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Theorem caucvgprprlemexbt 7244
Description: Lemma for caucvgprpr 7250. 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 7243 . . . . . . 7  |-  ( ph  ->  L  e.  P. )
7 caucvgprprlemexbt.q . . . . . . . 8  |-  ( ph  ->  Q  e.  Q. )
8 nqprlu 7085 . . . . . . . 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 7075 . . . . . . 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 403 . . . . . 6  |-  ( ph  ->  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )  e.  P. )
12 caucvgprprlemexbt.t . . . . . 6  |-  ( ph  ->  T  e.  P. )
13 ltdfpr 7044 . . . . . 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 403 . . . . 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 145 . . . 4  |-  ( ph  ->  E. x  e.  Q.  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  {
q  |  Q  <Q  q } >. ) )  /\  x  e.  ( 1st `  T ) ) )
166adantr 270 . . . . . . . 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 270 . . . . . . . 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 506 . . . . . . . 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 7158 . . . . . . 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 498 . . . . . . . . . 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 3841 . . . . . . . . . . . . . . . . 17  |-  ( u  =  y  ->  (
p  <Q  u  <->  p  <Q  y ) )
2221abbidv 2205 . . . . . . . . . . . . . . . 16  |-  ( u  =  y  ->  { p  |  p  <Q  u }  =  { p  |  p 
<Q  y } )
23 breq1 3840 . . . . . . . . . . . . . . . . 17  |-  ( u  =  y  ->  (
u  <Q  q  <->  y  <Q  q ) )
2423abbidv 2205 . . . . . . . . . . . . . . . 16  |-  ( u  =  y  ->  { q  |  u  <Q  q }  =  { q  |  y  <Q  q } )
2522, 24opeq12d 3625 . . . . . . . . . . . . . . 15  |-  ( u  =  y  ->  <. { p  |  p  <Q  u } ,  { q  |  u 
<Q  q } >.  =  <. { p  |  p  <Q  y } ,  { q  |  y  <Q  q } >. )
2625breq2d 3849 . . . . . . . . . . . . . 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 2381 . . . . . . . . . . . . 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 5292 . . . . . . . . . . . . . 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 6901 . . . . . . . . . . . . . . . 16  |-  Q.  e.  _V
3029rabex 3975 . . . . . . . . . . . . . . 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 3975 . . . . . . . . . . . . . . 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 5900 . . . . . . . . . . . . . 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 2108 . . . . . . . . . . . . 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 2772 . . . . . . . . . . . 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 118 . . . . . . . . . . 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 112 . . . . . . . . . 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 5289 . . . . . . . . . . . 12  |-  ( r  =  b  ->  ( F `  r )  =  ( F `  b ) )
39 opeq1 3617 . . . . . . . . . . . . . . . . 17  |-  ( r  =  b  ->  <. r ,  1o >.  =  <. b ,  1o >. )
4039eceq1d 6308 . . . . . . . . . . . . . . . 16  |-  ( r  =  b  ->  [ <. r ,  1o >. ]  ~Q  =  [ <. b ,  1o >. ]  ~Q  )
4140fveq2d 5293 . . . . . . . . . . . . . . 15  |-  ( r  =  b  ->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )
4241breq2d 3849 . . . . . . . . . . . . . 14  |-  ( r  =  b  ->  (
p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <->  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) )
4342abbidv 2205 . . . . . . . . . . . . 13  |-  ( r  =  b  ->  { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) }  =  { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } )
4441breq1d 3847 . . . . . . . . . . . . . 14  |-  ( r  =  b  ->  (
( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q ) )
4544abbidv 2205 . . . . . . . . . . . . 13  |-  ( r  =  b  ->  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q }  =  {
q  |  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )  <Q  q } )
4643, 45opeq12d 3625 . . . . . . . . . . . 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 5652 . . . . . . . . . . 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 3847 . . . . . . . . . 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 2591 . . . . . . . . 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 120 . . . . . . . 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 108 . . . . . . . . . . . . . . 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 7157 . . . . . . . . . . . . . . . . 17  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
f  <P  g  <->  ( h  +P.  f )  <P  (
h  +P.  g )
) )
5352adantl 271 . . . . . . . . . . . . . . . 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 478 . . . . . . . . . . . . . . . . . 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 497 . . . . . . . . . . . . . . . . . 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 5419 . . . . . . . . . . . . . . . . 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 7086 . . . . . . . . . . . . . . . . . 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 7075 . . . . . . . . . . . . . . . . 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 403 . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . . . 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 110 . . . . . . . . . . . . . . . . . 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 7085 . . . . . . . . . . . . . . . . 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 478 . . . . . . . . . . . . . . . 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 7116 . . . . . . . . . . . . . . . . 17  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
6867adantl 271 . . . . . . . . . . . . . . . 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 5796 . . . . . . . . . . . . . . 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 145 . . . . . . . . . . . . . 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 478 . . . . . . . . . . . . . . 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 7099 . . . . . . . . . . . . . . 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 403 . . . . . . . . . . . . . 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 3863 . . . . . . . . . . . . 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 503 . . . . . . . . . . . . . . 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 270 . . . . . . . . . . . . . 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 3841 . . . . . . . . . . . . . . . . 17  |-  ( ( y  +Q  Q )  =  x  ->  (
p  <Q  ( y  +Q  Q )  <->  p  <Q  x ) )
7877abbidv 2205 . . . . . . . . . . . . . . . 16  |-  ( ( y  +Q  Q )  =  x  ->  { p  |  p  <Q  ( y  +Q  Q ) }  =  { p  |  p  <Q  x }
)
79 breq1 3840 . . . . . . . . . . . . . . . . 17  |-  ( ( y  +Q  Q )  =  x  ->  (
( y  +Q  Q
)  <Q  q  <->  x  <Q  q ) )
8079abbidv 2205 . . . . . . . . . . . . . . . 16  |-  ( ( y  +Q  Q )  =  x  ->  { q  |  ( y  +Q  Q )  <Q  q }  =  { q  |  x  <Q  q } )
8178, 80opeq12d 3625 . . . . . . . . . . . . . . 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 3849 . . . . . . . . . . . . . 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 145 . . . . . . . . . . . 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 502 . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . 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 7075 . . . . . . . . . . . . . 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 403 . . . . . . . . . . . . 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 7090 . . . . . . . . . . . . 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 403 . . . . . . . . . . . 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 165 . . . . . . . . . . 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 507 . . . . . . . . . . . 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 476 . . . . . . . . . . 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 300 . . . . . . . . . 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 113 . . . . . . . . 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 2475 . . . . . . . 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 2493 . . . . . 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 367 . . . . 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 2475 . . . 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 2531 . . 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 120 . 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 5418 . . . . . 6  |-  ( (
ph  /\  b  e.  N. )  ->  ( F `
 b )  e. 
P. )
10557adantl 271 . . . . . 6  |-  ( (
ph  /\  b  e.  N. )  ->  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >.  e.  P. )
106104, 105, 59syl2anc 403 . . . . 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 270 . . . . 5  |-  ( (
ph  /\  b  e.  N. )  ->  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >.  e.  P. )
108106, 107, 87syl2anc 403 . . . 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 270 . . . 4  |-  ( (
ph  /\  b  e.  N. )  ->  T  e. 
P. )
110 ltdfpr 7044 . . . 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 403 . . 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 2377 . 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 165 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 102    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438   {cab 2074   A.wral 2359   E.wrex 2360   {crab 2363   <.cop 3444   class class class wbr 3837   -->wf 4998   ` cfv 5002  (class class class)co 5634   1stc1st 5891   2ndc2nd 5892   1oc1o 6156   [cec 6270   N.cnpi 6810    <N clti 6813    ~Q ceq 6817   Q.cnq 6818    +Q cplq 6820   *Qcrq 6822    <Q cltq 6823   P.cnp 6829    +P. cpp 6831    <P cltp 6833
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-eprel 4107  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-1o 6163  df-2o 6164  df-oadd 6167  df-omul 6168  df-er 6272  df-ec 6274  df-qs 6278  df-ni 6842  df-pli 6843  df-mi 6844  df-lti 6845  df-plpq 6882  df-mpq 6883  df-enq 6885  df-nqqs 6886  df-plqqs 6887  df-mqqs 6888  df-1nqqs 6889  df-rq 6890  df-ltnqqs 6891  df-enq0 6962  df-nq0 6963  df-0nq0 6964  df-plq0 6965  df-mq0 6966  df-inp 7004  df-iplp 7006  df-iltp 7008
This theorem is referenced by:  caucvgprprlemexb  7245
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