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Theorem caucvgprlemladdrl 7216
Description: Lemma for caucvgpr 7220. 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 3617 . . . . . . . . 9  |-  ( j  =  a  ->  <. j ,  1o >.  =  <. a ,  1o >. )
21eceq1d 6308 . . . . . . . 8  |-  ( j  =  a  ->  [ <. j ,  1o >. ]  ~Q  =  [ <. a ,  1o >. ]  ~Q  )
32fveq2d 5293 . . . . . . 7  |-  ( j  =  a  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )
43oveq2d 5650 . . . . . 6  |-  ( j  =  a  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( l  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )
5 fveq2 5289 . . . . . . 7  |-  ( j  =  a  ->  ( F `  j )  =  ( F `  a ) )
65oveq1d 5649 . . . . . 6  |-  ( j  =  a  ->  (
( F `  j
)  +Q  S )  =  ( ( F `
 a )  +Q  S ) )
74, 6breq12d 3850 . . . . 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 2591 . . . 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 2604 . 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 5641 . . . . . . 7  |-  ( l  =  r  ->  (
l  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
)  =  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )
1211breq1d 3847 . . . . . 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 2381 . . . . 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 2769 . . . 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 478 . . . . . . . . . . . . . 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 478 . . . . . . . . . . . . . 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 108 . . . . . . . . . . . . . 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 501 . . . . . . . . . . . . . 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 7205 . . . . . . . . . . . . 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 476 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  ->  F : N. --> Q. )
2322ffvelrnda 5418 . . . . . . . . . . . . . . . . 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 6960 . . . . . . . . . . . . . . . . . 18  |-  ( b  e.  N.  ->  [ <. b ,  1o >. ]  ~Q  e.  Q. )
25 recclnq 6930 . . . . . . . . . . . . . . . . . 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 6913 . . . . . . . . . . . . . . . . 17  |-  ( ( ( F `  b
)  e.  Q.  /\  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( ( F `  b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  e.  Q. )
2823, 26, 27syl2anc 403 . . . . . . . . . . . . . . . 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 6960 . . . . . . . . . . . . . . . . 17  |-  ( a  e.  N.  ->  [ <. a ,  1o >. ]  ~Q  e.  Q. )
30 recclnq 6930 . . . . . . . . . . . . . . . . 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 478 . . . . . . . . . . . . . . . 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 6920 . . . . . . . . . . . . . . . 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 1174 . . . . . . . . . . . . . . 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 3847 . . . . . . . . . . . . . 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 6938 . . . . . . . . . . . . . . . 16  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
3837adantl 271 . . . . . . . . . . . . . . 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 6913 . . . . . . . . . . . . . . . 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 403 . . . . . . . . . . . . . . 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 5419 . . . . . . . . . . . . . . 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 6919 . . . . . . . . . . . . . . . 16  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
4342adantl 271 . . . . . . . . . . . . . . 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 5796 . . . . . . . . . . . . . 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 6919 . . . . . . . . . . . . . . . . 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 403 . . . . . . . . . . . . . . . 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 5650 . . . . . . . . . . . . . . 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 3847 . . . . . . . . . . . . . 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 219 . . . . . . . . . . . . 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 633 . . . . . . . . . . . 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 2466 . . . . . . . . . . 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 878 . . . . . . . . . 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 476 . . . . . . . . . . . . 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 5289 . . . . . . . . . . . . . . . . 17  |-  ( j  =  b  ->  ( F `  j )  =  ( F `  b ) )
5655breq2d 3849 . . . . . . . . . . . . . . . 16  |-  ( j  =  b  ->  ( A  <Q  ( F `  j )  <->  A  <Q  ( F `  b ) ) )
5756cbvralv 2590 . . . . . . . . . . . . . . 15  |-  ( A. j  e.  N.  A  <Q  ( F `  j
)  <->  A. b  e.  N.  A  <Q  ( F `  b ) )
5854, 57sylib 120 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. b  e.  N.  A  <Q  ( F `  b ) )
5958ad3antrrr 476 . . . . . . . . . . . . 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 3617 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( j  =  b  ->  <. j ,  1o >.  =  <. b ,  1o >. )
6261eceq1d 6308 . . . . . . . . . . . . . . . . . . . . 21  |-  ( j  =  b  ->  [ <. j ,  1o >. ]  ~Q  =  [ <. b ,  1o >. ]  ~Q  )
6362fveq2d 5293 . . . . . . . . . . . . . . . . . . . 20  |-  ( j  =  b  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )
6463oveq2d 5650 . . . . . . . . . . . . . . . . . . 19  |-  ( j  =  b  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( l  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) )
6564, 55breq12d 3850 . . . . . . . . . . . . . . . . . 18  |-  ( j  =  b  ->  (
( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( l  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q 
( F `  b
) ) )
6665cbvrexv 2591 . . . . . . . . . . . . . . . . 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 2604 . . . . . . . . . . . . . . 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 5652 . . . . . . . . . . . . . . . . . . 19  |-  ( j  =  b  ->  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( ( F `  b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) )
7069breq1d 3847 . . . . . . . . . . . . . . . . . 18  |-  ( j  =  b  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  ( ( F `  b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q  u ) )
7170cbvrexv 2591 . . . . . . . . . . . . . . . . 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 2604 . . . . . . . . . . . . . . 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 3622 . . . . . . . . . . . . . 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 2108 . . . . . . . . . . . . 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 476 . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . 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 6913 . . . . . . . . . . . . . 14  |-  ( ( S  e.  Q.  /\  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  e.  Q. )
8076, 78, 79syl2anc 403 . . . . . . . . . . . . 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 7215 . . . . . . . . . . . 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 3022 . . . . . . . . . . 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 3841 . . . . . . . . . . . . 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 2381 . . . . . . . . . . . 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 2769 . . . . . . . . . . 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 159 . . . . . . . . . 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 624 . . . . . . . . 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 7214 . . . . . . . . . . . 12  |-  ( ph  ->  L  e.  P. )
8988ad3antrrr 476 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  ->  L  e.  P. )
90 nqprlu 7085 . . . . . . . . . . . 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 7075 . . . . . . . . . . 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 403 . . . . . . . . . 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 7013 . . . . . . . . . . 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 7029 . . . . . . . . . . 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 277 . . . . . . . . . 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 411 . . . . . . . . 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 1285 . . . . . . . 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 501 . . . . . . . . 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 7182 . . . . . . . 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 145 . . . . . . 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 113 . . . . . 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 2489 . . . . 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 355 . . . 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 150 . . 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 3029 . 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 3068 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 102    <-> wb 103    \/ wo 664    /\ w3a 924    = wceq 1289    e. wcel 1438   {cab 2074   A.wral 2359   E.wrex 2360   {crab 2363    C_ wss 2997   <.cop 3444   class class class wbr 3837   -->wf 4998   ` cfv 5002  (class class class)co 5634   1stc1st 5891   2ndc2nd 5892   1oc1o 6156   [cec 6270   N.cnpi 6810    <N clti 6813    ~Q ceq 6817   Q.cnq 6818    +Q cplq 6820   *Qcrq 6822    <Q cltq 6823   P.cnp 6829    +P. cpp 6831
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-eprel 4107  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-1o 6163  df-2o 6164  df-oadd 6167  df-omul 6168  df-er 6272  df-ec 6274  df-qs 6278  df-ni 6842  df-pli 6843  df-mi 6844  df-lti 6845  df-plpq 6882  df-mpq 6883  df-enq 6885  df-nqqs 6886  df-plqqs 6887  df-mqqs 6888  df-1nqqs 6889  df-rq 6890  df-ltnqqs 6891  df-enq0 6962  df-nq0 6963  df-0nq0 6964  df-plq0 6965  df-mq0 6966  df-inp 7004  df-iplp 7006  df-iltp 7008
This theorem is referenced by:  caucvgprlem1  7217
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