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Theorem caucvgprprlemloc 6782
Description: Lemma for caucvgprpr 6791. The putative limit is located. (Contributed by Jim Kingdon, 21-Dec-2020.)
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 } >. } >.
Assertion
Ref Expression
caucvgprprlemloc  |-  ( ph  ->  A. s  e.  Q.  A. t  e.  Q.  (
s  <Q  t  ->  (
s  e.  ( 1st `  L )  \/  t  e.  ( 2nd `  L
) ) ) )
Distinct variable groups:    A, m    m, F    F, l, r    u, F, r    q, p, s, t    ph, s, t    p, l, q, s, t, r   
u, p, q, s, t
Allowed substitution hints:    ph( u, k, m, n, r, q, p, l)    A( u, t, k, n, s, r, q, p, l)    F( t, k, n, s, q, p)    L( u, t, k, m, n, s, r, q, p, l)

Proof of Theorem caucvgprprlemloc
Dummy variables  a  b  f  g  h  c  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltexnqi 6488 . . . . 5  |-  ( s 
<Q  t  ->  E. y  e.  Q.  ( s  +Q  y )  =  t )
21adantl 262 . . . 4  |-  ( ( ( ph  /\  (
s  e.  Q.  /\  t  e.  Q. )
)  /\  s  <Q  t )  ->  E. y  e.  Q.  ( s  +Q  y )  =  t )
3 subhalfnqq 6493 . . . . . 6  |-  ( y  e.  Q.  ->  E. x  e.  Q.  ( x  +Q  x )  <Q  y
)
43ad2antrl 459 . . . . 5  |-  ( ( ( ( ph  /\  ( s  e.  Q.  /\  t  e.  Q. )
)  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  t ) )  ->  E. x  e.  Q.  ( x  +Q  x
)  <Q  y )
5 archrecnq 6742 . . . . . . 7  |-  ( x  e.  Q.  ->  E. c  e.  N.  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x )
65ad2antrl 459 . . . . . 6  |-  ( ( ( ( ( ph  /\  ( s  e.  Q.  /\  t  e.  Q. )
)  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  t ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  ->  E. c  e.  N.  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x )
7 simpllr 486 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  ( s  e.  Q.  /\  t  e.  Q. )
)  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  t ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  ->  s  <Q  t )
87adantr 261 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  t  e. 
Q. ) )  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  t ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x
) )  ->  s  <Q  t )
9 simplrl 487 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  ( s  e.  Q.  /\  t  e.  Q. )
)  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  t ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  ->  y  e.  Q. )
109adantr 261 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  t  e. 
Q. ) )  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  t ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x
) )  ->  y  e.  Q. )
11 simplrr 488 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  ( s  e.  Q.  /\  t  e.  Q. )
)  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  t ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  ->  (
s  +Q  y )  =  t )
1211adantr 261 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  t  e. 
Q. ) )  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  t ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x
) )  ->  (
s  +Q  y )  =  t )
13 simplrl 487 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  t  e. 
Q. ) )  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  t ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x
) )  ->  x  e.  Q. )
14 simplrr 488 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  t  e. 
Q. ) )  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  t ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x
) )  ->  (
x  +Q  x ) 
<Q  y )
15 simprl 483 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  t  e. 
Q. ) )  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  t ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x
) )  ->  c  e.  N. )
16 simprr 484 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  t  e. 
Q. ) )  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  t ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x
) )  ->  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x )
178, 10, 12, 13, 14, 15, 16caucvgprprlemloccalc 6763 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  t  e. 
Q. ) )  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  t ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x
) )  ->  ( <. { p  |  p 
<Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >. )
18 simplrl 487 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
s  e.  Q.  /\  t  e.  Q. )
)  /\  s  <Q  t )  ->  s  e.  Q. )
1918ad3antrrr 461 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  t  e. 
Q. ) )  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  t ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x
) )  ->  s  e.  Q. )
20 nnnq 6501 . . . . . . . . . . . . . 14  |-  ( c  e.  N.  ->  [ <. c ,  1o >. ]  ~Q  e.  Q. )
2120ad2antrl 459 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  t  e. 
Q. ) )  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  t ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x
) )  ->  [ <. c ,  1o >. ]  ~Q  e.  Q. )
22 recclnq 6471 . . . . . . . . . . . . 13  |-  ( [
<. c ,  1o >. ]  ~Q  e.  Q.  ->  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  e.  Q. )
2321, 22syl 14 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  t  e. 
Q. ) )  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  t ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x
) )  ->  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  e.  Q. )
24 addclnq 6454 . . . . . . . . . . . 12  |-  ( ( s  e.  Q.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  )
)  e.  Q. )
2519, 23, 24syl2anc 391 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  t  e. 
Q. ) )  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  t ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x
) )  ->  (
s  +Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  )
)  e.  Q. )
26 nqprlu 6626 . . . . . . . . . . 11  |-  ( ( s  +Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  )
)  e.  Q.  ->  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  e.  P. )
2725, 26syl 14 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  t  e. 
Q. ) )  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  t ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x
) )  ->  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )  <Q 
q } >.  e.  P. )
28 nqprlu 6626 . . . . . . . . . . 11  |-  ( ( *Q `  [ <. c ,  1o >. ]  ~Q  )  e.  Q.  ->  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  q } >.  e. 
P. )
2923, 28syl 14 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  t  e. 
Q. ) )  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  t ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x
) )  ->  <. { p  |  p  <Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >.  e.  P. )
30 addclpr 6616 . . . . . . . . . 10  |-  ( (
<. { p  |  p 
<Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  e.  P.  /\ 
<. { p  |  p 
<Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  q } >.  e. 
P. )  ->  ( <. { p  |  p 
<Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  e.  P. )
3127, 29, 30syl2anc 391 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  t  e. 
Q. ) )  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  t ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x
) )  ->  ( <. { p  |  p 
<Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  e.  P. )
32 simplrr 488 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
s  e.  Q.  /\  t  e.  Q. )
)  /\  s  <Q  t )  ->  t  e.  Q. )
3332ad3antrrr 461 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  t  e. 
Q. ) )  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  t ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x
) )  ->  t  e.  Q. )
34 nqprlu 6626 . . . . . . . . . 10  |-  ( t  e.  Q.  ->  <. { p  |  p  <Q  t } ,  { q  |  t  <Q  q } >.  e.  P. )
3533, 34syl 14 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  t  e. 
Q. ) )  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  t ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x
) )  ->  <. { p  |  p  <Q  t } ,  { q  |  t  <Q  q } >.  e.  P. )
36 caucvgprpr.f . . . . . . . . . . . 12  |-  ( ph  ->  F : N. --> P. )
3736ad5antr 465 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  t  e. 
Q. ) )  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  t ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x
) )  ->  F : N. --> P. )
3837, 15ffvelrnd 5290 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  t  e. 
Q. ) )  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  t ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x
) )  ->  ( F `  c )  e.  P. )
39 ltrelnq 6444 . . . . . . . . . . . . . 14  |-  <Q  C_  ( Q.  X.  Q. )
4039brel 4379 . . . . . . . . . . . . 13  |-  ( ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x  ->  (
( *Q `  [ <. c ,  1o >. ]  ~Q  )  e.  Q.  /\  x  e.  Q. )
)
4140simpld 105 . . . . . . . . . . . 12  |-  ( ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x  ->  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  e.  Q. )
4241ad2antll 460 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  t  e. 
Q. ) )  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  t ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x
) )  ->  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  e.  Q. )
4342, 28syl 14 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  t  e. 
Q. ) )  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  t ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x
) )  ->  <. { p  |  p  <Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >.  e.  P. )
44 addclpr 6616 . . . . . . . . . 10  |-  ( ( ( F `  c
)  e.  P.  /\  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  q } >.  e. 
P. )  ->  (
( F `  c
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  e.  P. )
4538, 43, 44syl2anc 391 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  t  e. 
Q. ) )  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  t ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x
) )  ->  (
( F `  c
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  e.  P. )
46 ltsopr 6675 . . . . . . . . . 10  |-  <P  Or  P.
47 sowlin 4054 . . . . . . . . . 10  |-  ( ( 
<P  Or  P.  /\  (
( <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  )
)  <Q  q } >.  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  q } >. )  e.  P.  /\  <. { p  |  p  <Q  t } ,  { q  |  t  <Q  q } >.  e.  P.  /\  ( ( F `  c )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  e.  P. ) )  -> 
( ( <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )  <Q 
q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >.  ->  (
( <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  )
)  <Q  q } >.  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  ( ( F `  c )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  \/  ( ( F `  c )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >. )
) )
4846, 47mpan 400 . . . . . . . . 9  |-  ( ( ( <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  )
)  <Q  q } >.  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  q } >. )  e.  P.  /\  <. { p  |  p  <Q  t } ,  { q  |  t  <Q  q } >.  e.  P.  /\  ( ( F `  c )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  e.  P. )  ->  (
( <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  )
)  <Q  q } >.  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  t } ,  { q  |  t 
<Q  q } >.  ->  (
( <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  )
)  <Q  q } >.  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  ( ( F `  c )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  \/  ( ( F `  c )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >. )
) )
4931, 35, 45, 48syl3anc 1135 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  t  e. 
Q. ) )  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  t ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x
) )  ->  (
( <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  )
)  <Q  q } >.  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  t } ,  { q  |  t 
<Q  q } >.  ->  (
( <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  )
)  <Q  q } >.  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  ( ( F `  c )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  \/  ( ( F `  c )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >. )
) )
5017, 49mpd 13 . . . . . . 7  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  t  e. 
Q. ) )  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  t ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x
) )  ->  (
( <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  )
)  <Q  q } >.  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  ( ( F `  c )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  \/  ( ( F `  c )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >. )
)
5119adantr 261 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ph  /\  (
s  e.  Q.  /\  t  e.  Q. )
)  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  t ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  /\  (
c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x ) )  /\  ( <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )  <Q 
q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  c )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )
)  ->  s  e.  Q. )
52 simplrl 487 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ph  /\  (
s  e.  Q.  /\  t  e.  Q. )
)  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  t ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  /\  (
c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x ) )  /\  ( <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )  <Q 
q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  c )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )
)  ->  c  e.  N. )
53 simpr 103 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ph  /\  (
s  e.  Q.  /\  t  e.  Q. )
)  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  t ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  /\  (
c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x ) )  /\  ( <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )  <Q 
q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  c )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )
)  ->  ( <. { p  |  p  <Q  ( s  +Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  c )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )
)
54 ltaprg 6698 . . . . . . . . . . . . . 14  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
f  <P  g  <->  ( h  +P.  f )  <P  (
h  +P.  g )
) )
5554adantl 262 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ph  /\  ( s  e.  Q.  /\  t  e.  Q. )
)  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  t ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  /\  (
c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x ) )  /\  ( <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )  <Q 
q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  c )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )
)  /\  ( f  e.  P.  /\  g  e. 
P.  /\  h  e.  P. ) )  ->  (
f  <P  g  <->  ( h  +P.  f )  <P  (
h  +P.  g )
) )
5642adantr 261 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ph  /\  (
s  e.  Q.  /\  t  e.  Q. )
)  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  t ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  /\  (
c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x ) )  /\  ( <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )  <Q 
q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  c )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )
)  ->  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  e. 
Q. )
5751, 56, 24syl2anc 391 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ph  /\  (
s  e.  Q.  /\  t  e.  Q. )
)  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  t ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  /\  (
c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x ) )  /\  ( <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )  <Q 
q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  c )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )
)  ->  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )  e. 
Q. )
5857, 26syl 14 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ph  /\  (
s  e.  Q.  /\  t  e.  Q. )
)  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  t ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  /\  (
c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x ) )  /\  ( <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )  <Q 
q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  c )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )
)  ->  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )  <Q 
q } >.  e.  P. )
5938adantr 261 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ph  /\  (
s  e.  Q.  /\  t  e.  Q. )
)  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  t ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  /\  (
c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x ) )  /\  ( <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )  <Q 
q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  c )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )
)  ->  ( F `  c )  e.  P. )
6056, 28syl 14 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ph  /\  (
s  e.  Q.  /\  t  e.  Q. )
)  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  t ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  /\  (
c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x ) )  /\  ( <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )  <Q 
q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  c )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )
)  ->  <. { p  |  p  <Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >.  e.  P. )
61 addcomprg 6657 . . . . . . . . . . . . . 14  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
6261adantl 262 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ph  /\  ( s  e.  Q.  /\  t  e.  Q. )
)  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  t ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  /\  (
c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x ) )  /\  ( <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )  <Q 
q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  c )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )
)  /\  ( f  e.  P.  /\  g  e. 
P. ) )  -> 
( f  +P.  g
)  =  ( g  +P.  f ) )
6355, 58, 59, 60, 62caovord2d 5657 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ph  /\  (
s  e.  Q.  /\  t  e.  Q. )
)  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  t ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  /\  (
c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x ) )  /\  ( <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )  <Q 
q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  c )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )
)  ->  ( <. { p  |  p  <Q  ( s  +Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  c )  <->  (
<. { p  |  p 
<Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  c )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )
) )
6453, 63mpbird 156 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ph  /\  (
s  e.  Q.  /\  t  e.  Q. )
)  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  t ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  /\  (
c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x ) )  /\  ( <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )  <Q 
q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  c )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )
)  ->  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )  <Q 
q } >.  <P  ( F `  c )
)
65 opeq1 3546 . . . . . . . . . . . . . . . . . . 19  |-  ( a  =  c  ->  <. a ,  1o >.  =  <. c ,  1o >. )
6665eceq1d 6129 . . . . . . . . . . . . . . . . . 18  |-  ( a  =  c  ->  [ <. a ,  1o >. ]  ~Q  =  [ <. c ,  1o >. ]  ~Q  )
6766fveq2d 5169 . . . . . . . . . . . . . . . . 17  |-  ( a  =  c  ->  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )
6867oveq2d 5515 . . . . . . . . . . . . . . . 16  |-  ( a  =  c  ->  (
s  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) )
6968breq2d 3773 . . . . . . . . . . . . . . 15  |-  ( a  =  c  ->  (
p  <Q  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <->  p  <Q  ( s  +Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  )
) ) )
7069abbidv 2155 . . . . . . . . . . . . . 14  |-  ( a  =  c  ->  { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) }  =  { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) } )
7168breq1d 3771 . . . . . . . . . . . . . . 15  |-  ( a  =  c  ->  (
( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  <Q  q  <->  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )  <Q 
q ) )
7271abbidv 2155 . . . . . . . . . . . . . 14  |-  ( a  =  c  ->  { q  |  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
q }  =  {
q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) 
<Q  q } )
7370, 72opeq12d 3554 . . . . . . . . . . . . 13  |-  ( a  =  c  ->  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
q } >.  =  <. { p  |  p  <Q  ( s  +Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) 
<Q  q } >. )
74 fveq2 5165 . . . . . . . . . . . . 13  |-  ( a  =  c  ->  ( F `  a )  =  ( F `  c ) )
7573, 74breq12d 3774 . . . . . . . . . . . 12  |-  ( a  =  c  ->  ( <. { p  |  p 
<Q  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  a )  <->  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  c )
) )
7675rspcev 2653 . . . . . . . . . . 11  |-  ( ( c  e.  N.  /\  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  c )
)  ->  E. a  e.  N.  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P 
( F `  a
) )
7752, 64, 76syl2anc 391 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ph  /\  (
s  e.  Q.  /\  t  e.  Q. )
)  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  t ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  /\  (
c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x ) )  /\  ( <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )  <Q 
q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  c )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )
)  ->  E. a  e.  N.  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P 
( F `  a
) )
78 caucvgprpr.lim . . . . . . . . . . 11  |-  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 } >. } >.
7978caucvgprprlemell 6764 . . . . . . . . . 10  |-  ( s  e.  ( 1st `  L
)  <->  ( s  e. 
Q.  /\  E. a  e.  N.  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P 
( F `  a
) ) )
8051, 77, 79sylanbrc 394 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ph  /\  (
s  e.  Q.  /\  t  e.  Q. )
)  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  t ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  /\  (
c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x ) )  /\  ( <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )  <Q 
q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  c )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )
)  ->  s  e.  ( 1st `  L ) )
8180ex 108 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  t  e. 
Q. ) )  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  t ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x
) )  ->  (
( <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  )
)  <Q  q } >.  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  ( ( F `  c )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  ->  s  e.  ( 1st `  L ) ) )
8233adantr 261 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ph  /\  (
s  e.  Q.  /\  t  e.  Q. )
)  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  t ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  /\  (
c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x ) )  /\  ( ( F `
 c )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  t } ,  { q  |  t 
<Q  q } >. )  ->  t  e.  Q. )
83 fveq2 5165 . . . . . . . . . . . . . 14  |-  ( b  =  c  ->  ( F `  b )  =  ( F `  c ) )
84 opeq1 3546 . . . . . . . . . . . . . . . . . . 19  |-  ( b  =  c  ->  <. b ,  1o >.  =  <. c ,  1o >. )
8584eceq1d 6129 . . . . . . . . . . . . . . . . . 18  |-  ( b  =  c  ->  [ <. b ,  1o >. ]  ~Q  =  [ <. c ,  1o >. ]  ~Q  )
8685fveq2d 5169 . . . . . . . . . . . . . . . . 17  |-  ( b  =  c  ->  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )
8786breq2d 3773 . . . . . . . . . . . . . . . 16  |-  ( b  =  c  ->  (
p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <->  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) )
8887abbidv 2155 . . . . . . . . . . . . . . 15  |-  ( b  =  c  ->  { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) }  =  { p  |  p  <Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  ) } )
8986breq1d 3771 . . . . . . . . . . . . . . . 16  |-  ( b  =  c  ->  (
( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  q ) )
9089abbidv 2155 . . . . . . . . . . . . . . 15  |-  ( b  =  c  ->  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q }  =  {
q  |  ( *Q
`  [ <. c ,  1o >. ]  ~Q  )  <Q  q } )
9188, 90opeq12d 3554 . . . . . . . . . . . . . 14  |-  ( b  =  c  ->  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >.  =  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )
9283, 91oveq12d 5517 . . . . . . . . . . . . 13  |-  ( b  =  c  ->  (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  =  ( ( F `
 c )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  q } >. ) )
9392breq1d 3771 . . . . . . . . . . . 12  |-  ( b  =  c  ->  (
( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >.  <->  ( ( F `  c )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  t } ,  { q  |  t  <Q  q } >. ) )
9493rspcev 2653 . . . . . . . . . . 11  |-  ( ( c  e.  N.  /\  ( ( F `  c )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >. )  ->  E. b  e.  N.  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >. )
9515, 94sylan 267 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ph  /\  (
s  e.  Q.  /\  t  e.  Q. )
)  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  t ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  /\  (
c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x ) )  /\  ( ( F `
 c )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  t } ,  { q  |  t 
<Q  q } >. )  ->  E. b  e.  N.  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >. )
9678caucvgprprlemelu 6765 . . . . . . . . . 10  |-  ( t  e.  ( 2nd `  L
)  <->  ( t  e. 
Q.  /\  E. b  e.  N.  ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  t } ,  { q  |  t 
<Q  q } >. )
)
9782, 95, 96sylanbrc 394 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ph  /\  (
s  e.  Q.  /\  t  e.  Q. )
)  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  t ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  /\  (
c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x ) )  /\  ( ( F `
 c )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  t } ,  { q  |  t 
<Q  q } >. )  ->  t  e.  ( 2nd `  L ) )
9897ex 108 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  t  e. 
Q. ) )  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  t ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x
) )  ->  (
( ( F `  c )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >.  ->  t  e.  ( 2nd `  L
) ) )
9981, 98orim12d 700 . . . . . . 7  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  t  e. 
Q. ) )  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  t ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x
) )  ->  (
( ( <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )  <Q 
q } >.  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( ( F `  c )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  \/  ( ( F `  c )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  t } ,  {
q  |  t  <Q 
q } >. )  ->  ( s  e.  ( 1st `  L )  \/  t  e.  ( 2nd `  L ) ) ) )
10050, 99mpd 13 . . . . . 6  |-  ( ( ( ( ( (
ph  /\  ( s  e.  Q.  /\  t  e. 
Q. ) )  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  ( s  +Q  y
)  =  t ) )  /\  ( x  e.  Q.  /\  (
x  +Q  x ) 
<Q  y ) )  /\  ( c  e.  N.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x
) )  ->  (
s  e.  ( 1st `  L )  \/  t  e.  ( 2nd `  L
) ) )
1016, 100rexlimddv 2434 . . . . 5  |-  ( ( ( ( ( ph  /\  ( s  e.  Q.  /\  t  e.  Q. )
)  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  t ) )  /\  ( x  e. 
Q.  /\  ( x  +Q  x )  <Q  y
) )  ->  (
s  e.  ( 1st `  L )  \/  t  e.  ( 2nd `  L
) ) )
1024, 101rexlimddv 2434 . . . 4  |-  ( ( ( ( ph  /\  ( s  e.  Q.  /\  t  e.  Q. )
)  /\  s  <Q  t )  /\  ( y  e.  Q.  /\  (
s  +Q  y )  =  t ) )  ->  ( s  e.  ( 1st `  L
)  \/  t  e.  ( 2nd `  L
) ) )
1032, 102rexlimddv 2434 . . 3  |-  ( ( ( ph  /\  (
s  e.  Q.  /\  t  e.  Q. )
)  /\  s  <Q  t )  ->  ( s  e.  ( 1st `  L
)  \/  t  e.  ( 2nd `  L
) ) )
104103ex 108 . 2  |-  ( (
ph  /\  ( s  e.  Q.  /\  t  e. 
Q. ) )  -> 
( s  <Q  t  ->  ( s  e.  ( 1st `  L )  \/  t  e.  ( 2nd `  L ) ) ) )
105104ralrimivva 2398 1  |-  ( ph  ->  A. s  e.  Q.  A. t  e.  Q.  (
s  <Q  t  ->  (
s  e.  ( 1st `  L )  \/  t  e.  ( 2nd `  L
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    \/ wo 629    /\ w3a 885    = wceq 1243    e. wcel 1393   {cab 2026   A.wral 2303   E.wrex 2304   {crab 2307   <.cop 3375   class class class wbr 3761    Or wor 4029   -->wf 4885   ` cfv 4889  (class class class)co 5499   1stc1st 5752   2ndc2nd 5753   1oc1o 5981   [cec 6091   N.cnpi 6351    <N clti 6354    ~Q ceq 6358   Q.cnq 6359    +Q cplq 6361   *Qcrq 6363    <Q cltq 6364   P.cnp 6370    +P. cpp 6372    <P cltp 6374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3869  ax-sep 3872  ax-nul 3880  ax-pow 3924  ax-pr 3941  ax-un 4166  ax-setind 4256  ax-iinf 4298
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2308  df-rex 2309  df-reu 2310  df-rab 2312  df-v 2556  df-sbc 2762  df-csb 2850  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-int 3613  df-iun 3656  df-br 3762  df-opab 3816  df-mpt 3817  df-tr 3852  df-eprel 4023  df-id 4027  df-po 4030  df-iso 4031  df-iord 4099  df-on 4101  df-suc 4104  df-iom 4301  df-xp 4338  df-rel 4339  df-cnv 4340  df-co 4341  df-dm 4342  df-rn 4343  df-res 4344  df-ima 4345  df-iota 4854  df-fun 4891  df-fn 4892  df-f 4893  df-f1 4894  df-fo 4895  df-f1o 4896  df-fv 4897  df-ov 5502  df-oprab 5503  df-mpt2 5504  df-1st 5754  df-2nd 5755  df-recs 5907  df-irdg 5944  df-1o 5988  df-2o 5989  df-oadd 5992  df-omul 5993  df-er 6093  df-ec 6095  df-qs 6099  df-ni 6383  df-pli 6384  df-mi 6385  df-lti 6386  df-plpq 6423  df-mpq 6424  df-enq 6426  df-nqqs 6427  df-plqqs 6428  df-mqqs 6429  df-1nqqs 6430  df-rq 6431  df-ltnqqs 6432  df-enq0 6503  df-nq0 6504  df-0nq0 6505  df-plq0 6506  df-mq0 6507  df-inp 6545  df-iplp 6547  df-iltp 6549
This theorem is referenced by:  caucvgprprlemcl  6783
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