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Theorem caucvgprlemladdrl 6757
Description: Lemma for caucvgpr 6761. 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 3546 . . . . . . . . 9  |-  ( j  =  a  ->  <. j ,  1o >.  =  <. a ,  1o >. )
21eceq1d 6129 . . . . . . . 8  |-  ( j  =  a  ->  [ <. j ,  1o >. ]  ~Q  =  [ <. a ,  1o >. ]  ~Q  )
32fveq2d 5169 . . . . . . 7  |-  ( j  =  a  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )
43oveq2d 5515 . . . . . 6  |-  ( j  =  a  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( l  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )
5 fveq2 5165 . . . . . . 7  |-  ( j  =  a  ->  ( F `  j )  =  ( F `  a ) )
65oveq1d 5514 . . . . . 6  |-  ( j  =  a  ->  (
( F `  j
)  +Q  S )  =  ( ( F `
 a )  +Q  S ) )
74, 6breq12d 3774 . . . . 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 2531 . . . 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 2544 . 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 5506 . . . . . . 7  |-  ( l  =  r  ->  (
l  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
)  =  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )
1211breq1d 3771 . . . . . 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 2324 . . . . 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 2695 . . . 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 463 . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . 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 103 . . . . . . . . . . . . . 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 486 . . . . . . . . . . . . . 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 6746 . . . . . . . . . . . . 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 461 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  ->  F : N. --> Q. )
2322ffvelrnda 5289 . . . . . . . . . . . . . . . . 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 6501 . . . . . . . . . . . . . . . . . 18  |-  ( b  e.  N.  ->  [ <. b ,  1o >. ]  ~Q  e.  Q. )
25 recclnq 6471 . . . . . . . . . . . . . . . . . 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 6454 . . . . . . . . . . . . . . . . 17  |-  ( ( ( F `  b
)  e.  Q.  /\  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( ( F `  b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  e.  Q. )
2823, 26, 27syl2anc 391 . . . . . . . . . . . . . . . 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 6501 . . . . . . . . . . . . . . . . 17  |-  ( a  e.  N.  ->  [ <. a ,  1o >. ]  ~Q  e.  Q. )
30 recclnq 6471 . . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . . 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 6461 . . . . . . . . . . . . . . . 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 1135 . . . . . . . . . . . . . . 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 3771 . . . . . . . . . . . . . 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 6479 . . . . . . . . . . . . . . . 16  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
3837adantl 262 . . . . . . . . . . . . . . 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 6454 . . . . . . . . . . . . . . . 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 391 . . . . . . . . . . . . . . 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 5290 . . . . . . . . . . . . . . 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 6460 . . . . . . . . . . . . . . . 16  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
4342adantl 262 . . . . . . . . . . . . . . 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 5657 . . . . . . . . . . . . . 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 6460 . . . . . . . . . . . . . . . . 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 391 . . . . . . . . . . . . . . . 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 5515 . . . . . . . . . . . . . . 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 3771 . . . . . . . . . . . . . 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 210 . . . . . . . . . . . . 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 598 . . . . . . . . . . . 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 2409 . . . . . . . . . . 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 840 . . . . . . . . . 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 461 . . . . . . . . . . . . 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 5165 . . . . . . . . . . . . . . . . 17  |-  ( j  =  b  ->  ( F `  j )  =  ( F `  b ) )
5655breq2d 3773 . . . . . . . . . . . . . . . 16  |-  ( j  =  b  ->  ( A  <Q  ( F `  j )  <->  A  <Q  ( F `  b ) ) )
5756cbvralv 2530 . . . . . . . . . . . . . . 15  |-  ( A. j  e.  N.  A  <Q  ( F `  j
)  <->  A. b  e.  N.  A  <Q  ( F `  b ) )
5854, 57sylib 127 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. b  e.  N.  A  <Q  ( F `  b ) )
5958ad3antrrr 461 . . . . . . . . . . . . 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 3546 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( j  =  b  ->  <. j ,  1o >.  =  <. b ,  1o >. )
6261eceq1d 6129 . . . . . . . . . . . . . . . . . . . . 21  |-  ( j  =  b  ->  [ <. j ,  1o >. ]  ~Q  =  [ <. b ,  1o >. ]  ~Q  )
6362fveq2d 5169 . . . . . . . . . . . . . . . . . . . 20  |-  ( j  =  b  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )
6463oveq2d 5515 . . . . . . . . . . . . . . . . . . 19  |-  ( j  =  b  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( l  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) )
6564, 55breq12d 3774 . . . . . . . . . . . . . . . . . 18  |-  ( j  =  b  ->  (
( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( l  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q 
( F `  b
) ) )
6665cbvrexv 2531 . . . . . . . . . . . . . . . . 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 2544 . . . . . . . . . . . . . . 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 5517 . . . . . . . . . . . . . . . . . . 19  |-  ( j  =  b  ->  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( ( F `  b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) )
7069breq1d 3771 . . . . . . . . . . . . . . . . . 18  |-  ( j  =  b  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  ( ( F `  b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q  u ) )
7170cbvrexv 2531 . . . . . . . . . . . . . . . . 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 2544 . . . . . . . . . . . . . . 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 3551 . . . . . . . . . . . . . 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 2060 . . . . . . . . . . . . 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 461 . . . . . . . . . . . . . 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 458 . . . . . . . . . . . . . 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 6454 . . . . . . . . . . . . . 14  |-  ( ( S  e.  Q.  /\  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  e.  Q. )
8076, 78, 79syl2anc 391 . . . . . . . . . . . . 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 6756 . . . . . . . . . . . 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 2941 . . . . . . . . . . 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 3765 . . . . . . . . . . . . 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 2324 . . . . . . . . . . . 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 2695 . . . . . . . . . . 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 150 . . . . . . . . . 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 589 . . . . . . . . 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 6755 . . . . . . . . . . . 12  |-  ( ph  ->  L  e.  P. )
8988ad3antrrr 461 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  ->  L  e.  P. )
90 nqprlu 6626 . . . . . . . . . . . 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 6616 . . . . . . . . . . 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 391 . . . . . . . . . 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 6554 . . . . . . . . . . 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 6570 . . . . . . . . . . 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 267 . . . . . . . . . 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 397 . . . . . . . . 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 1239 . . . . . . . 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 486 . . . . . . . . 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 6723 . . . . . . . 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 135 . . . . . . 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 108 . . . . . 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 2430 . . . . 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 345 . . . 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 141 . . 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 2948 . 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 2986 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 97    <-> wb 98    \/ wo 629    /\ w3a 885    = wceq 1243    e. wcel 1393   {cab 2026   A.wral 2303   E.wrex 2304   {crab 2307    C_ wss 2914   <.cop 3375   class class class wbr 3761   -->wf 4885   ` cfv 4889  (class class class)co 5499   1stc1st 5752   2ndc2nd 5753   1oc1o 5981   [cec 6091   N.cnpi 6351    <N clti 6354    ~Q ceq 6358   Q.cnq 6359    +Q cplq 6361   *Qcrq 6363    <Q cltq 6364   P.cnp 6370    +P. cpp 6372
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3869  ax-sep 3872  ax-nul 3880  ax-pow 3924  ax-pr 3941  ax-un 4166  ax-setind 4256  ax-iinf 4298
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2308  df-rex 2309  df-reu 2310  df-rab 2312  df-v 2556  df-sbc 2762  df-csb 2850  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-int 3613  df-iun 3656  df-br 3762  df-opab 3816  df-mpt 3817  df-tr 3852  df-eprel 4023  df-id 4027  df-po 4030  df-iso 4031  df-iord 4099  df-on 4101  df-suc 4104  df-iom 4301  df-xp 4338  df-rel 4339  df-cnv 4340  df-co 4341  df-dm 4342  df-rn 4343  df-res 4344  df-ima 4345  df-iota 4854  df-fun 4891  df-fn 4892  df-f 4893  df-f1 4894  df-fo 4895  df-f1o 4896  df-fv 4897  df-ov 5502  df-oprab 5503  df-mpt2 5504  df-1st 5754  df-2nd 5755  df-recs 5907  df-irdg 5944  df-1o 5988  df-2o 5989  df-oadd 5992  df-omul 5993  df-er 6093  df-ec 6095  df-qs 6099  df-ni 6383  df-pli 6384  df-mi 6385  df-lti 6386  df-plpq 6423  df-mpq 6424  df-enq 6426  df-nqqs 6427  df-plqqs 6428  df-mqqs 6429  df-1nqqs 6430  df-rq 6431  df-ltnqqs 6432  df-enq0 6503  df-nq0 6504  df-0nq0 6505  df-plq0 6506  df-mq0 6507  df-inp 6545  df-iplp 6547  df-iltp 6549
This theorem is referenced by:  caucvgprlem1  6758
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