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Theorem caucvgprlemladdrl 7665
Description: Lemma for caucvgpr 7669. 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 3776 . . . . . . . . 9  |-  ( j  =  a  ->  <. j ,  1o >.  =  <. a ,  1o >. )
21eceq1d 6565 . . . . . . . 8  |-  ( j  =  a  ->  [ <. j ,  1o >. ]  ~Q  =  [ <. a ,  1o >. ]  ~Q  )
32fveq2d 5515 . . . . . . 7  |-  ( j  =  a  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )
43oveq2d 5885 . . . . . 6  |-  ( j  =  a  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( l  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )
5 fveq2 5511 . . . . . . 7  |-  ( j  =  a  ->  ( F `  j )  =  ( F `  a ) )
65oveq1d 5884 . . . . . 6  |-  ( j  =  a  ->  (
( F `  j
)  +Q  S )  =  ( ( F `
 a )  +Q  S ) )
74, 6breq12d 4013 . . . . 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 2704 . . . 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 2722 . 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 5876 . . . . . . 7  |-  ( l  =  r  ->  (
l  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
)  =  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )
1211breq1d 4010 . . . . . 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 2478 . . . . 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 2893 . . . 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 494 . . . . . . . . . . . . . 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 494 . . . . . . . . . . . . . 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 110 . . . . . . . . . . . . . 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 534 . . . . . . . . . . . . . 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 7654 . . . . . . . . . . . . 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 492 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  ->  F : N. --> Q. )
2322ffvelcdmda 5647 . . . . . . . . . . . . . . . . 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 7409 . . . . . . . . . . . . . . . . . 18  |-  ( b  e.  N.  ->  [ <. b ,  1o >. ]  ~Q  e.  Q. )
25 recclnq 7379 . . . . . . . . . . . . . . . . . 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 7362 . . . . . . . . . . . . . . . . 17  |-  ( ( ( F `  b
)  e.  Q.  /\  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( ( F `  b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  e.  Q. )
2823, 26, 27syl2anc 411 . . . . . . . . . . . . . . . 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 7409 . . . . . . . . . . . . . . . . 17  |-  ( a  e.  N.  ->  [ <. a ,  1o >. ]  ~Q  e.  Q. )
30 recclnq 7379 . . . . . . . . . . . . . . . . 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 494 . . . . . . . . . . . . . . . 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 7369 . . . . . . . . . . . . . . . 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 1238 . . . . . . . . . . . . . . 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 4010 . . . . . . . . . . . . . 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 7387 . . . . . . . . . . . . . . . 16  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
3837adantl 277 . . . . . . . . . . . . . . 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 7362 . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . . 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, 20ffvelcdmd 5648 . . . . . . . . . . . . . . 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 7368 . . . . . . . . . . . . . . . 16  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
4342adantl 277 . . . . . . . . . . . . . . 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 6038 . . . . . . . . . . . . . 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 7368 . . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . . . 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 5885 . . . . . . . . . . . . . . 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 4010 . . . . . . . . . . . . . 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 221 . . . . . . . . . . . . 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 673 . . . . . . . . . . . 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 2570 . . . . . . . . . . 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 931 . . . . . . . . . 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 492 . . . . . . . . . . . . 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 5511 . . . . . . . . . . . . . . . . 17  |-  ( j  =  b  ->  ( F `  j )  =  ( F `  b ) )
5655breq2d 4012 . . . . . . . . . . . . . . . 16  |-  ( j  =  b  ->  ( A  <Q  ( F `  j )  <->  A  <Q  ( F `  b ) ) )
5756cbvralv 2703 . . . . . . . . . . . . . . 15  |-  ( A. j  e.  N.  A  <Q  ( F `  j
)  <->  A. b  e.  N.  A  <Q  ( F `  b ) )
5854, 57sylib 122 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. b  e.  N.  A  <Q  ( F `  b ) )
5958ad3antrrr 492 . . . . . . . . . . . . 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 3776 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( j  =  b  ->  <. j ,  1o >.  =  <. b ,  1o >. )
6261eceq1d 6565 . . . . . . . . . . . . . . . . . . . . 21  |-  ( j  =  b  ->  [ <. j ,  1o >. ]  ~Q  =  [ <. b ,  1o >. ]  ~Q  )
6362fveq2d 5515 . . . . . . . . . . . . . . . . . . . 20  |-  ( j  =  b  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )
6463oveq2d 5885 . . . . . . . . . . . . . . . . . . 19  |-  ( j  =  b  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( l  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) )
6564, 55breq12d 4013 . . . . . . . . . . . . . . . . . 18  |-  ( j  =  b  ->  (
( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( l  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q 
( F `  b
) ) )
6665cbvrexv 2704 . . . . . . . . . . . . . . . . 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 2722 . . . . . . . . . . . . . . 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 5887 . . . . . . . . . . . . . . . . . . 19  |-  ( j  =  b  ->  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( ( F `  b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) )
7069breq1d 4010 . . . . . . . . . . . . . . . . . 18  |-  ( j  =  b  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  ( ( F `  b )  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q  u ) )
7170cbvrexv 2704 . . . . . . . . . . . . . . . . 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 2722 . . . . . . . . . . . . . . 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 3781 . . . . . . . . . . . . . 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 2198 . . . . . . . . . . . . 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 492 . . . . . . . . . . . . . 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 489 . . . . . . . . . . . . . 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 7362 . . . . . . . . . . . . . 14  |-  ( ( S  e.  Q.  /\  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( S  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  e.  Q. )
8076, 78, 79syl2anc 411 . . . . . . . . . . . . 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 7664 . . . . . . . . . . . 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 3154 . . . . . . . . . . 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 4004 . . . . . . . . . . . . 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 2478 . . . . . . . . . . . 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 2893 . . . . . . . . . . 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 161 . . . . . . . . . 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 663 . . . . . . . . 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 7663 . . . . . . . . . . . 12  |-  ( ph  ->  L  e.  P. )
8988ad3antrrr 492 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  a  e.  N. )  /\  ( r  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  a )  +Q  S
) )  ->  L  e.  P. )
90 nqprlu 7534 . . . . . . . . . . . 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 7524 . . . . . . . . . . 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 411 . . . . . . . . . 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 7462 . . . . . . . . . . 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 7478 . . . . . . . . . . 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 283 . . . . . . . . . 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 420 . . . . . . . . 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 1349 . . . . . . . 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 534 . . . . . . . . 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 7631 . . . . . . . 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 147 . . . . . . 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 115 . . . . . 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 2594 . . . . 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 363 . . . 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, 104biimtrid 152 . . 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 3161 . 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, 106eqsstrid 3201 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 104    <-> wb 105    \/ wo 708    /\ w3a 978    = wceq 1353    e. wcel 2148   {cab 2163   A.wral 2455   E.wrex 2456   {crab 2459    C_ wss 3129   <.cop 3594   class class class wbr 4000   -->wf 5208   ` cfv 5212  (class class class)co 5869   1stc1st 6133   2ndc2nd 6134   1oc1o 6404   [cec 6527   N.cnpi 7259    <N clti 7262    ~Q ceq 7266   Q.cnq 7267    +Q cplq 7269   *Qcrq 7271    <Q cltq 7272   P.cnp 7278    +P. cpp 7280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-2o 6412  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7291  df-pli 7292  df-mi 7293  df-lti 7294  df-plpq 7331  df-mpq 7332  df-enq 7334  df-nqqs 7335  df-plqqs 7336  df-mqqs 7337  df-1nqqs 7338  df-rq 7339  df-ltnqqs 7340  df-enq0 7411  df-nq0 7412  df-0nq0 7413  df-plq0 7414  df-mq0 7415  df-inp 7453  df-iplp 7455  df-iltp 7457
This theorem is referenced by:  caucvgprlem1  7666
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