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Theorem caucvgprprlemloc 6859
Description: Lemma for caucvgprpr 6868. 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 6565 . . . . 5  |-  ( s 
<Q  t  ->  E. y  e.  Q.  ( s  +Q  y )  =  t )
21adantl 266 . . . 4  |-  ( ( ( ph  /\  (
s  e.  Q.  /\  t  e.  Q. )
)  /\  s  <Q  t )  ->  E. y  e.  Q.  ( s  +Q  y )  =  t )
3 subhalfnqq 6570 . . . . . 6  |-  ( y  e.  Q.  ->  E. x  e.  Q.  ( x  +Q  x )  <Q  y
)
43ad2antrl 467 . . . . 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 6819 . . . . . . 7  |-  ( x  e.  Q.  ->  E. c  e.  N.  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x )
65ad2antrl 467 . . . . . 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 494 . . . . . . . . . 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 265 . . . . . . . . 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 495 . . . . . . . . . 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 265 . . . . . . . . 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 496 . . . . . . . . . 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 265 . . . . . . . . 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 495 . . . . . . . . 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 496 . . . . . . . . 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 491 . . . . . . . . 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 492 . . . . . . . . 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 6840 . . . . . . . 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 495 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
s  e.  Q.  /\  t  e.  Q. )
)  /\  s  <Q  t )  ->  s  e.  Q. )
1918ad3antrrr 469 . . . . . . . . . . . 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 6578 . . . . . . . . . . . . . 14  |-  ( c  e.  N.  ->  [ <. c ,  1o >. ]  ~Q  e.  Q. )
2120ad2antrl 467 . . . . . . . . . . . . 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 6548 . . . . . . . . . . . . 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 6531 . . . . . . . . . . . 12  |-  ( ( s  e.  Q.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  )
)  e.  Q. )
2519, 23, 24syl2anc 397 . . . . . . . . . . 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 6703 . . . . . . . . . . 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 6703 . . . . . . . . . . 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 6693 . . . . . . . . . 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 397 . . . . . . . . 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 496 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
s  e.  Q.  /\  t  e.  Q. )
)  /\  s  <Q  t )  ->  t  e.  Q. )
3332ad3antrrr 469 . . . . . . . . . 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 6703 . . . . . . . . . 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 473 . . . . . . . . . . 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 5331 . . . . . . . . . 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 6521 . . . . . . . . . . . . . 14  |-  <Q  C_  ( Q.  X.  Q. )
4039brel 4420 . . . . . . . . . . . . 13  |-  ( ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x  ->  (
( *Q `  [ <. c ,  1o >. ]  ~Q  )  e.  Q.  /\  x  e.  Q. )
)
4140simpld 109 . . . . . . . . . . . 12  |-  ( ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x  ->  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  e.  Q. )
4241ad2antll 468 . . . . . . . . . . 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 6693 . . . . . . . . . 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 397 . . . . . . . . 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 6752 . . . . . . . . . 10  |-  <P  Or  P.
47 sowlin 4085 . . . . . . . . . 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 408 . . . . . . . . 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 1146 . . . . . . . 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 265 . . . . . . . . . 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 495 . . . . . . . . . . 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 107 . . . . . . . . . . . 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 6775 . . . . . . . . . . . . . 14  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
f  <P  g  <->  ( h  +P.  f )  <P  (
h  +P.  g )
) )
5554adantl 266 . . . . . . . . . . . . 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 265 . . . . . . . . . . . . . . 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 397 . . . . . . . . . . . . . 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 265 . . . . . . . . . . . . 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 6734 . . . . . . . . . . . . . 14  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
6261adantl 266 . . . . . . . . . . . . 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 5698 . . . . . . . . . . . 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 160 . . . . . . . . . . 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 3577 . . . . . . . . . . . . . . . . . . 19  |-  ( a  =  c  ->  <. a ,  1o >.  =  <. c ,  1o >. )
6665eceq1d 6173 . . . . . . . . . . . . . . . . . 18  |-  ( a  =  c  ->  [ <. a ,  1o >. ]  ~Q  =  [ <. c ,  1o >. ]  ~Q  )
6766fveq2d 5210 . . . . . . . . . . . . . . . . 17  |-  ( a  =  c  ->  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )
6867oveq2d 5556 . . . . . . . . . . . . . . . 16  |-  ( a  =  c  ->  (
s  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) )
6968breq2d 3804 . . . . . . . . . . . . . . 15  |-  ( a  =  c  ->  (
p  <Q  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <->  p  <Q  ( s  +Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  )
) ) )
7069abbidv 2171 . . . . . . . . . . . . . 14  |-  ( a  =  c  ->  { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) }  =  { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) } )
7168breq1d 3802 . . . . . . . . . . . . . . 15  |-  ( a  =  c  ->  (
( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  <Q  q  <->  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )  <Q 
q ) )
7271abbidv 2171 . . . . . . . . . . . . . 14  |-  ( a  =  c  ->  { q  |  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
q }  =  {
q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) 
<Q  q } )
7370, 72opeq12d 3585 . . . . . . . . . . . . 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 5206 . . . . . . . . . . . . 13  |-  ( a  =  c  ->  ( F `  a )  =  ( F `  c ) )
7573, 74breq12d 3805 . . . . . . . . . . . 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 2673 . . . . . . . . . . 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 397 . . . . . . . . . 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 6841 . . . . . . . . . 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 402 . . . . . . . . 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 112 . . . . . . . 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 265 . . . . . . . . . 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 5206 . . . . . . . . . . . . . 14  |-  ( b  =  c  ->  ( F `  b )  =  ( F `  c ) )
84 opeq1 3577 . . . . . . . . . . . . . . . . . . 19  |-  ( b  =  c  ->  <. b ,  1o >.  =  <. c ,  1o >. )
8584eceq1d 6173 . . . . . . . . . . . . . . . . . 18  |-  ( b  =  c  ->  [ <. b ,  1o >. ]  ~Q  =  [ <. c ,  1o >. ]  ~Q  )
8685fveq2d 5210 . . . . . . . . . . . . . . . . 17  |-  ( b  =  c  ->  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )
8786breq2d 3804 . . . . . . . . . . . . . . . 16  |-  ( b  =  c  ->  (
p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <->  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) )
8887abbidv 2171 . . . . . . . . . . . . . . 15  |-  ( b  =  c  ->  { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) }  =  { p  |  p  <Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  ) } )
8986breq1d 3802 . . . . . . . . . . . . . . . 16  |-  ( b  =  c  ->  (
( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  q ) )
9089abbidv 2171 . . . . . . . . . . . . . . 15  |-  ( b  =  c  ->  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q }  =  {
q  |  ( *Q
`  [ <. c ,  1o >. ]  ~Q  )  <Q  q } )
9188, 90opeq12d 3585 . . . . . . . . . . . . . 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 5558 . . . . . . . . . . . . 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 3802 . . . . . . . . . . . 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 2673 . . . . . . . . . . 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 271 . . . . . . . . . 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 6842 . . . . . . . . . 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 402 . . . . . . . . 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 112 . . . . . . . 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 710 . . . . . . 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 2454 . . . . 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 2454 . . . 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 2454 . . 3  |-  ( ( ( ph  /\  (
s  e.  Q.  /\  t  e.  Q. )
)  /\  s  <Q  t )  ->  ( s  e.  ( 1st `  L
)  \/  t  e.  ( 2nd `  L
) ) )
104103ex 112 . 2  |-  ( (
ph  /\  ( s  e.  Q.  /\  t  e. 
Q. ) )  -> 
( s  <Q  t  ->  ( s  e.  ( 1st `  L )  \/  t  e.  ( 2nd `  L ) ) ) )
105104ralrimivva 2418 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 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   1stc1st 5793   2ndc2nd 5794   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:  caucvgprprlemcl  6860
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