  e.  N.  ( ( F `  b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  <Q  u } >.
7560, 74eqtri 2076 . . . . . . . . . . . . 13  |-  L  = 
<. { l  e.  Q.  |  E. b  e.  N.  ( l  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  <Q  ( F `  b ) } ,  { u  e.  Q.  |  E. b  e.  N.  ( ( F `  b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  <Q  u } >.
7632ad3antrrr 469 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  ->  S  e.  Q. )
7729, 30syl 14 . . . . . . . . . . . . . . 15  |-  ( a  e.  N.  ->  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  e.  Q. )
7877ad2antlr 466 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  ->  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  e.  Q. )
79 addclnq 6531 . . . . . . . . . . . . . 14  |-  ( ( S  e.  Q.  /\  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  e.  Q. )
8076, 78, 79syl2anc 397 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  ->  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  e.  Q. )
8122, 53, 59, 75, 80caucvgprlemladdfu 6833 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  ->  ( 2nd `  ( L  +P.  <. { l  |  l 
<Q  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
) } ,  {
u  |  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) 
<Q  u } >. )
)  C_  { u  e.  Q.  |  E. b  e.  N.  ( ( ( F `  b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  +Q  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )  <Q  u } )
8281sseld 2972 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  ->  (
( ( F `  a )  +Q  S
)  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q 
( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
) } ,  {
u  |  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) 
<Q  u } >. )
)  ->  ( ( F `  a )  +Q  S )  e.  {
u  e.  Q.  |  E. b  e.  N.  ( ( ( F `
 b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  +Q  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
) )  <Q  u } ) )
83 breq2 3796 . . . . . . . . . . . . 13  |-  ( u  =  ( ( F `
 a )  +Q  S )  ->  (
( ( ( F `
 b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  +Q  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
) )  <Q  u  <->  ( ( ( F `  b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  +Q  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )  <Q  ( ( F `  a )  +Q  S ) ) )
8483rexbidv 2344 . . . . . . . . . . . 12  |-  ( u  =  ( ( F `
 a )  +Q  S )  ->  ( E. b  e.  N.  ( ( ( F `
 b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  +Q  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
) )  <Q  u  <->  E. b  e.  N.  (
( ( F `  b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  +Q  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )  <Q  ( ( F `  a )  +Q  S ) ) )
8584elrab 2721 . . . . . . . . . . 11  |-  ( ( ( F `  a
)  +Q  S )  e.  { u  e. 
Q.  |  E. b  e.  N.  ( ( ( F `  b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  +Q  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )  <Q  u }  <->  ( ( ( F `  a )  +Q  S )  e. 
Q.  /\  E. b  e.  N.  ( ( ( F `  b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  +Q  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )  <Q 
( ( F `  a )  +Q  S
) ) )
8682, 85syl6ib 154 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  ->  (
( ( F `  a )  +Q  S
)  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q 
( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
) } ,  {
u  |  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) 
<Q  u } >. )
)  ->  ( (
( F `  a
)  +Q  S )  e.  Q.  /\  E. b  e.  N.  (
( ( F `  b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  +Q  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )  <Q  ( ( F `  a )  +Q  S ) ) ) )
8752, 86mtod 599 . . . . . . . . 9  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  ->  -.  ( ( F `  a )  +Q  S
)  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q 
( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
) } ,  {
u  |  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) 
<Q  u } >. )
) )
8815, 17, 54, 60caucvgprlemcl 6832 . . . . . . . . . . . 12  |-  ( ph  ->  L  e.  P. )
8988ad3antrrr 469 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  ->  L  e.  P. )
90 nqprlu 6703 . . . . . . . . . . . 12  |-  ( ( S  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
)  e.  Q.  ->  <. { l  |  l 
<Q  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
) } ,  {
u  |  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) 
<Q  u } >.  e.  P. )
9180, 90syl 14 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  ->  <. { l  |  l  <Q  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) } ,  { u  |  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q  u } >.  e.  P. )
92 addclpr 6693 . . . . . . . . . . 11  |-  ( ( L  e.  P.  /\  <. { l  |  l 
<Q  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
) } ,  {
u  |  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) 
<Q  u } >.  e.  P. )  ->  ( L  +P.  <. { l  |  l 
<Q  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
) } ,  {
u  |  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) 
<Q  u } >. )  e.  P. )
9389, 91, 92syl2anc 397 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  ->  ( L  +P.  <. { l  |  l  <Q  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) } ,  { u  |  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  <Q  u } >. )  e.  P. )
94 prop 6631 . . . . . . . . . . 11  |-  ( ( L  +P.  <. { l  |  l  <Q  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) } ,  { u  |  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q  u } >. )  e.  P.  -> 
<. ( 1st `  ( L  +P.  <. { l  |  l  <Q  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) } ,  { u  |  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  <Q  u } >. ) ) ,  ( 2nd `  ( L  +P.  <. { l  |  l  <Q 
( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
) } ,  {
u  |  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) 
<Q  u } >. )
) >.  e.  P. )
95 prloc 6647 . . . . . . . . . . 11  |-  ( (
<. ( 1st `  ( L  +P.  <. { l  |  l  <Q  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) } ,  { u  |  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  <Q  u } >. ) ) ,  ( 2nd `  ( L  +P.  <. { l  |  l  <Q 
( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
) } ,  {
u  |  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) 
<Q  u } >. )
) >.  e.  P.  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 a )  +Q  S ) )  -> 
( ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) } ,  { u  |  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  <Q  u } >. ) )  \/  ( ( F `  a )  +Q  S )  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) } ,  { u  |  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  <Q  u } >. ) ) ) )
9694, 95sylan 271 . . . . . . . . . 10  |-  ( ( ( L  +P.  <. { l  |  l  <Q 
( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
) } ,  {
u  |  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) 
<Q  u } >. )  e.  P.  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) 
<Q  ( ( F `  a )  +Q  S
) )  ->  (
( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q 
( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
) } ,  {
u  |  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) 
<Q  u } >. )
)  \/  ( ( F `  a )  +Q  S )  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) } ,  { u  |  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  <Q  u } >. ) ) ) )
9793, 96sylancom 405 . . . . . . . . 9  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  ->  (
( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q 
( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
) } ,  {
u  |  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) 
<Q  u } >. )
)  \/  ( ( F `  a )  +Q  S )  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) } ,  { u  |  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  <Q  u } >. ) ) ) )
9887, 97ecased 1255 . . . . . . . 8  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  ->  (
r  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
)  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q 
( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
) } ,  {
u  |  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) 
<Q  u } >. )
) )
99 simpllr 494 . . . . . . . . 9  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  ->  r  e.  Q. )
10089, 76, 99, 78caucvgprlemcanl 6800 . . . . . . . 8  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  ->  (
( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q 
( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
) } ,  {
u  |  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) 
<Q  u } >. )
)  <->  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) ) )
10198, 100mpbid 139 . . . . . . 7  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  ->  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )
102101ex 112 . . . . . 6  |-  ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  ->  (
( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 a )  +Q  S )  ->  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) ) )
103102rexlimdva 2450 . . . . 5  |-  ( (
ph  /\  r  e.  Q. )  ->  ( E. a  e.  N.  (
r  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 a )  +Q  S )  ->  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) ) )
104103expimpd 349 . . . 4  |-  ( ph  ->  ( ( r  e. 
Q.  /\  E. a  e.  N.  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  ->  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) ) )
10514, 104syl5bi 145 . . 3  |-  ( ph  ->  ( r  e.  {
l  e.  Q.  |  E. a  e.  N.  ( l  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 a )  +Q  S ) }  ->  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) ) )
106105ssrdv 2979 . 2  |-  ( ph  ->  { l  e.  Q.  |  E. a  e.  N.  ( l  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 a )  +Q  S ) }  C_  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )
10710, 106syl5eqss 3017 1  |-  ( ph  ->  { l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 j )  +Q  S ) }  C_  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    \/ wo 639    /\ w3a 896    = wceq 1259    e. wcel 1409   {cab 2042   A.wral 2323   E.wrex 2324   {crab 2327    C_ wss 2945   <.cop 3406   class class class wbr 3792   -->wf 4926   ` cfv 4930  (class class class)co 5540   1stc1st 5793   2ndc2nd 5794   1oc1o 6025   [cec 6135   N.cnpi 6428    <N clti 6431    ~Q ceq 6435   Q.cnq 6436    +Q cplq 6438   *Qcrq 6440    <Q cltq 6441   P.cnp 6447    +P. cpp 6449
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-iplp 6624  df-iltp 6626
This theorem is referenced by:  caucvgprlem1  6835
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