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Theorem caucvgprprlemaddq 7246
Description: Lemma for caucvgprpr 7250. 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 1466 . . 3  |-  F/ r
ph
3 nfcv 2228 . . . 4  |-  F/_ r X
4 nfcv 2228 . . . 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 2419 . . . . . . . 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 2228 . . . . . . . 8  |-  F/_ r Q.
86, 7nfrabxy 2547 . . . . . . 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 2419 . . . . . . . 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 2547 . . . . . . 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 3633 . . . . . 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 2225 . . . . 5  |-  F/_ r L
13 nfcv 2228 . . . . 5  |-  F/_ r  +P.
14 nfcv 2228 . . . . 5  |-  F/_ r Q
1512, 13, 14nfov 5661 . . . 4  |-  F/_ r
( L  +P.  Q
)
163, 4, 15nfbr 3881 . . 3  |-  F/ r  X  <P  ( L  +P.  Q )
17 caucvgprpr.f . . . . . . . . . . . 12  |-  ( ph  ->  F : N. --> P. )
1817ad2antrr 472 . . . . . . . . . . 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 472 . . . . . . . . . . 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 108 . . . . . . . . . . 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 502 . . . . . . . . . . 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 7231 . . . . . . . . . 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 5419 . . . . . . . . . . . . . 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 7086 . . . . . . . . . . . . . . 15  |-  ( b  e.  N.  ->  <. { l  |  l  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  u } >.  e.  P. )
2625adantl 271 . . . . . . . . . . . . . 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 7075 . . . . . . . . . . . . . 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 403 . . . . . . . . . . . . 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 7086 . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . 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 7117 . . . . . . . . . . . . 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 1174 . . . . . . . . . . . 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 3847 . . . . . . . . . . 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 7157 . . . . . . . . . . . . 13  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
f  <P  g  <->  ( h  +P.  f )  <P  (
h  +P.  g )
) )
3736adantl 271 . . . . . . . . . . . 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 7075 . . . . . . . . . . . . 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 403 . . . . . . . . . . . 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 5419 . . . . . . . . . . . 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 7116 . . . . . . . . . . . . 13  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
4241adantl 271 . . . . . . . . . . . 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 5796 . . . . . . . . . . 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 7116 . . . . . . . . . . . . . 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 403 . . . . . . . . . . . . 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 5650 . . . . . . . . . . . 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 3847 . . . . . . . . . . 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 219 . . . . . . . . . 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 633 . . . . . . . . 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 2466 . . . . . . . 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 3840 . . . . . . . . . . . . . 14  |-  ( p  =  l  ->  (
p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <->  l  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) )
5251cbvabv 2211 . . . . . . . . . . . . 13  |-  { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) }  =  { l  |  l  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) }
53 breq2 3841 . . . . . . . . . . . . . 14  |-  ( q  =  u  ->  (
( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  u ) )
5453cbvabv 2211 . . . . . . . . . . . . 13  |-  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q }  =  {
u  |  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )  <Q  u }
5552, 54opeq12i 3622 . . . . . . . . . . . 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 5645 . . . . . . . . . . 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 3840 . . . . . . . . . . . . . 14  |-  ( p  =  l  ->  (
p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <->  l  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) )
5857cbvabv 2211 . . . . . . . . . . . . 13  |-  { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) }  =  { l  |  l  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) }
59 breq2 3841 . . . . . . . . . . . . . 14  |-  ( q  =  u  ->  (
( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  u ) )
6059cbvabv 2211 . . . . . . . . . . . . 13  |-  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q }  =  {
u  |  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )  <Q  u }
6158, 60opeq12i 3622 . . . . . . . . . . . 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 5645 . . . . . . . . . . 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 5646 . . . . . . . . . 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 3844 . . . . . . . . 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 2385 . . . . . . . 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 637 . . . . . . 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 270 . . . . . . . 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 270 . . . . . . . 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 270 . . . . . . . 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 270 . . . . . . . 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 498 . . . . . . . 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 7245 . . . . . . 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 624 . . . . . 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 499 . . . . . . 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 270 . . . . . . . . 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 7086 . . . . . . . . . 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 7075 . . . . . . . . 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 403 . . . . . . . 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 5419 . . . . . . . . 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 7075 . . . . . . . . 9  |-  ( ( ( F `  r
)  e.  P.  /\  Q  e.  P. )  ->  ( ( F `  r )  +P.  Q
)  e.  P. )
8482, 71, 83syl2anc 403 . . . . . . . 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 7242 . . . . . . . . . . 11  |-  ( ph  ->  L  e.  P. )
8685adantr 270 . . . . . . . . . 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 7075 . . . . . . . . . 10  |-  ( ( L  e.  P.  /\  Q  e.  P. )  ->  ( L  +P.  Q
)  e.  P. )
8886, 71, 87syl2anc 403 . . . . . . . . 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 7075 . . . . . . . . 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 403 . . . . . . . 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 7134 . . . . . . . . 9  |-  <P  Or  P.
92 sowlin 4138 . . . . . . . . 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 415 . . . . . . . 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 1174 . . . . . . 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 1285 . . . . 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 271 . . . . . 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 271 . . . . . 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 5796 . . . . 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 165 . . . 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 357 . . 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 2486 . 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 102    <-> wb 103    \/ wo 664    /\ w3a 924    = wceq 1289    e. wcel 1438   {cab 2074   A.wral 2359   E.wrex 2360   {crab 2363   <.cop 3444   class class class wbr 3837    Or wor 4113   -->wf 4998   ` cfv 5002  (class class class)co 5634   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    <P cltp 6833
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:  caucvgprprlem1  7247
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