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Theorem caucvgprprlemopl 6776
Description: Lemma for caucvgprpr 6791. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 21-Dec-2020.)
Hypotheses
Ref Expression
caucvgprpr.f  |-  ( ph  ->  F : N. --> P. )
caucvgprpr.cau  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <P  ( ( F `
 k )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( F `  k
)  <P  ( ( F `
 n )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )
) ) )
caucvgprpr.bnd  |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )
caucvgprpr.lim  |-  L  = 
<. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  r ) } ,  { u  e.  Q.  |  E. r  e.  N.  ( ( F `
 r )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  u } ,  { q  |  u 
<Q  q } >. } >.
Assertion
Ref Expression
caucvgprprlemopl  |-  ( (
ph  /\  s  e.  ( 1st `  L ) )  ->  E. t  e.  Q.  ( s  <Q 
t  /\  t  e.  ( 1st `  L ) ) )
Distinct variable groups:    A, m    m, F    F, l, t, r   
u, F, t    t, L    p, l, q, r, s, t    u, p, q, r, s    ph, r,
t
Allowed substitution hints:    ph( u, k, m, n, s, q, p, l)    A( u, t, k, n, s, r, q, p, l)    F( k, n, s, q, p)    L( u, k, m, n, s, r, q, p, l)

Proof of Theorem caucvgprprlemopl
Dummy variables  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 caucvgprpr.lim . . . . 5  |-  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 } >. } >.
21caucvgprprlemell 6764 . . . 4  |-  ( s  e.  ( 1st `  L
)  <->  ( s  e. 
Q.  /\  E. b  e.  N.  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P 
( F `  b
) ) )
32simprbi 260 . . 3  |-  ( s  e.  ( 1st `  L
)  ->  E. b  e.  N.  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P 
( F `  b
) )
43adantl 262 . 2  |-  ( (
ph  /\  s  e.  ( 1st `  L ) )  ->  E. b  e.  N.  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P 
( F `  b
) )
5 caucvgprpr.f . . . . . . 7  |-  ( ph  ->  F : N. --> P. )
65ad2antrr 457 . . . . . 6  |-  ( ( ( ph  /\  s  e.  ( 1st `  L
) )  /\  (
b  e.  N.  /\  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  ->  F : N. --> P. )
7 simprl 483 . . . . . 6  |-  ( ( ( ph  /\  s  e.  ( 1st `  L
) )  /\  (
b  e.  N.  /\  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  ->  b  e.  N. )
86, 7ffvelrnd 5290 . . . . 5  |-  ( ( ( ph  /\  s  e.  ( 1st `  L
) )  /\  (
b  e.  N.  /\  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  ->  ( F `  b )  e.  P. )
9 prop 6554 . . . . 5  |-  ( ( F `  b )  e.  P.  ->  <. ( 1st `  ( F `  b ) ) ,  ( 2nd `  ( F `  b )
) >.  e.  P. )
108, 9syl 14 . . . 4  |-  ( ( ( ph  /\  s  e.  ( 1st `  L
) )  /\  (
b  e.  N.  /\  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  ->  <. ( 1st `  ( F `  b ) ) ,  ( 2nd `  ( F `  b )
) >.  e.  P. )
11 simprr 484 . . . . 5  |-  ( ( ( ph  /\  s  e.  ( 1st `  L
) )  /\  (
b  e.  N.  /\  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  ->  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q 
q } >.  <P  ( F `  b )
)
121caucvgprprlemell 6764 . . . . . . . . 9  |-  ( s  e.  ( 1st `  L
)  <->  ( s  e. 
Q.  /\  E. r  e.  N.  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P 
( F `  r
) ) )
1312simplbi 259 . . . . . . . 8  |-  ( s  e.  ( 1st `  L
)  ->  s  e.  Q. )
1413ad2antlr 458 . . . . . . 7  |-  ( ( ( ph  /\  s  e.  ( 1st `  L
) )  /\  (
b  e.  N.  /\  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  ->  s  e.  Q. )
15 nnnq 6501 . . . . . . . . 9  |-  ( b  e.  N.  ->  [ <. b ,  1o >. ]  ~Q  e.  Q. )
16 recclnq 6471 . . . . . . . . 9  |-  ( [
<. b ,  1o >. ]  ~Q  e.  Q.  ->  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  e.  Q. )
1715, 16syl 14 . . . . . . . 8  |-  ( b  e.  N.  ->  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  e.  Q. )
1817ad2antrl 459 . . . . . . 7  |-  ( ( ( ph  /\  s  e.  ( 1st `  L
) )  /\  (
b  e.  N.  /\  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  ->  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  e.  Q. )
19 addclnq 6454 . . . . . . 7  |-  ( ( s  e.  Q.  /\  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  e.  Q. )
2014, 18, 19syl2anc 391 . . . . . 6  |-  ( ( ( ph  /\  s  e.  ( 1st `  L
) )  /\  (
b  e.  N.  /\  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  ->  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  e.  Q. )
21 nqprl 6630 . . . . . 6  |-  ( ( ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  e.  Q.  /\  ( F `  b )  e.  P. )  -> 
( ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  e.  ( 1st `  ( F `  b )
)  <->  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P 
( F `  b
) ) )
2220, 8, 21syl2anc 391 . . . . 5  |-  ( ( ( ph  /\  s  e.  ( 1st `  L
) )  /\  (
b  e.  N.  /\  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  ->  (
( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  e.  ( 1st `  ( F `  b
) )  <->  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q 
q } >.  <P  ( F `  b )
) )
2311, 22mpbird 156 . . . 4  |-  ( ( ( ph  /\  s  e.  ( 1st `  L
) )  /\  (
b  e.  N.  /\  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  ->  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  e.  ( 1st `  ( F `  b
) ) )
24 prnmaxl 6567 . . . 4  |-  ( (
<. ( 1st `  ( F `  b )
) ,  ( 2nd `  ( F `  b
) ) >.  e.  P.  /\  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  e.  ( 1st `  ( F `  b
) ) )  ->  E. a  e.  ( 1st `  ( F `  b ) ) ( s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a )
2510, 23, 24syl2anc 391 . . 3  |-  ( ( ( ph  /\  s  e.  ( 1st `  L
) )  /\  (
b  e.  N.  /\  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  ->  E. a  e.  ( 1st `  ( F `  b )
) ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q 
a )
2618adantr 261 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\ 
<. { p  |  p 
<Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  ->  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  e. 
Q. )
2714adantr 261 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\ 
<. { p  |  p 
<Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  ->  s  e.  Q. )
28 ltaddnq 6486 . . . . . . . 8  |-  ( ( ( *Q `  [ <. b ,  1o >. ]  ~Q  )  e.  Q.  /\  s  e.  Q. )  ->  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  (
( *Q `  [ <. b ,  1o >. ]  ~Q  )  +Q  s
) )
2926, 27, 28syl2anc 391 . . . . . . 7  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\ 
<. { p  |  p 
<Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  ->  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
( ( *Q `  [ <. b ,  1o >. ]  ~Q  )  +Q  s ) )
30 addcomnqg 6460 . . . . . . . 8  |-  ( ( ( *Q `  [ <. b ,  1o >. ]  ~Q  )  e.  Q.  /\  s  e.  Q. )  ->  ( ( *Q `  [ <. b ,  1o >. ]  ~Q  )  +Q  s )  =  ( s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) )
3126, 27, 30syl2anc 391 . . . . . . 7  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\ 
<. { p  |  p 
<Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  ->  ( ( *Q
`  [ <. b ,  1o >. ]  ~Q  )  +Q  s )  =  ( s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) )
3229, 31breqtrd 3785 . . . . . 6  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\ 
<. { p  |  p 
<Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  ->  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) )
33 simprr 484 . . . . . 6  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\ 
<. { p  |  p 
<Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  ->  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q 
a )
34 ltsonq 6477 . . . . . . 7  |-  <Q  Or  Q.
35 ltrelnq 6444 . . . . . . 7  |-  <Q  C_  ( Q.  X.  Q. )
3634, 35sotri 4707 . . . . . 6  |-  ( ( ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  /\  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q 
a )  ->  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  a )
3732, 33, 36syl2anc 391 . . . . 5  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\ 
<. { p  |  p 
<Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  ->  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
a )
3810adantr 261 . . . . . . 7  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\ 
<. { p  |  p 
<Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  ->  <. ( 1st `  ( F `  b )
) ,  ( 2nd `  ( F `  b
) ) >.  e.  P. )
39 simprl 483 . . . . . . 7  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\ 
<. { p  |  p 
<Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  ->  a  e.  ( 1st `  ( F `
 b ) ) )
40 elprnql 6560 . . . . . . 7  |-  ( (
<. ( 1st `  ( F `  b )
) ,  ( 2nd `  ( F `  b
) ) >.  e.  P.  /\  a  e.  ( 1st `  ( F `  b
) ) )  -> 
a  e.  Q. )
4138, 39, 40syl2anc 391 . . . . . 6  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\ 
<. { p  |  p 
<Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  ->  a  e.  Q. )
42 ltexnqq 6487 . . . . . 6  |-  ( ( ( *Q `  [ <. b ,  1o >. ]  ~Q  )  e.  Q.  /\  a  e.  Q. )  ->  ( ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
a  <->  E. t  e.  Q.  ( ( *Q `  [ <. b ,  1o >. ]  ~Q  )  +Q  t )  =  a ) )
4326, 41, 42syl2anc 391 . . . . 5  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\ 
<. { p  |  p 
<Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  ->  ( ( *Q
`  [ <. b ,  1o >. ]  ~Q  )  <Q  a  <->  E. t  e.  Q.  ( ( *Q `  [ <. b ,  1o >. ]  ~Q  )  +Q  t )  =  a ) )
4437, 43mpbid 135 . . . 4  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\ 
<. { p  |  p 
<Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  ->  E. t  e.  Q.  ( ( *Q `  [ <. b ,  1o >. ]  ~Q  )  +Q  t )  =  a )
4527ad2antrr 457 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\  <. { p  |  p  <Q  ( s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  /\  t  e.  Q. )  /\  ( ( *Q
`  [ <. b ,  1o >. ]  ~Q  )  +Q  t )  =  a )  ->  s  e.  Q. )
4626ad2antrr 457 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\  <. { p  |  p  <Q  ( s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  /\  t  e.  Q. )  /\  ( ( *Q
`  [ <. b ,  1o >. ]  ~Q  )  +Q  t )  =  a )  ->  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  e. 
Q. )
47 addcomnqg 6460 . . . . . . . . . . 11  |-  ( ( s  e.  Q.  /\  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  =  ( ( *Q `  [ <. b ,  1o >. ]  ~Q  )  +Q  s ) )
4845, 46, 47syl2anc 391 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\  <. { p  |  p  <Q  ( s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  /\  t  e.  Q. )  /\  ( ( *Q
`  [ <. b ,  1o >. ]  ~Q  )  +Q  t )  =  a )  ->  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  =  ( ( *Q `  [ <. b ,  1o >. ]  ~Q  )  +Q  s ) )
4933ad2antrr 457 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\  <. { p  |  p  <Q  ( s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  /\  t  e.  Q. )  /\  ( ( *Q
`  [ <. b ,  1o >. ]  ~Q  )  +Q  t )  =  a )  ->  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q 
a )
5048, 49eqbrtrrd 3783 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\  <. { p  |  p  <Q  ( s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  /\  t  e.  Q. )  /\  ( ( *Q
`  [ <. b ,  1o >. ]  ~Q  )  +Q  t )  =  a )  ->  ( ( *Q `  [ <. b ,  1o >. ]  ~Q  )  +Q  s )  <Q  a
)
51 simpr 103 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\  <. { p  |  p  <Q  ( s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  /\  t  e.  Q. )  /\  ( ( *Q
`  [ <. b ,  1o >. ]  ~Q  )  +Q  t )  =  a )  ->  ( ( *Q `  [ <. b ,  1o >. ]  ~Q  )  +Q  t )  =  a )
5250, 51breqtrrd 3787 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\  <. { p  |  p  <Q  ( s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  /\  t  e.  Q. )  /\  ( ( *Q
`  [ <. b ,  1o >. ]  ~Q  )  +Q  t )  =  a )  ->  ( ( *Q `  [ <. b ,  1o >. ]  ~Q  )  +Q  s )  <Q  (
( *Q `  [ <. b ,  1o >. ]  ~Q  )  +Q  t
) )
53 simplr 482 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\  <. { p  |  p  <Q  ( s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  /\  t  e.  Q. )  /\  ( ( *Q
`  [ <. b ,  1o >. ]  ~Q  )  +Q  t )  =  a )  ->  t  e.  Q. )
54 ltanqg 6479 . . . . . . . . 9  |-  ( ( s  e.  Q.  /\  t  e.  Q.  /\  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  e.  Q. )  ->  (
s  <Q  t  <->  ( ( *Q `  [ <. b ,  1o >. ]  ~Q  )  +Q  s )  <Q  (
( *Q `  [ <. b ,  1o >. ]  ~Q  )  +Q  t
) ) )
5545, 53, 46, 54syl3anc 1135 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\  <. { p  |  p  <Q  ( s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  /\  t  e.  Q. )  /\  ( ( *Q
`  [ <. b ,  1o >. ]  ~Q  )  +Q  t )  =  a )  ->  ( s  <Q  t  <->  ( ( *Q
`  [ <. b ,  1o >. ]  ~Q  )  +Q  s )  <Q  (
( *Q `  [ <. b ,  1o >. ]  ~Q  )  +Q  t
) ) )
5652, 55mpbird 156 . . . . . . 7  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\  <. { p  |  p  <Q  ( s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  /\  t  e.  Q. )  /\  ( ( *Q
`  [ <. b ,  1o >. ]  ~Q  )  +Q  t )  =  a )  ->  s  <Q  t )
577ad3antrrr 461 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\  <. { p  |  p  <Q  ( s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  /\  t  e.  Q. )  /\  ( ( *Q
`  [ <. b ,  1o >. ]  ~Q  )  +Q  t )  =  a )  ->  b  e.  N. )
58 addcomnqg 6460 . . . . . . . . . . . . 13  |-  ( ( ( *Q `  [ <. b ,  1o >. ]  ~Q  )  e.  Q.  /\  t  e.  Q. )  ->  ( ( *Q `  [ <. b ,  1o >. ]  ~Q  )  +Q  t )  =  ( t  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) )
5946, 53, 58syl2anc 391 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\  <. { p  |  p  <Q  ( s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  /\  t  e.  Q. )  /\  ( ( *Q
`  [ <. b ,  1o >. ]  ~Q  )  +Q  t )  =  a )  ->  ( ( *Q `  [ <. b ,  1o >. ]  ~Q  )  +Q  t )  =  ( t  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) )
6059, 51eqtr3d 2074 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\  <. { p  |  p  <Q  ( s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  /\  t  e.  Q. )  /\  ( ( *Q
`  [ <. b ,  1o >. ]  ~Q  )  +Q  t )  =  a )  ->  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  =  a )
6139ad2antrr 457 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\  <. { p  |  p  <Q  ( s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  /\  t  e.  Q. )  /\  ( ( *Q
`  [ <. b ,  1o >. ]  ~Q  )  +Q  t )  =  a )  ->  a  e.  ( 1st `  ( F `
 b ) ) )
6260, 61eqeltrd 2114 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\  <. { p  |  p  <Q  ( s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  /\  t  e.  Q. )  /\  ( ( *Q
`  [ <. b ,  1o >. ]  ~Q  )  +Q  t )  =  a )  ->  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  e.  ( 1st `  ( F `  b )
) )
63 addclnq 6454 . . . . . . . . . . . 12  |-  ( ( t  e.  Q.  /\  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  e.  Q. )
6453, 46, 63syl2anc 391 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\  <. { p  |  p  <Q  ( s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  /\  t  e.  Q. )  /\  ( ( *Q
`  [ <. b ,  1o >. ]  ~Q  )  +Q  t )  =  a )  ->  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  e. 
Q. )
658ad3antrrr 461 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\  <. { p  |  p  <Q  ( s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  /\  t  e.  Q. )  /\  ( ( *Q
`  [ <. b ,  1o >. ]  ~Q  )  +Q  t )  =  a )  ->  ( F `  b )  e.  P. )
66 nqprl 6630 . . . . . . . . . . 11  |-  ( ( ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  e.  Q.  /\  ( F `  b )  e.  P. )  -> 
( ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  e.  ( 1st `  ( F `  b )
)  <->  <. { p  |  p  <Q  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P 
( F `  b
) ) )
6764, 65, 66syl2anc 391 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\  <. { p  |  p  <Q  ( s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  /\  t  e.  Q. )  /\  ( ( *Q
`  [ <. b ,  1o >. ]  ~Q  )  +Q  t )  =  a )  ->  ( (
t  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  e.  ( 1st `  ( F `  b
) )  <->  <. { p  |  p  <Q  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q 
q } >.  <P  ( F `  b )
) )
6862, 67mpbid 135 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\  <. { p  |  p  <Q  ( s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  /\  t  e.  Q. )  /\  ( ( *Q
`  [ <. b ,  1o >. ]  ~Q  )  +Q  t )  =  a )  ->  <. { p  |  p  <Q  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q 
q } >.  <P  ( F `  b )
)
69 opeq1 3546 . . . . . . . . . . . . . . . . 17  |-  ( r  =  b  ->  <. r ,  1o >.  =  <. b ,  1o >. )
7069eceq1d 6129 . . . . . . . . . . . . . . . 16  |-  ( r  =  b  ->  [ <. r ,  1o >. ]  ~Q  =  [ <. b ,  1o >. ]  ~Q  )
7170fveq2d 5169 . . . . . . . . . . . . . . 15  |-  ( r  =  b  ->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )
7271oveq2d 5515 . . . . . . . . . . . . . 14  |-  ( r  =  b  ->  (
t  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
)  =  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) )
7372breq2d 3773 . . . . . . . . . . . . 13  |-  ( r  =  b  ->  (
p  <Q  ( t  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <->  p  <Q  ( t  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) ) )
7473abbidv 2155 . . . . . . . . . . . 12  |-  ( r  =  b  ->  { p  |  p  <Q  ( t  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) }  =  { p  |  p  <Q  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) } )
7572breq1d 3771 . . . . . . . . . . . . 13  |-  ( r  =  b  ->  (
( t  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  <Q  q  <->  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q 
q ) )
7675abbidv 2155 . . . . . . . . . . . 12  |-  ( r  =  b  ->  { q  |  ( t  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q }  =  {
q  |  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } )
7774, 76opeq12d 3554 . . . . . . . . . . 11  |-  ( r  =  b  ->  <. { p  |  p  <Q  ( t  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( t  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q } >.  =  <. { p  |  p  <Q  ( t  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >. )
78 fveq2 5165 . . . . . . . . . . 11  |-  ( r  =  b  ->  ( F `  r )  =  ( F `  b ) )
7977, 78breq12d 3774 . . . . . . . . . 10  |-  ( r  =  b  ->  ( <. { p  |  p 
<Q  ( t  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( t  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  r )  <->  <. { p  |  p  <Q  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )
8079rspcev 2653 . . . . . . . . 9  |-  ( ( b  e.  N.  /\  <. { p  |  p  <Q  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
)  ->  E. r  e.  N.  <. { p  |  p  <Q  ( t  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( t  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P 
( F `  r
) )
8157, 68, 80syl2anc 391 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\  <. { p  |  p  <Q  ( s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  /\  t  e.  Q. )  /\  ( ( *Q
`  [ <. b ,  1o >. ]  ~Q  )  +Q  t )  =  a )  ->  E. r  e.  N.  <. { p  |  p  <Q  ( t  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( t  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P 
( F `  r
) )
821caucvgprprlemell 6764 . . . . . . . 8  |-  ( t  e.  ( 1st `  L
)  <->  ( t  e. 
Q.  /\  E. r  e.  N.  <. { p  |  p  <Q  ( t  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( t  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P 
( F `  r
) ) )
8353, 81, 82sylanbrc 394 . . . . . . 7  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\  <. { p  |  p  <Q  ( s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  /\  t  e.  Q. )  /\  ( ( *Q
`  [ <. b ,  1o >. ]  ~Q  )  +Q  t )  =  a )  ->  t  e.  ( 1st `  L ) )
8456, 83jca 290 . . . . . 6  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\  <. { p  |  p  <Q  ( s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  /\  t  e.  Q. )  /\  ( ( *Q
`  [ <. b ,  1o >. ]  ~Q  )  +Q  t )  =  a )  ->  ( s  <Q  t  /\  t  e.  ( 1st `  L
) ) )
8584ex 108 . . . . 5  |-  ( ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\ 
<. { p  |  p 
<Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  /\  t  e.  Q. )  ->  ( ( ( *Q `  [ <. b ,  1o >. ]  ~Q  )  +Q  t )  =  a  ->  ( s  <Q  t  /\  t  e.  ( 1st `  L
) ) ) )
8685reximdva 2418 . . . 4  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\ 
<. { p  |  p 
<Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  ->  ( E. t  e.  Q.  ( ( *Q
`  [ <. b ,  1o >. ]  ~Q  )  +Q  t )  =  a  ->  E. t  e.  Q.  ( s  <Q  t  /\  t  e.  ( 1st `  L ) ) ) )
8744, 86mpd 13 . . 3  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( b  e.  N.  /\ 
<. { p  |  p 
<Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  /\  (
a  e.  ( 1st `  ( F `  b
) )  /\  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  a ) )  ->  E. t  e.  Q.  ( s  <Q  t  /\  t  e.  ( 1st `  L ) ) )
8825, 87rexlimddv 2434 . 2  |-  ( ( ( ph  /\  s  e.  ( 1st `  L
) )  /\  (
b  e.  N.  /\  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )  ->  E. t  e.  Q.  ( s  <Q 
t  /\  t  e.  ( 1st `  L ) ) )
894, 88rexlimddv 2434 1  |-  ( (
ph  /\  s  e.  ( 1st `  L ) )  ->  E. t  e.  Q.  ( s  <Q 
t  /\  t  e.  ( 1st `  L ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    = wceq 1243    e. wcel 1393   {cab 2026   A.wral 2303   E.wrex 2304   {crab 2307   <.cop 3375   class class class wbr 3761   -->wf 4885   ` cfv 4889  (class class class)co 5499   1stc1st 5752   2ndc2nd 5753   1oc1o 5981   [cec 6091   N.cnpi 6351    <N clti 6354    ~Q ceq 6358   Q.cnq 6359    +Q cplq 6361   *Qcrq 6363    <Q cltq 6364   P.cnp 6370    +P. cpp 6372    <P cltp 6374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3869  ax-sep 3872  ax-nul 3880  ax-pow 3924  ax-pr 3941  ax-un 4166  ax-setind 4256  ax-iinf 4298
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2308  df-rex 2309  df-reu 2310  df-rab 2312  df-v 2556  df-sbc 2762  df-csb 2850  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-int 3613  df-iun 3656  df-br 3762  df-opab 3816  df-mpt 3817  df-tr 3852  df-eprel 4023  df-id 4027  df-po 4030  df-iso 4031  df-iord 4099  df-on 4101  df-suc 4104  df-iom 4301  df-xp 4338  df-rel 4339  df-cnv 4340  df-co 4341  df-dm 4342  df-rn 4343  df-res 4344  df-ima 4345  df-iota 4854  df-fun 4891  df-fn 4892  df-f 4893  df-f1 4894  df-fo 4895  df-f1o 4896  df-fv 4897  df-ov 5502  df-oprab 5503  df-mpt2 5504  df-1st 5754  df-2nd 5755  df-recs 5907  df-irdg 5944  df-1o 5988  df-oadd 5992  df-omul 5993  df-er 6093  df-ec 6095  df-qs 6099  df-ni 6383  df-pli 6384  df-mi 6385  df-lti 6386  df-plpq 6423  df-mpq 6424  df-enq 6426  df-nqqs 6427  df-plqqs 6428  df-mqqs 6429  df-1nqqs 6430  df-rq 6431  df-ltnqqs 6432  df-inp 6545  df-iltp 6549
This theorem is referenced by:  caucvgprprlemrnd  6780
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