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Theorem caucvgprprlemaddq 6864
Description: Lemma for caucvgprpr 6868. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 5-Jun-2021.)
Hypotheses
Ref Expression
caucvgprpr.f  |-  ( ph  ->  F : N. --> P. )
caucvgprpr.cau  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <P  ( ( F `
 k )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( F `  k
)  <P  ( ( F `
 n )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )
) ) )
caucvgprpr.bnd  |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )
caucvgprpr.lim  |-  L  = 
<. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  r ) } ,  { u  e.  Q.  |  E. r  e.  N.  ( ( F `
 r )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  u } ,  { q  |  u 
<Q  q } >. } >.
caucvgprprlemaddq.x  |-  ( ph  ->  X  e.  P. )
caucvgprprlemaddq.q  |-  ( ph  ->  Q  e.  P. )
caucvgprprlemaddq.ex  |-  ( ph  ->  E. r  e.  N.  ( X  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  r )  +P.  Q
) )
Assertion
Ref Expression
caucvgprprlemaddq  |-  ( ph  ->  X  <P  ( L  +P.  Q ) )
Distinct variable groups:    A, m    m, F    A, r, m    F, l, r, u, k, n   
k, L    Q, r    X, r    p, l, q, r, u    ph, r    k, p, q
Allowed substitution hints:    ph( u, k, m, n, q, p, l)    A( u, k, n, q, p, l)    Q( u, k, m, n, q, p, l)    F( q, p)    L( u, m, n, r, q, p, l)    X( u, k, m, n, q, p, l)

Proof of Theorem caucvgprprlemaddq
Dummy variables  b  f  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 caucvgprprlemaddq.ex . 2  |-  ( ph  ->  E. r  e.  N.  ( X  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  r )  +P.  Q
) )
2 nfv 1437 . . 3  |-  F/ r
ph
3 nfcv 2194 . . . 4  |-  F/_ r X
4 nfcv 2194 . . . 4  |-  F/_ r  <P
5 caucvgprpr.lim . . . . . 6  |-  L  = 
<. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  r ) } ,  { u  e.  Q.  |  E. r  e.  N.  ( ( F `
 r )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  u } ,  { q  |  u 
<Q  q } >. } >.
6 nfre1 2382 . . . . . . . 8  |-  F/ r E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  r )
7 nfcv 2194 . . . . . . . 8  |-  F/_ r Q.
86, 7nfrabxy 2507 . . . . . . 7  |-  F/_ r { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  r ) }
9 nfre1 2382 . . . . . . . 8  |-  F/ r E. r  e.  N.  ( ( F `  r )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  u } ,  {
q  |  u  <Q  q } >.
109, 7nfrabxy 2507 . . . . . . 7  |-  F/_ r { u  e.  Q.  |  E. r  e.  N.  ( ( F `  r )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  u } ,  {
q  |  u  <Q  q } >. }
118, 10nfop 3593 . . . . . 6  |-  F/_ r <. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  r ) } ,  { u  e.  Q.  |  E. r  e.  N.  ( ( F `
 r )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  u } ,  { q  |  u 
<Q  q } >. } >.
125, 11nfcxfr 2191 . . . . 5  |-  F/_ r L
13 nfcv 2194 . . . . 5  |-  F/_ r  +P.
14 nfcv 2194 . . . . 5  |-  F/_ r Q
1512, 13, 14nfov 5563 . . . 4  |-  F/_ r
( L  +P.  Q
)
163, 4, 15nfbr 3836 . . 3  |-  F/ r  X  <P  ( L  +P.  Q )
17 caucvgprpr.f . . . . . . . . . . . 12  |-  ( ph  ->  F : N. --> P. )
1817ad2antrr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  r )  +P.  Q
) ) )  /\  b  e.  N. )  ->  F : N. --> P. )
19 caucvgprpr.cau . . . . . . . . . . . 12  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <P  ( ( F `
 k )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( F `  k
)  <P  ( ( F `
 n )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )
) ) )
2019ad2antrr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  r )  +P.  Q
) ) )  /\  b  e.  N. )  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <P  ( ( F `
 k )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( F `  k
)  <P  ( ( F `
 n )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )
) ) )
21 simpr 107 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  r )  +P.  Q
) ) )  /\  b  e.  N. )  ->  b  e.  N. )
22 simplrl 495 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  r )  +P.  Q
) ) )  /\  b  e.  N. )  ->  r  e.  N. )
2318, 20, 21, 22caucvgprprlemnbj 6849 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  r )  +P.  Q
) ) )  /\  b  e.  N. )  ->  -.  ( ( ( F `  b )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >. )  <P  ( F `  r
) )
2418, 21ffvelrnd 5331 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  r )  +P.  Q
) ) )  /\  b  e.  N. )  ->  ( F `  b
)  e.  P. )
25 recnnpr 6704 . . . . . . . . . . . . . . 15  |-  ( b  e.  N.  ->  <. { l  |  l  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  u } >.  e.  P. )
2625adantl 266 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  r )  +P.  Q
) ) )  /\  b  e.  N. )  -> 
<. { l  |  l 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  u } >.  e.  P. )
27 addclpr 6693 . . . . . . . . . . . . . 14  |-  ( ( ( F `  b
)  e.  P.  /\  <. { l  |  l 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  u } >.  e.  P. )  ->  ( ( F `
 b )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  u } >. )  e.  P. )
2824, 26, 27syl2anc 397 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  r )  +P.  Q
) ) )  /\  b  e.  N. )  ->  ( ( F `  b )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  u } >. )  e.  P. )
29 recnnpr 6704 . . . . . . . . . . . . . 14  |-  ( r  e.  N.  ->  <. { l  |  l  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >.  e.  P. )
3022, 29syl 14 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  r )  +P.  Q
) ) )  /\  b  e.  N. )  -> 
<. { l  |  l 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >.  e.  P. )
31 caucvgprprlemaddq.q . . . . . . . . . . . . . 14  |-  ( ph  ->  Q  e.  P. )
3231ad2antrr 465 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  r )  +P.  Q
) ) )  /\  b  e.  N. )  ->  Q  e.  P. )
33 addassprg 6735 . . . . . . . . . . . . 13  |-  ( ( ( ( F `  b )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  u } >. )  e.  P.  /\  <. { l  |  l  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >.  e.  P.  /\  Q  e.  P. )  ->  ( ( ( ( F `  b )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  Q )  =  ( ( ( F `  b )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  ( <. { l  |  l  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  Q )
) )
3428, 30, 32, 33syl3anc 1146 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  r )  +P.  Q
) ) )  /\  b  e.  N. )  ->  ( ( ( ( F `  b )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  Q )  =  ( ( ( F `  b )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  ( <. { l  |  l  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  Q )
) )
3534breq1d 3802 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  r )  +P.  Q
) ) )  /\  b  e.  N. )  ->  ( ( ( ( ( F `  b
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  Q )  <P  (
( F `  r
)  +P.  Q )  <->  ( ( ( F `  b )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  ( <. { l  |  l  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  Q )
)  <P  ( ( F `
 r )  +P. 
Q ) ) )
36 ltaprg 6775 . . . . . . . . . . . . 13  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
f  <P  g  <->  ( h  +P.  f )  <P  (
h  +P.  g )
) )
3736adantl 266 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  r )  +P.  Q
) ) )  /\  b  e.  N. )  /\  ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )
)  ->  ( f  <P  g  <->  ( h  +P.  f )  <P  (
h  +P.  g )
) )
38 addclpr 6693 . . . . . . . . . . . . 13  |-  ( ( ( ( F `  b )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  u } >. )  e.  P.  /\  <. { l  |  l  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >.  e.  P. )  ->  ( ( ( F `  b )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >. )  e.  P. )
3928, 30, 38syl2anc 397 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  r )  +P.  Q
) ) )  /\  b  e.  N. )  ->  ( ( ( F `
 b )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >. )  e.  P. )
4018, 22ffvelrnd 5331 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  r )  +P.  Q
) ) )  /\  b  e.  N. )  ->  ( F `  r
)  e.  P. )
41 addcomprg 6734 . . . . . . . . . . . . 13  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
4241adantl 266 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  r )  +P.  Q
) ) )  /\  b  e.  N. )  /\  ( f  e.  P.  /\  g  e.  P. )
)  ->  ( f  +P.  g )  =  ( g  +P.  f ) )
4337, 39, 40, 32, 42caovord2d 5698 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  r )  +P.  Q
) ) )  /\  b  e.  N. )  ->  ( ( ( ( F `  b )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >. )  <P  ( F `  r
)  <->  ( ( ( ( F `  b
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  Q )  <P  (
( F `  r
)  +P.  Q )
) )
44 addcomprg 6734 . . . . . . . . . . . . . 14  |-  ( ( Q  e.  P.  /\  <. { l  |  l 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >.  e.  P. )  ->  ( Q  +P.  <. { l  |  l 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >. )  =  ( <. { l  |  l  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  Q ) )
4532, 30, 44syl2anc 397 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  r )  +P.  Q
) ) )  /\  b  e.  N. )  ->  ( Q  +P.  <. { l  |  l  <Q 
( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >. )  =  ( <. { l  |  l  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  Q ) )
4645oveq2d 5556 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  r )  +P.  Q
) ) )  /\  b  e.  N. )  ->  ( ( ( F `
 b )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  ( Q  +P.  <. { l  |  l  <Q 
( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >. )
)  =  ( ( ( F `  b
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  ( <. { l  |  l 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  Q
) ) )
4746breq1d 3802 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  r )  +P.  Q
) ) )  /\  b  e.  N. )  ->  ( ( ( ( F `  b )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  ( Q  +P.  <. { l  |  l  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >. ) )  <P 
( ( F `  r )  +P.  Q
)  <->  ( ( ( F `  b )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  ( <. { l  |  l 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  Q
) )  <P  (
( F `  r
)  +P.  Q )
) )
4835, 43, 473bitr4rd 214 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  r )  +P.  Q
) ) )  /\  b  e.  N. )  ->  ( ( ( ( F `  b )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  ( Q  +P.  <. { l  |  l  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >. ) )  <P 
( ( F `  r )  +P.  Q
)  <->  ( ( ( F `  b )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >. )  <P  ( F `  r
) ) )
4923, 48mtbird 608 . . . . . . . . 9  |-  ( ( ( ph  /\  (
r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  r )  +P.  Q
) ) )  /\  b  e.  N. )  ->  -.  ( ( ( F `  b )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  ( Q  +P.  <. { l  |  l  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >. ) )  <P 
( ( F `  r )  +P.  Q
) )
5049nrexdv 2429 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  (
( F `  r
)  +P.  Q )
) )  ->  -.  E. b  e.  N.  (
( ( F `  b )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  ( Q  +P.  <. { l  |  l  <Q 
( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >. )
)  <P  ( ( F `
 r )  +P. 
Q ) )
51 breq1 3795 . . . . . . . . . . . . . 14  |-  ( p  =  l  ->  (
p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <->  l  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) )
5251cbvabv 2177 . . . . . . . . . . . . 13  |-  { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) }  =  { l  |  l  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) }
53 breq2 3796 . . . . . . . . . . . . . 14  |-  ( q  =  u  ->  (
( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  u ) )
5453cbvabv 2177 . . . . . . . . . . . . 13  |-  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q }  =  {
u  |  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )  <Q  u }
5552, 54opeq12i 3582 . . . . . . . . . . . 12  |-  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >.  =  <. { l  |  l  <Q 
( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  u } >.
5655oveq2i 5551 . . . . . . . . . . 11  |-  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  =  ( ( F `  b
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  u } >. )
57 breq1 3795 . . . . . . . . . . . . . 14  |-  ( p  =  l  ->  (
p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <->  l  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) )
5857cbvabv 2177 . . . . . . . . . . . . 13  |-  { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) }  =  { l  |  l  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) }
59 breq2 3796 . . . . . . . . . . . . . 14  |-  ( q  =  u  ->  (
( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u ) )
6059cbvabv 2177 . . . . . . . . . . . . 13  |-  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q }  =  {
u  |  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )  <Q  u }
6158, 60opeq12i 3582 . . . . . . . . . . . 12  |-  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >.  =  <. { l  |  l  <Q 
( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >.
6261oveq2i 5551 . . . . . . . . . . 11  |-  ( Q  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  =  ( Q  +P.  <. { l  |  l  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >. )
6356, 62oveq12i 5552 . . . . . . . . . 10  |-  ( ( ( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  ( Q  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )
)  =  ( ( ( F `  b
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  ( Q  +P.  <. { l  |  l  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >. ) )
6463breq1i 3799 . . . . . . . . 9  |-  ( ( ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  ( Q  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )
)  <P  ( ( F `
 r )  +P. 
Q )  <->  ( (
( F `  b
)  +P.  <. { l  |  l  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  ( Q  +P.  <. { l  |  l  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >. ) )  <P 
( ( F `  r )  +P.  Q
) )
6564rexbii 2348 . . . . . . . 8  |-  ( E. b  e.  N.  (
( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  ( Q  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )
)  <P  ( ( F `
 r )  +P. 
Q )  <->  E. b  e.  N.  ( ( ( F `  b )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  u } >. )  +P.  ( Q  +P.  <. { l  |  l  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u } >. ) )  <P 
( ( F `  r )  +P.  Q
) )
6650, 65sylnibr 612 . . . . . . 7  |-  ( (
ph  /\  ( r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  (
( F `  r
)  +P.  Q )
) )  ->  -.  E. b  e.  N.  (
( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  ( Q  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )
)  <P  ( ( F `
 r )  +P. 
Q ) )
6717adantr 265 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  (
( F `  r
)  +P.  Q )
) )  ->  F : N. --> P. )
6819adantr 265 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  (
( F `  r
)  +P.  Q )
) )  ->  A. n  e.  N.  A. k  e. 
N.  ( n  <N  k  ->  ( ( F `
 n )  <P 
( ( F `  k )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( F `  k
)  <P  ( ( F `
 n )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )
) ) )
69 caucvgprpr.bnd . . . . . . . . 9  |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )
7069adantr 265 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  (
( F `  r
)  +P.  Q )
) )  ->  A. m  e.  N.  A  <P  ( F `  m )
)
7131adantr 265 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  (
( F `  r
)  +P.  Q )
) )  ->  Q  e.  P. )
72 simprl 491 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  (
( F `  r
)  +P.  Q )
) )  ->  r  e.  N. )
7367, 68, 70, 5, 71, 72caucvgprprlemexb 6863 . . . . . . 7  |-  ( (
ph  /\  ( r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  (
( F `  r
)  +P.  Q )
) )  ->  (
( ( L  +P.  Q )  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  r )  +P.  Q
)  ->  E. b  e.  N.  ( ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  ( Q  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) )  <P 
( ( F `  r )  +P.  Q
) ) )
7466, 73mtod 599 . . . . . 6  |-  ( (
ph  /\  ( r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  (
( F `  r
)  +P.  Q )
) )  ->  -.  ( ( L  +P.  Q )  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  r )  +P.  Q
) )
75 simprr 492 . . . . . . 7  |-  ( (
ph  /\  ( r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  (
( F `  r
)  +P.  Q )
) )  ->  ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  (
( F `  r
)  +P.  Q )
)
76 caucvgprprlemaddq.x . . . . . . . . . 10  |-  ( ph  ->  X  e.  P. )
7776adantr 265 . . . . . . . . 9  |-  ( (
ph  /\  ( r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  (
( F `  r
)  +P.  Q )
) )  ->  X  e.  P. )
78 recnnpr 6704 . . . . . . . . . 10  |-  ( r  e.  N.  ->  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >.  e.  P. )
7972, 78syl 14 . . . . . . . . 9  |-  ( (
ph  /\  ( r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  (
( F `  r
)  +P.  Q )
) )  ->  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >.  e.  P. )
80 addclpr 6693 . . . . . . . . 9  |-  ( ( X  e.  P.  /\  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >.  e. 
P. )  ->  ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  e.  P. )
8177, 79, 80syl2anc 397 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  (
( F `  r
)  +P.  Q )
) )  ->  ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  e.  P. )
8267, 72ffvelrnd 5331 . . . . . . . . 9  |-  ( (
ph  /\  ( r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  (
( F `  r
)  +P.  Q )
) )  ->  ( F `  r )  e.  P. )
83 addclpr 6693 . . . . . . . . 9  |-  ( ( ( F `  r
)  e.  P.  /\  Q  e.  P. )  ->  ( ( F `  r )  +P.  Q
)  e.  P. )
8482, 71, 83syl2anc 397 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  (
( F `  r
)  +P.  Q )
) )  ->  (
( F `  r
)  +P.  Q )  e.  P. )
8517, 19, 69, 5caucvgprprlemcl 6860 . . . . . . . . . . 11  |-  ( ph  ->  L  e.  P. )
8685adantr 265 . . . . . . . . . 10  |-  ( (
ph  /\  ( r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  (
( F `  r
)  +P.  Q )
) )  ->  L  e.  P. )
87 addclpr 6693 . . . . . . . . . 10  |-  ( ( L  e.  P.  /\  Q  e.  P. )  ->  ( L  +P.  Q
)  e.  P. )
8886, 71, 87syl2anc 397 . . . . . . . . 9  |-  ( (
ph  /\  ( r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  (
( F `  r
)  +P.  Q )
) )  ->  ( L  +P.  Q )  e. 
P. )
89 addclpr 6693 . . . . . . . . 9  |-  ( ( ( L  +P.  Q
)  e.  P.  /\  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >.  e. 
P. )  ->  (
( L  +P.  Q
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  e.  P. )
9088, 79, 89syl2anc 397 . . . . . . . 8  |-  ( (
ph  /\  ( r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  (
( F `  r
)  +P.  Q )
) )  ->  (
( L  +P.  Q
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  e.  P. )
91 ltsopr 6752 . . . . . . . . 9  |-  <P  Or  P.
92 sowlin 4085 . . . . . . . . 9  |-  ( ( 
<P  Or  P.  /\  (
( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  e.  P.  /\  ( ( F `  r )  +P.  Q )  e. 
P.  /\  ( ( L  +P.  Q )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  e.  P. ) )  ->  ( ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  (
( F `  r
)  +P.  Q )  ->  ( ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  ( ( L  +P.  Q )  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  \/  ( ( L  +P.  Q )  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  r )  +P.  Q
) ) ) )
9391, 92mpan 408 . . . . . . . 8  |-  ( ( ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  e.  P.  /\  ( ( F `  r )  +P.  Q )  e. 
P.  /\  ( ( L  +P.  Q )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  e.  P. )  -> 
( ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  ( ( F `  r )  +P.  Q
)  ->  ( ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  (
( L  +P.  Q
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  \/  ( ( L  +P.  Q )  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  r )  +P.  Q
) ) ) )
9481, 84, 90, 93syl3anc 1146 . . . . . . 7  |-  ( (
ph  /\  ( r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  (
( F `  r
)  +P.  Q )
) )  ->  (
( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  r )  +P.  Q
)  ->  ( ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  (
( L  +P.  Q
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  \/  ( ( L  +P.  Q )  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  r )  +P.  Q
) ) ) )
9575, 94mpd 13 . . . . . 6  |-  ( (
ph  /\  ( r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  (
( F `  r
)  +P.  Q )
) )  ->  (
( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( L  +P.  Q )  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  \/  ( ( L  +P.  Q )  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  r )  +P.  Q
) ) )
9674, 95ecased 1255 . . . . 5  |-  ( (
ph  /\  ( r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  (
( F `  r
)  +P.  Q )
) )  ->  ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  (
( L  +P.  Q
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )
)
9736adantl 266 . . . . . 6  |-  ( ( ( ph  /\  (
r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  r )  +P.  Q
) ) )  /\  ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )
)  ->  ( f  <P  g  <->  ( h  +P.  f )  <P  (
h  +P.  g )
) )
9841adantl 266 . . . . . 6  |-  ( ( ( ph  /\  (
r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  r )  +P.  Q
) ) )  /\  ( f  e.  P.  /\  g  e.  P. )
)  ->  ( f  +P.  g )  =  ( g  +P.  f ) )
9997, 77, 88, 79, 98caovord2d 5698 . . . . 5  |-  ( (
ph  /\  ( r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  (
( F `  r
)  +P.  Q )
) )  ->  ( X  <P  ( L  +P.  Q )  <->  ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  ( ( L  +P.  Q )  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )
) )
10096, 99mpbird 160 . . . 4  |-  ( (
ph  /\  ( r  e.  N.  /\  ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  (
( F `  r
)  +P.  Q )
) )  ->  X  <P  ( L  +P.  Q
) )
101100exp32 351 . . 3  |-  ( ph  ->  ( r  e.  N.  ->  ( ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  ( ( F `  r )  +P.  Q
)  ->  X  <P  ( L  +P.  Q ) ) ) )
1022, 16, 101rexlimd 2447 . 2  |-  ( ph  ->  ( E. r  e. 
N.  ( X  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  ( ( F `  r )  +P.  Q
)  ->  X  <P  ( L  +P.  Q ) ) )
1031, 102mpd 13 1  |-  ( ph  ->  X  <P  ( L  +P.  Q ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    \/ wo 639    /\ w3a 896    = wceq 1259    e. wcel 1409   {cab 2042   A.wral 2323   E.wrex 2324   {crab 2327   <.cop 3406   class class class wbr 3792    Or wor 4060   -->wf 4926   ` cfv 4930  (class class class)co 5540   1oc1o 6025   [cec 6135   N.cnpi 6428    <N clti 6431    ~Q ceq 6435   Q.cnq 6436    +Q cplq 6438   *Qcrq 6440    <Q cltq 6441   P.cnp 6447    +P. cpp 6449    <P cltp 6451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-iplp 6624  df-iltp 6626
This theorem is referenced by:  caucvgprprlem1  6865
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