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Theorem caucvgprprlemloc 7412
Description: Lemma for caucvgprpr 7421. 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 7118 . . . . 5  |-  ( s 
<Q  t  ->  E. y  e.  Q.  ( s  +Q  y )  =  t )
21adantl 273 . . . 4  |-  ( ( ( ph  /\  (
s  e.  Q.  /\  t  e.  Q. )
)  /\  s  <Q  t )  ->  E. y  e.  Q.  ( s  +Q  y )  =  t )
3 subhalfnqq 7123 . . . . . 6  |-  ( y  e.  Q.  ->  E. x  e.  Q.  ( x  +Q  x )  <Q  y
)
43ad2antrl 477 . . . . 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 7372 . . . . . . 7  |-  ( x  e.  Q.  ->  E. c  e.  N.  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x )
65ad2antrl 477 . . . . . 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 504 . . . . . . . . . 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 272 . . . . . . . . 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 505 . . . . . . . . . 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 272 . . . . . . . . 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 506 . . . . . . . . . 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 272 . . . . . . . . 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 505 . . . . . . . . 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 506 . . . . . . . . 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 501 . . . . . . . . 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 502 . . . . . . . . 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 7393 . . . . . . . 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 505 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
s  e.  Q.  /\  t  e.  Q. )
)  /\  s  <Q  t )  ->  s  e.  Q. )
1918ad3antrrr 479 . . . . . . . . . . . 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 7131 . . . . . . . . . . . . . 14  |-  ( c  e.  N.  ->  [ <. c ,  1o >. ]  ~Q  e.  Q. )
2120ad2antrl 477 . . . . . . . . . . . . 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 7101 . . . . . . . . . . . . 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 7084 . . . . . . . . . . . 12  |-  ( ( s  e.  Q.  /\  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  )
)  e.  Q. )
2519, 23, 24syl2anc 406 . . . . . . . . . . 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 7256 . . . . . . . . . . 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 7256 . . . . . . . . . . 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 7246 . . . . . . . . . 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 406 . . . . . . . . 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 506 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
s  e.  Q.  /\  t  e.  Q. )
)  /\  s  <Q  t )  ->  t  e.  Q. )
3332ad3antrrr 479 . . . . . . . . . 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 7256 . . . . . . . . . 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 483 . . . . . . . . . . 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 5488 . . . . . . . . . 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 7074 . . . . . . . . . . . . . 14  |-  <Q  C_  ( Q.  X.  Q. )
4039brel 4529 . . . . . . . . . . . . 13  |-  ( ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x  ->  (
( *Q `  [ <. c ,  1o >. ]  ~Q  )  e.  Q.  /\  x  e.  Q. )
)
4140simpld 111 . . . . . . . . . . . 12  |-  ( ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  x  ->  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  e.  Q. )
4241ad2antll 478 . . . . . . . . . . 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 7246 . . . . . . . . . 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 406 . . . . . . . . 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 7305 . . . . . . . . . 10  |-  <P  Or  P.
47 sowlin 4180 . . . . . . . . . 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 418 . . . . . . . . 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 1184 . . . . . . . 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 272 . . . . . . . . . 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 505 . . . . . . . . . . 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 109 . . . . . . . . . . . 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 7328 . . . . . . . . . . . . . 14  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
f  <P  g  <->  ( h  +P.  f )  <P  (
h  +P.  g )
) )
5554adantl 273 . . . . . . . . . . . . 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 272 . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . 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 272 . . . . . . . . . . . . 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 7287 . . . . . . . . . . . . . 14  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
6261adantl 273 . . . . . . . . . . . . 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 5872 . . . . . . . . . . . 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 166 . . . . . . . . . . 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 3652 . . . . . . . . . . . . . . . . . . 19  |-  ( a  =  c  ->  <. a ,  1o >.  =  <. c ,  1o >. )
6665eceq1d 6395 . . . . . . . . . . . . . . . . . 18  |-  ( a  =  c  ->  [ <. a ,  1o >. ]  ~Q  =  [ <. c ,  1o >. ]  ~Q  )
6766fveq2d 5357 . . . . . . . . . . . . . . . . 17  |-  ( a  =  c  ->  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )
6867oveq2d 5722 . . . . . . . . . . . . . . . 16  |-  ( a  =  c  ->  (
s  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) )
6968breq2d 3887 . . . . . . . . . . . . . . 15  |-  ( a  =  c  ->  (
p  <Q  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <->  p  <Q  ( s  +Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  )
) ) )
7069abbidv 2217 . . . . . . . . . . . . . 14  |-  ( a  =  c  ->  { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) }  =  { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) } )
7168breq1d 3885 . . . . . . . . . . . . . . 15  |-  ( a  =  c  ->  (
( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  <Q  q  <->  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )  <Q 
q ) )
7271abbidv 2217 . . . . . . . . . . . . . 14  |-  ( a  =  c  ->  { q  |  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
q }  =  {
q  |  ( s  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) 
<Q  q } )
7370, 72opeq12d 3660 . . . . . . . . . . . . 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 5353 . . . . . . . . . . . . 13  |-  ( a  =  c  ->  ( F `  a )  =  ( F `  c ) )
7573, 74breq12d 3888 . . . . . . . . . . . 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 2744 . . . . . . . . . . 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 406 . . . . . . . . . 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 7394 . . . . . . . . . 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 411 . . . . . . . . 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 114 . . . . . . . 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 272 . . . . . . . . . 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 5353 . . . . . . . . . . . . . 14  |-  ( b  =  c  ->  ( F `  b )  =  ( F `  c ) )
84 opeq1 3652 . . . . . . . . . . . . . . . . . . 19  |-  ( b  =  c  ->  <. b ,  1o >.  =  <. c ,  1o >. )
8584eceq1d 6395 . . . . . . . . . . . . . . . . . 18  |-  ( b  =  c  ->  [ <. b ,  1o >. ]  ~Q  =  [ <. c ,  1o >. ]  ~Q  )
8685fveq2d 5357 . . . . . . . . . . . . . . . . 17  |-  ( b  =  c  ->  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )
8786breq2d 3887 . . . . . . . . . . . . . . . 16  |-  ( b  =  c  ->  (
p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <->  p  <Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) )
8887abbidv 2217 . . . . . . . . . . . . . . 15  |-  ( b  =  c  ->  { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) }  =  { p  |  p  <Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  ) } )
8986breq1d 3885 . . . . . . . . . . . . . . . 16  |-  ( b  =  c  ->  (
( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. c ,  1o >. ]  ~Q  )  <Q  q ) )
9089abbidv 2217 . . . . . . . . . . . . . . 15  |-  ( b  =  c  ->  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q }  =  {
q  |  ( *Q
`  [ <. c ,  1o >. ]  ~Q  )  <Q  q } )
9188, 90opeq12d 3660 . . . . . . . . . . . . . 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 5724 . . . . . . . . . . . . 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 3885 . . . . . . . . . . . 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 2744 . . . . . . . . . . 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 279 . . . . . . . . . 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 7395 . . . . . . . . . 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 411 . . . . . . . . 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 114 . . . . . . . 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 741 . . . . . . 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 2513 . . . . 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 2513 . . . 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 2513 . . 3  |-  ( ( ( ph  /\  (
s  e.  Q.  /\  t  e.  Q. )
)  /\  s  <Q  t )  ->  ( s  e.  ( 1st `  L
)  \/  t  e.  ( 2nd `  L
) ) )
104103ex 114 . 2  |-  ( (
ph  /\  ( s  e.  Q.  /\  t  e. 
Q. ) )  -> 
( s  <Q  t  ->  ( s  e.  ( 1st `  L )  \/  t  e.  ( 2nd `  L ) ) ) )
105104ralrimivva 2473 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 103    <-> wb 104    \/ wo 670    /\ w3a 930    = wceq 1299    e. wcel 1448   {cab 2086   A.wral 2375   E.wrex 2376   {crab 2379   <.cop 3477   class class class wbr 3875    Or wor 4155   -->wf 5055   ` cfv 5059  (class class class)co 5706   1stc1st 5967   2ndc2nd 5968   1oc1o 6236   [cec 6357   N.cnpi 6981    <N clti 6984    ~Q ceq 6988   Q.cnq 6989    +Q cplq 6991   *Qcrq 6993    <Q cltq 6994   P.cnp 7000    +P. cpp 7002    <P cltp 7004
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-eprel 4149  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-irdg 6197  df-1o 6243  df-2o 6244  df-oadd 6247  df-omul 6248  df-er 6359  df-ec 6361  df-qs 6365  df-ni 7013  df-pli 7014  df-mi 7015  df-lti 7016  df-plpq 7053  df-mpq 7054  df-enq 7056  df-nqqs 7057  df-plqqs 7058  df-mqqs 7059  df-1nqqs 7060  df-rq 7061  df-ltnqqs 7062  df-enq0 7133  df-nq0 7134  df-0nq0 7135  df-plq0 7136  df-mq0 7137  df-inp 7175  df-iplp 7177  df-iltp 7179
This theorem is referenced by:  caucvgprprlemcl  7413
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