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Theorem caucvgprprlemexbt 7534
Description: Lemma for caucvgprpr 7540. 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 7533 . . . . . . 7  |-  ( ph  ->  L  e.  P. )
7 caucvgprprlemexbt.q . . . . . . . 8  |-  ( ph  ->  Q  e.  Q. )
8 nqprlu 7375 . . . . . . . 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 7365 . . . . . . 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 409 . . . . . 6  |-  ( ph  ->  ( L  +P.  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >. )  e.  P. )
12 caucvgprprlemexbt.t . . . . . 6  |-  ( ph  ->  T  e.  P. )
13 ltdfpr 7334 . . . . . 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 409 . . . . 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 146 . . . 4  |-  ( ph  ->  E. x  e.  Q.  ( x  e.  ( 2nd `  ( L  +P.  <. { p  |  p  <Q  Q } ,  {
q  |  Q  <Q  q } >. ) )  /\  x  e.  ( 1st `  T ) ) )
166adantr 274 . . . . . . . 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 274 . . . . . . . 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 529 . . . . . . . 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 7448 . . . . . . 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 521 . . . . . . . . . 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 3937 . . . . . . . . . . . . . . . . 17  |-  ( u  =  y  ->  (
p  <Q  u  <->  p  <Q  y ) )
2221abbidv 2258 . . . . . . . . . . . . . . . 16  |-  ( u  =  y  ->  { p  |  p  <Q  u }  =  { p  |  p 
<Q  y } )
23 breq1 3936 . . . . . . . . . . . . . . . . 17  |-  ( u  =  y  ->  (
u  <Q  q  <->  y  <Q  q ) )
2423abbidv 2258 . . . . . . . . . . . . . . . 16  |-  ( u  =  y  ->  { q  |  u  <Q  q }  =  { q  |  y  <Q  q } )
2522, 24opeq12d 3717 . . . . . . . . . . . . . . 15  |-  ( u  =  y  ->  <. { p  |  p  <Q  u } ,  { q  |  u 
<Q  q } >.  =  <. { p  |  p  <Q  y } ,  { q  |  y  <Q  q } >. )
2625breq2d 3945 . . . . . . . . . . . . . 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 2439 . . . . . . . . . . . . 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 5428 . . . . . . . . . . . . . 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 7191 . . . . . . . . . . . . . . . 16  |-  Q.  e.  _V
3029rabex 4076 . . . . . . . . . . . . . . 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 4076 . . . . . . . . . . . . . . 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 6049 . . . . . . . . . . . . . 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 2161 . . . . . . . . . . . . 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 2844 . . . . . . . . . . . 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 119 . . . . . . . . . . 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 113 . . . . . . . . . 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 5425 . . . . . . . . . . . 12  |-  ( r  =  b  ->  ( F `  r )  =  ( F `  b ) )
39 opeq1 3709 . . . . . . . . . . . . . . . . 17  |-  ( r  =  b  ->  <. r ,  1o >.  =  <. b ,  1o >. )
4039eceq1d 6469 . . . . . . . . . . . . . . . 16  |-  ( r  =  b  ->  [ <. r ,  1o >. ]  ~Q  =  [ <. b ,  1o >. ]  ~Q  )
4140fveq2d 5429 . . . . . . . . . . . . . . 15  |-  ( r  =  b  ->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )
4241breq2d 3945 . . . . . . . . . . . . . 14  |-  ( r  =  b  ->  (
p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <->  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) )
4342abbidv 2258 . . . . . . . . . . . . 13  |-  ( r  =  b  ->  { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) }  =  { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } )
4441breq1d 3943 . . . . . . . . . . . . . 14  |-  ( r  =  b  ->  (
( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q ) )
4544abbidv 2258 . . . . . . . . . . . . 13  |-  ( r  =  b  ->  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q }  =  {
q  |  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )  <Q  q } )
4643, 45opeq12d 3717 . . . . . . . . . . . 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 5796 . . . . . . . . . . 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 3943 . . . . . . . . . 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 2656 . . . . . . . . 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 121 . . . . . . . 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 109 . . . . . . . . . . . . . . 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 7447 . . . . . . . . . . . . . . . . 17  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
f  <P  g  <->  ( h  +P.  f )  <P  (
h  +P.  g )
) )
5352adantl 275 . . . . . . . . . . . . . . . 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 486 . . . . . . . . . . . . . . . . . 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 520 . . . . . . . . . . . . . . . . . 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 5560 . . . . . . . . . . . . . . . . 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 7376 . . . . . . . . . . . . . . . . . 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 7365 . . . . . . . . . . . . . . . . 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 409 . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . 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 111 . . . . . . . . . . . . . . . . . 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 7375 . . . . . . . . . . . . . . . . 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 486 . . . . . . . . . . . . . . . 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 7406 . . . . . . . . . . . . . . . . 17  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
6867adantl 275 . . . . . . . . . . . . . . . 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 5944 . . . . . . . . . . . . . . 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 146 . . . . . . . . . . . . . 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 486 . . . . . . . . . . . . . . 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 7389 . . . . . . . . . . . . . . 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 409 . . . . . . . . . . . . . 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 3960 . . . . . . . . . . . . 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 526 . . . . . . . . . . . . . . 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 274 . . . . . . . . . . . . . 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 3937 . . . . . . . . . . . . . . . . 17  |-  ( ( y  +Q  Q )  =  x  ->  (
p  <Q  ( y  +Q  Q )  <->  p  <Q  x ) )
7877abbidv 2258 . . . . . . . . . . . . . . . 16  |-  ( ( y  +Q  Q )  =  x  ->  { p  |  p  <Q  ( y  +Q  Q ) }  =  { p  |  p  <Q  x }
)
79 breq1 3936 . . . . . . . . . . . . . . . . 17  |-  ( ( y  +Q  Q )  =  x  ->  (
( y  +Q  Q
)  <Q  q  <->  x  <Q  q ) )
8079abbidv 2258 . . . . . . . . . . . . . . . 16  |-  ( ( y  +Q  Q )  =  x  ->  { q  |  ( y  +Q  Q )  <Q  q }  =  { q  |  x  <Q  q } )
8178, 80opeq12d 3717 . . . . . . . . . . . . . . 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 3945 . . . . . . . . . . . . . 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 146 . . . . . . . . . . . 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 525 . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . 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 7365 . . . . . . . . . . . . . 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 409 . . . . . . . . . . . . 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 7380 . . . . . . . . . . . . 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 409 . . . . . . . . . . . 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 166 . . . . . . . . . . 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 530 . . . . . . . . . . . 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 484 . . . . . . . . . . 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 304 . . . . . . . . . 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 114 . . . . . . . . 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 2535 . . . . . . . 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 2555 . . . . . 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 373 . . . . 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 2535 . . . 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 2596 . . 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 121 . 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 5559 . . . . . 6  |-  ( (
ph  /\  b  e.  N. )  ->  ( F `
 b )  e. 
P. )
10557adantl 275 . . . . . 6  |-  ( (
ph  /\  b  e.  N. )  ->  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >.  e.  P. )
106104, 105, 59syl2anc 409 . . . . 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 274 . . . . 5  |-  ( (
ph  /\  b  e.  N. )  ->  <. { p  |  p  <Q  Q } ,  { q  |  Q  <Q  q } >.  e.  P. )
108106, 107, 87syl2anc 409 . . . 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 274 . . . 4  |-  ( (
ph  /\  b  e.  N. )  ->  T  e. 
P. )
110 ltdfpr 7334 . . . 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 409 . . 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 2435 . 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 166 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 103    <-> wb 104    /\ w3a 963    = wceq 1332    e. wcel 1481   {cab 2126   A.wral 2417   E.wrex 2418   {crab 2421   <.cop 3531   class class class wbr 3933   -->wf 5123   ` cfv 5127  (class class class)co 5778   1stc1st 6040   2ndc2nd 6041   1oc1o 6310   [cec 6431   N.cnpi 7100    <N clti 7103    ~Q ceq 7107   Q.cnq 7108    +Q cplq 7110   *Qcrq 7112    <Q cltq 7113   P.cnp 7119    +P. cpp 7121    <P cltp 7123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4047  ax-sep 4050  ax-nul 4058  ax-pow 4102  ax-pr 4135  ax-un 4359  ax-setind 4456  ax-iinf 4506
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2689  df-sbc 2911  df-csb 3005  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-nul 3365  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-int 3776  df-iun 3819  df-br 3934  df-opab 3994  df-mpt 3995  df-tr 4031  df-eprel 4215  df-id 4219  df-po 4222  df-iso 4223  df-iord 4292  df-on 4294  df-suc 4297  df-iom 4509  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555  df-ima 4556  df-iota 5092  df-fun 5129  df-fn 5130  df-f 5131  df-f1 5132  df-fo 5133  df-f1o 5134  df-fv 5135  df-ov 5781  df-oprab 5782  df-mpo 5783  df-1st 6042  df-2nd 6043  df-recs 6206  df-irdg 6271  df-1o 6317  df-2o 6318  df-oadd 6321  df-omul 6322  df-er 6433  df-ec 6435  df-qs 6439  df-ni 7132  df-pli 7133  df-mi 7134  df-lti 7135  df-plpq 7172  df-mpq 7173  df-enq 7175  df-nqqs 7176  df-plqqs 7177  df-mqqs 7178  df-1nqqs 7179  df-rq 7180  df-ltnqqs 7181  df-enq0 7252  df-nq0 7253  df-0nq0 7254  df-plq0 7255  df-mq0 7256  df-inp 7294  df-iplp 7296  df-iltp 7298
This theorem is referenced by:  caucvgprprlemexb  7535
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