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Theorem caucvgprprlemexbt 6862
Description: Lemma for caucvgprpr 6868. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 16-Jun-2021.)
Hypotheses
Ref Expression
caucvgprpr.f  |-  ( ph  ->  F : N. --> P. )
caucvgprpr.cau  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <P  ( ( F `
 k )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( F `  k
)  <P  ( ( F `
 n )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )
) ) )
caucvgprpr.bnd  |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )
caucvgprpr.lim  |-  L  = 
<. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  r ) } ,  { u  e.  Q.  |  E. r  e.  N.  ( ( F `
 r )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  u } ,  { q  |  u 
<Q  q } >. } >.
caucvgprprlemexbt.q  |-  ( ph  ->  Q  e.  Q. )
caucvgprprlemexbt.t  |-  ( ph  ->  T  e.  P. )
caucvgprprlemexbt.lt  |-  ( ph  ->  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )  <P  T )
Assertion
Ref Expression
caucvgprprlemexbt  |-  ( ph  ->  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  Q } ,  { q  |  Q  <Q  q } >. )  <P  T )
Distinct variable groups:    A, m    m, F    A, r, m    F, b    k, F, l, n, u    F, r    L, b   
k, L    Q, b, p, q    T, b    ph, b    r, b, p, q    k, p, q, r, l, u
Allowed substitution hints:    ph( u, k, m, n, r, q, p, l)    A( u, k, n, q, p, b, l)    Q( u, k, m, n, r, l)    T( u, k, m, n, r, q, p, l)    F( q, p)    L( u, m, n, r, q, p, l)

Proof of Theorem caucvgprprlemexbt
Dummy variables  f  g  h  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 caucvgprprlemexbt.lt . . . . 5  |-  ( ph  ->  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )  <P  T )
2 caucvgprpr.f . . . . . . . 8  |-  ( ph  ->  F : N. --> P. )
3 caucvgprpr.cau . . . . . . . 8  |-  ( 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 } >. )
) ) )
4 caucvgprpr.bnd . . . . . . . 8  |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )
5 caucvgprpr.lim . . . . . . . 8  |-  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 } >. } >.
62, 3, 4, 5caucvgprprlemclphr 6861 . . . . . . 7  |-  ( ph  ->  L  e.  P. )
7 caucvgprprlemexbt.q . . . . . . . 8  |-  ( ph  ->  Q  e.  Q. )
8 nqprlu 6703 . . . . . . . 8  |-  ( Q  e.  Q.  ->  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >.  e.  P. )
97, 8syl 14 . . . . . . 7  |-  ( ph  -> 
<. { p  |  p 
<Q  Q } ,  {
q  |  Q  <Q  q } >.  e.  P. )
10 addclpr 6693 . . . . . . 7  |-  ( ( L  e.  P.  /\  <. { p  |  p  <Q  Q } ,  {
q  |  Q  <Q  q } >.  e.  P. )  ->  ( L  +P.  <. { p  |  p  <Q  Q } ,  {
q  |  Q  <Q  q } >. )  e.  P. )
116, 9, 10syl2anc 397 . . . . . 6  |-  ( ph  ->  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )  e.  P. )
12 caucvgprprlemexbt.t . . . . . 6  |-  ( ph  ->  T  e.  P. )
13 ltdfpr 6662 . . . . . 6  |-  ( ( ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )  e.  P.  /\  T  e.  P. )  ->  ( ( L  +P.  <. { p  |  p  <Q  Q } ,  {
q  |  Q  <Q  q } >. )  <P  T  <->  E. x  e.  Q.  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )
1411, 12, 13syl2anc 397 . . . . 5  |-  ( ph  ->  ( ( L  +P.  <. { p  |  p  <Q  Q } ,  {
q  |  Q  <Q  q } >. )  <P  T  <->  E. x  e.  Q.  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )
151, 14mpbid 139 . . . 4  |-  ( ph  ->  E. x  e.  Q.  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  {
q  |  Q  <Q  q } >. ) )  /\  x  e.  ( 1st `  T ) ) )
166adantr 265 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )  ->  L  e.  P. )
177adantr 265 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )  ->  Q  e.  Q. )
18 simprrl 499 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )  ->  x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. ) ) )
1916, 17, 18prplnqu 6776 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )  ->  E. y  e.  ( 2nd `  L ) ( y  +Q  Q )  =  x )
20 simprl 491 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  {
q  |  Q  <Q  q } >. ) )  /\  x  e.  ( 1st `  T ) ) ) )  /\  ( y  e.  ( 2nd `  L
)  /\  ( y  +Q  Q )  =  x ) )  ->  y  e.  ( 2nd `  L
) )
21 breq2 3796 . . . . . . . . . . . . . . . . 17  |-  ( u  =  y  ->  (
p  <Q  u  <->  p  <Q  y ) )
2221abbidv 2171 . . . . . . . . . . . . . . . 16  |-  ( u  =  y  ->  { p  |  p  <Q  u }  =  { p  |  p 
<Q  y } )
23 breq1 3795 . . . . . . . . . . . . . . . . 17  |-  ( u  =  y  ->  (
u  <Q  q  <->  y  <Q  q ) )
2423abbidv 2171 . . . . . . . . . . . . . . . 16  |-  ( u  =  y  ->  { q  |  u  <Q  q }  =  { q  |  y  <Q  q } )
2522, 24opeq12d 3585 . . . . . . . . . . . . . . 15  |-  ( u  =  y  ->  <. { p  |  p  <Q  u } ,  { q  |  u 
<Q  q } >.  =  <. { p  |  p  <Q  y } ,  { q  |  y  <Q  q } >. )
2625breq2d 3804 . . . . . . . . . . . . . 14  |-  ( u  =  y  ->  (
( ( 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 } >.  <->  ( ( F `
 r )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  y } ,  { q  |  y 
<Q  q } >. )
)
2726rexbidv 2344 . . . . . . . . . . . . 13  |-  ( u  =  y  ->  ( 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 } >.  <->  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  y } ,  {
q  |  y  <Q 
q } >. )
)
285fveq2i 5209 . . . . . . . . . . . . . 14  |-  ( 2nd `  L )  =  ( 2nd `  <. { 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 } >. } >. )
29 nqex 6519 . . . . . . . . . . . . . . . 16  |-  Q.  e.  _V
3029rabex 3929 . . . . . . . . . . . . . . 15  |-  { 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 ) }  e.  _V
3129rabex 3929 . . . . . . . . . . . . . . 15  |-  { 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 } >. }  e.  _V
3230, 31op2nd 5802 . . . . . . . . . . . . . 14  |-  ( 2nd `  <. { 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 } >. } >. )  =  { 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 } >. }
3328, 32eqtri 2076 . . . . . . . . . . . . 13  |-  ( 2nd `  L )  =  {
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 } >. }
3427, 33elrab2 2723 . . . . . . . . . . . 12  |-  ( y  e.  ( 2nd `  L
)  <->  ( y  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  y } ,  { q  |  y 
<Q  q } >. )
)
3534biimpi 117 . . . . . . . . . . 11  |-  ( y  e.  ( 2nd `  L
)  ->  ( y  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  y } ,  { q  |  y 
<Q  q } >. )
)
3635simprd 111 . . . . . . . . . 10  |-  ( y  e.  ( 2nd `  L
)  ->  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  y } ,  { q  |  y 
<Q  q } >. )
3720, 36syl 14 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  {
q  |  Q  <Q  q } >. ) )  /\  x  e.  ( 1st `  T ) ) ) )  /\  ( y  e.  ( 2nd `  L
)  /\  ( y  +Q  Q )  =  x ) )  ->  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  y } ,  { q  |  y 
<Q  q } >. )
38 fveq2 5206 . . . . . . . . . . . 12  |-  ( r  =  b  ->  ( F `  r )  =  ( F `  b ) )
39 opeq1 3577 . . . . . . . . . . . . . . . . 17  |-  ( r  =  b  ->  <. r ,  1o >.  =  <. b ,  1o >. )
4039eceq1d 6173 . . . . . . . . . . . . . . . 16  |-  ( r  =  b  ->  [ <. r ,  1o >. ]  ~Q  =  [ <. b ,  1o >. ]  ~Q  )
4140fveq2d 5210 . . . . . . . . . . . . . . 15  |-  ( r  =  b  ->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )
4241breq2d 3804 . . . . . . . . . . . . . 14  |-  ( r  =  b  ->  (
p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <->  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) )
4342abbidv 2171 . . . . . . . . . . . . 13  |-  ( r  =  b  ->  { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) }  =  { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } )
4441breq1d 3802 . . . . . . . . . . . . . 14  |-  ( r  =  b  ->  (
( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q ) )
4544abbidv 2171 . . . . . . . . . . . . 13  |-  ( r  =  b  ->  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q }  =  {
q  |  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )  <Q  q } )
4643, 45opeq12d 3585 . . . . . . . . . . . 12  |-  ( r  =  b  ->  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >.  =  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )
4738, 46oveq12d 5558 . . . . . . . . . . 11  |-  ( r  =  b  ->  (
( F `  r
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  =  ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. ) )
4847breq1d 3802 . . . . . . . . . 10  |-  ( r  =  b  ->  (
( ( F `  r )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  y } ,  {
q  |  y  <Q 
q } >.  <->  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  <. { p  |  p  <Q  y } ,  { q  |  y  <Q  q } >. ) )
4948cbvrexv 2551 . . . . . . . . 9  |-  ( 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  y } ,  {
q  |  y  <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  y } ,  { q  |  y 
<Q  q } >. )
5037, 49sylib 131 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  {
q  |  Q  <Q  q } >. ) )  /\  x  e.  ( 1st `  T ) ) ) )  /\  ( y  e.  ( 2nd `  L
)  /\  ( y  +Q  Q )  =  x ) )  ->  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  y } ,  { q  |  y 
<Q  q } >. )
51 simpr 107 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )  /\  ( y  e.  ( 2nd `  L )  /\  ( y  +Q  Q )  =  x ) )  /\  b  e.  N. )  /\  (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  y } ,  {
q  |  y  <Q 
q } >. )  ->  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  y } ,  {
q  |  y  <Q 
q } >. )
52 ltaprg 6775 . . . . . . . . . . . . . . . . 17  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
f  <P  g  <->  ( h  +P.  f )  <P  (
h  +P.  g )
) )
5352adantl 266 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( (
ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )  /\  ( y  e.  ( 2nd `  L )  /\  ( y  +Q  Q )  =  x ) )  /\  b  e.  N. )  /\  (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  y } ,  {
q  |  y  <Q 
q } >. )  /\  ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )
)  ->  ( f  <P  g  <->  ( h  +P.  f )  <P  (
h  +P.  g )
) )
542ad4antr 471 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )  /\  ( y  e.  ( 2nd `  L )  /\  ( y  +Q  Q )  =  x ) )  /\  b  e.  N. )  /\  (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  y } ,  {
q  |  y  <Q 
q } >. )  ->  F : N. --> P. )
55 simplr 490 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )  /\  ( y  e.  ( 2nd `  L )  /\  ( y  +Q  Q )  =  x ) )  /\  b  e.  N. )  /\  (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  y } ,  {
q  |  y  <Q 
q } >. )  ->  b  e.  N. )
5654, 55ffvelrnd 5331 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )  /\  ( y  e.  ( 2nd `  L )  /\  ( y  +Q  Q )  =  x ) )  /\  b  e.  N. )  /\  (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  y } ,  {
q  |  y  <Q 
q } >. )  ->  ( F `  b
)  e.  P. )
57 recnnpr 6704 . . . . . . . . . . . . . . . . . 18  |-  ( b  e.  N.  ->  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >.  e.  P. )
5855, 57syl 14 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )  /\  ( y  e.  ( 2nd `  L )  /\  ( y  +Q  Q )  =  x ) )  /\  b  e.  N. )  /\  (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  y } ,  {
q  |  y  <Q 
q } >. )  -> 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >.  e. 
P. )
59 addclpr 6693 . . . . . . . . . . . . . . . . 17  |-  ( ( ( F `  b
)  e.  P.  /\  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >.  e. 
P. )  ->  (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  e.  P. )
6056, 58, 59syl2anc 397 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )  /\  ( y  e.  ( 2nd `  L )  /\  ( y  +Q  Q )  =  x ) )  /\  b  e.  N. )  /\  (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  y } ,  {
q  |  y  <Q 
q } >. )  ->  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  e.  P. )
6120ad2antrr 465 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )  /\  ( y  e.  ( 2nd `  L )  /\  ( y  +Q  Q )  =  x ) )  /\  b  e.  N. )  /\  (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  y } ,  {
q  |  y  <Q 
q } >. )  ->  y  e.  ( 2nd `  L ) )
6235simpld 109 . . . . . . . . . . . . . . . . . 18  |-  ( y  e.  ( 2nd `  L
)  ->  y  e.  Q. )
6361, 62syl 14 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )  /\  ( y  e.  ( 2nd `  L )  /\  ( y  +Q  Q )  =  x ) )  /\  b  e.  N. )  /\  (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  y } ,  {
q  |  y  <Q 
q } >. )  ->  y  e.  Q. )
64 nqprlu 6703 . . . . . . . . . . . . . . . . 17  |-  ( y  e.  Q.  ->  <. { p  |  p  <Q  y } ,  { q  |  y  <Q  q } >.  e.  P. )
6563, 64syl 14 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )  /\  ( y  e.  ( 2nd `  L )  /\  ( y  +Q  Q )  =  x ) )  /\  b  e.  N. )  /\  (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  y } ,  {
q  |  y  <Q 
q } >. )  -> 
<. { p  |  p 
<Q  y } ,  {
q  |  y  <Q 
q } >.  e.  P. )
669ad4antr 471 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )  /\  ( y  e.  ( 2nd `  L )  /\  ( y  +Q  Q )  =  x ) )  /\  b  e.  N. )  /\  (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  y } ,  {
q  |  y  <Q 
q } >. )  -> 
<. { p  |  p 
<Q  Q } ,  {
q  |  Q  <Q  q } >.  e.  P. )
67 addcomprg 6734 . . . . . . . . . . . . . . . . 17  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
6867adantl 266 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( (
ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )  /\  ( y  e.  ( 2nd `  L )  /\  ( y  +Q  Q )  =  x ) )  /\  b  e.  N. )  /\  (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  y } ,  {
q  |  y  <Q 
q } >. )  /\  ( f  e.  P.  /\  g  e.  P. )
)  ->  ( f  +P.  g )  =  ( g  +P.  f ) )
6953, 60, 65, 66, 68caovord2d 5698 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )  /\  ( y  e.  ( 2nd `  L )  /\  ( y  +Q  Q )  =  x ) )  /\  b  e.  N. )  /\  (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  y } ,  {
q  |  y  <Q 
q } >. )  ->  ( ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  y } ,  { q  |  y 
<Q  q } >.  <->  ( (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )  <P  ( <. { p  |  p  <Q  y } ,  { q  |  y 
<Q  q } >.  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. ) ) )
7051, 69mpbid 139 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )  /\  ( y  e.  ( 2nd `  L )  /\  ( y  +Q  Q )  =  x ) )  /\  b  e.  N. )  /\  (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  y } ,  {
q  |  y  <Q 
q } >. )  ->  ( ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )  <P  ( <. { p  |  p  <Q  y } ,  { q  |  y 
<Q  q } >.  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. ) )
717ad4antr 471 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )  /\  ( y  e.  ( 2nd `  L )  /\  ( y  +Q  Q )  =  x ) )  /\  b  e.  N. )  /\  (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  y } ,  {
q  |  y  <Q 
q } >. )  ->  Q  e.  Q. )
72 addnqpr 6717 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  Q.  /\  Q  e.  Q. )  -> 
<. { p  |  p 
<Q  ( y  +Q  Q
) } ,  {
q  |  ( y  +Q  Q )  <Q 
q } >.  =  (
<. { p  |  p 
<Q  y } ,  {
q  |  y  <Q 
q } >.  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. ) )
7363, 71, 72syl2anc 397 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )  /\  ( y  e.  ( 2nd `  L )  /\  ( y  +Q  Q )  =  x ) )  /\  b  e.  N. )  /\  (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  y } ,  {
q  |  y  <Q 
q } >. )  -> 
<. { p  |  p 
<Q  ( y  +Q  Q
) } ,  {
q  |  ( y  +Q  Q )  <Q 
q } >.  =  (
<. { p  |  p 
<Q  y } ,  {
q  |  y  <Q 
q } >.  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. ) )
7470, 73breqtrrd 3818 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )  /\  ( y  e.  ( 2nd `  L )  /\  ( y  +Q  Q )  =  x ) )  /\  b  e.  N. )  /\  (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  y } ,  {
q  |  y  <Q 
q } >. )  ->  ( ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )  <P 
<. { p  |  p 
<Q  ( y  +Q  Q
) } ,  {
q  |  ( y  +Q  Q )  <Q 
q } >. )
75 simplrr 496 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )  /\  ( y  e.  ( 2nd `  L )  /\  ( y  +Q  Q )  =  x ) )  /\  b  e.  N. )  ->  (
y  +Q  Q )  =  x )
7675adantr 265 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )  /\  ( y  e.  ( 2nd `  L )  /\  ( y  +Q  Q )  =  x ) )  /\  b  e.  N. )  /\  (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  y } ,  {
q  |  y  <Q 
q } >. )  ->  ( y  +Q  Q
)  =  x )
77 breq2 3796 . . . . . . . . . . . . . . . . 17  |-  ( ( y  +Q  Q )  =  x  ->  (
p  <Q  ( y  +Q  Q )  <->  p  <Q  x ) )
7877abbidv 2171 . . . . . . . . . . . . . . . 16  |-  ( ( y  +Q  Q )  =  x  ->  { p  |  p  <Q  ( y  +Q  Q ) }  =  { p  |  p  <Q  x }
)
79 breq1 3795 . . . . . . . . . . . . . . . . 17  |-  ( ( y  +Q  Q )  =  x  ->  (
( y  +Q  Q
)  <Q  q  <->  x  <Q  q ) )
8079abbidv 2171 . . . . . . . . . . . . . . . 16  |-  ( ( y  +Q  Q )  =  x  ->  { q  |  ( y  +Q  Q )  <Q  q }  =  { q  |  x  <Q  q } )
8178, 80opeq12d 3585 . . . . . . . . . . . . . . 15  |-  ( ( y  +Q  Q )  =  x  ->  <. { p  |  p  <Q  ( y  +Q  Q ) } ,  { q  |  ( y  +Q  Q
)  <Q  q } >.  = 
<. { p  |  p 
<Q  x } ,  {
q  |  x  <Q  q } >. )
8281breq2d 3804 . . . . . . . . . . . . . 14  |-  ( ( y  +Q  Q )  =  x  ->  (
( ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )  <P 
<. { p  |  p 
<Q  ( y  +Q  Q
) } ,  {
q  |  ( y  +Q  Q )  <Q 
q } >.  <->  ( (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )  <P 
<. { p  |  p 
<Q  x } ,  {
q  |  x  <Q  q } >. ) )
8376, 82syl 14 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )  /\  ( y  e.  ( 2nd `  L )  /\  ( y  +Q  Q )  =  x ) )  /\  b  e.  N. )  /\  (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  y } ,  {
q  |  y  <Q 
q } >. )  ->  ( ( ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )  <P  <. { p  |  p  <Q  ( y  +Q  Q ) } ,  { q  |  ( y  +Q  Q
)  <Q  q } >.  <->  (
( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )  <P 
<. { p  |  p 
<Q  x } ,  {
q  |  x  <Q  q } >. ) )
8474, 83mpbid 139 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )  /\  ( y  e.  ( 2nd `  L )  /\  ( y  +Q  Q )  =  x ) )  /\  b  e.  N. )  /\  (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  y } ,  {
q  |  y  <Q 
q } >. )  ->  ( ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )  <P 
<. { p  |  p 
<Q  x } ,  {
q  |  x  <Q  q } >. )
85 simplrl 495 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  {
q  |  Q  <Q  q } >. ) )  /\  x  e.  ( 1st `  T ) ) ) )  /\  ( y  e.  ( 2nd `  L
)  /\  ( y  +Q  Q )  =  x ) )  ->  x  e.  Q. )
8685ad2antrr 465 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )  /\  ( y  e.  ( 2nd `  L )  /\  ( y  +Q  Q )  =  x ) )  /\  b  e.  N. )  /\  (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  y } ,  {
q  |  y  <Q 
q } >. )  ->  x  e.  Q. )
87 addclpr 6693 . . . . . . . . . . . . . 14  |-  ( ( ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  e.  P.  /\  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >.  e.  P. )  ->  ( ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )  e.  P. )
8860, 66, 87syl2anc 397 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )  /\  ( y  e.  ( 2nd `  L )  /\  ( y  +Q  Q )  =  x ) )  /\  b  e.  N. )  /\  (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  y } ,  {
q  |  y  <Q 
q } >. )  ->  ( ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )  e.  P. )
89 nqpru 6708 . . . . . . . . . . . . 13  |-  ( ( x  e.  Q.  /\  ( ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )  e.  P. )  ->  (
x  e.  ( 2nd `  ( ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  <->  ( ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )  <P  <. { p  |  p  <Q  x } ,  { q  |  x 
<Q  q } >. )
)
9086, 88, 89syl2anc 397 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )  /\  ( y  e.  ( 2nd `  L )  /\  ( y  +Q  Q )  =  x ) )  /\  b  e.  N. )  /\  (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  y } ,  {
q  |  y  <Q 
q } >. )  ->  ( x  e.  ( 2nd `  ( ( ( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  <->  ( ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )  <P  <. { p  |  p  <Q  x } ,  { q  |  x 
<Q  q } >. )
)
9184, 90mpbird 160 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )  /\  ( y  e.  ( 2nd `  L )  /\  ( y  +Q  Q )  =  x ) )  /\  b  e.  N. )  /\  (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  y } ,  {
q  |  y  <Q 
q } >. )  ->  x  e.  ( 2nd `  ( ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
) )
92 simprrr 500 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )  ->  x  e.  ( 1st `  T ) )
9392ad3antrrr 469 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )  /\  ( y  e.  ( 2nd `  L )  /\  ( y  +Q  Q )  =  x ) )  /\  b  e.  N. )  /\  (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  y } ,  {
q  |  y  <Q 
q } >. )  ->  x  e.  ( 1st `  T ) )
9491, 93jca 294 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )  /\  ( y  e.  ( 2nd `  L )  /\  ( y  +Q  Q )  =  x ) )  /\  b  e.  N. )  /\  (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  y } ,  {
q  |  y  <Q 
q } >. )  ->  ( x  e.  ( 2nd `  ( ( ( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) )
9594ex 112 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )  /\  ( y  e.  ( 2nd `  L )  /\  ( y  +Q  Q )  =  x ) )  /\  b  e.  N. )  ->  (
( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  y } ,  {
q  |  y  <Q 
q } >.  ->  (
x  e.  ( 2nd `  ( ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )
9695reximdva 2438 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  {
q  |  Q  <Q  q } >. ) )  /\  x  e.  ( 1st `  T ) ) ) )  /\  ( y  e.  ( 2nd `  L
)  /\  ( y  +Q  Q )  =  x ) )  ->  ( 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  y } ,  {
q  |  y  <Q 
q } >.  ->  E. b  e.  N.  ( x  e.  ( 2nd `  (
( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )
9750, 96mpd 13 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  {
q  |  Q  <Q  q } >. ) )  /\  x  e.  ( 1st `  T ) ) ) )  /\  ( y  e.  ( 2nd `  L
)  /\  ( y  +Q  Q )  =  x ) )  ->  E. b  e.  N.  ( x  e.  ( 2nd `  (
( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) )
9819, 97rexlimddv 2454 . . . . . 6  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )  ->  E. b  e.  N.  ( x  e.  ( 2nd `  ( ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. ) )  /\  x  e.  ( 1st `  T ) ) )
9998expr 361 . . . . 5  |-  ( (
ph  /\  x  e.  Q. )  ->  ( ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. ) )  /\  x  e.  ( 1st `  T ) )  ->  E. b  e.  N.  ( x  e.  ( 2nd `  ( ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. ) )  /\  x  e.  ( 1st `  T ) ) ) )
10099reximdva 2438 . . . 4  |-  ( ph  ->  ( E. x  e. 
Q.  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) )  ->  E. x  e.  Q.  E. b  e. 
N.  ( x  e.  ( 2nd `  (
( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )
10115, 100mpd 13 . . 3  |-  ( ph  ->  E. x  e.  Q.  E. b  e.  N.  (
x  e.  ( 2nd `  ( ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) )
102 rexcom 2491 . . 3  |-  ( E. x  e.  Q.  E. b  e.  N.  (
x  e.  ( 2nd `  ( ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) )  <->  E. b  e.  N.  E. x  e.  Q.  (
x  e.  ( 2nd `  ( ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) )
103101, 102sylib 131 . 2  |-  ( ph  ->  E. b  e.  N.  E. x  e.  Q.  (
x  e.  ( 2nd `  ( ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) )
1042ffvelrnda 5330 . . . . . 6  |-  ( (
ph  /\  b  e.  N. )  ->  ( F `
 b )  e. 
P. )
10557adantl 266 . . . . . 6  |-  ( (
ph  /\  b  e.  N. )  ->  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >.  e.  P. )
106104, 105, 59syl2anc 397 . . . . 5  |-  ( (
ph  /\  b  e.  N. )  ->  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  e.  P. )
1079adantr 265 . . . . 5  |-  ( (
ph  /\  b  e.  N. )  ->  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >.  e.  P. )
108106, 107, 87syl2anc 397 . . . 4  |-  ( (
ph  /\  b  e.  N. )  ->  ( ( ( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )  e.  P. )
10912adantr 265 . . . 4  |-  ( (
ph  /\  b  e.  N. )  ->  T  e. 
P. )
110 ltdfpr 6662 . . . 4  |-  ( ( ( ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )  e.  P.  /\  T  e. 
P. )  ->  (
( ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )  <P  T  <->  E. x  e.  Q.  ( x  e.  ( 2nd `  ( ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. ) )  /\  x  e.  ( 1st `  T ) ) ) )
111108, 109, 110syl2anc 397 . . 3  |-  ( (
ph  /\  b  e.  N. )  ->  ( ( ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )  <P  T  <->  E. x  e.  Q.  ( x  e.  ( 2nd `  ( ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. ) )  /\  x  e.  ( 1st `  T ) ) ) )
112111rexbidva 2340 . 2  |-  ( ph  ->  ( 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  Q } ,  { q  |  Q  <Q  q } >. )  <P  T  <->  E. b  e.  N.  E. x  e. 
Q.  ( x  e.  ( 2nd `  (
( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )
)  /\  x  e.  ( 1st `  T ) ) ) )
113103, 112mpbird 160 1  |-  ( ph  ->  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  Q } ,  { q  |  Q  <Q  q } >. )  <P  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    /\ w3a 896    = wceq 1259    e. wcel 1409   {cab 2042   A.wral 2323   E.wrex 2324   {crab 2327   <.cop 3406   class class class wbr 3792   -->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:  caucvgprprlemexb  6863
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