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Theorem caucvgprprlemopl 6985
Description: Lemma for caucvgprpr 7000. 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 6973 . . . 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 269 . . 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 271 . 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 472 . . . . . 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 498 . . . . . 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 5356 . . . . 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 6763 . . . . 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 499 . . . . 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 6973 . . . . . . . . 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 268 . . . . . . . 8  |-  ( s  e.  ( 1st `  L
)  ->  s  e.  Q. )
1413ad2antlr 473 . . . . . . 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 6710 . . . . . . . . 9  |-  ( b  e.  N.  ->  [ <. b ,  1o >. ]  ~Q  e.  Q. )
16 recclnq 6680 . . . . . . . . 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 474 . . . . . . 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 6663 . . . . . . 7  |-  ( ( s  e.  Q.  /\  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  e.  Q. )
2014, 18, 19syl2anc 403 . . . . . 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 6839 . . . . . 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 403 . . . . 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 165 . . . 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 6776 . . . 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 403 . . 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 270 . . . . . . . 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 270 . . . . . . . 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 6695 . . . . . . . 8  |-  ( ( ( *Q `  [ <. b ,  1o >. ]  ~Q  )  e.  Q.  /\  s  e.  Q. )  ->  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  (
( *Q `  [ <. b ,  1o >. ]  ~Q  )  +Q  s
) )
2926, 27, 28syl2anc 403 . . . . . . 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 6669 . . . . . . . 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 403 . . . . . . 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 3830 . . . . . 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 499 . . . . . 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 6686 . . . . . . 7  |-  <Q  Or  Q.
35 ltrelnq 6653 . . . . . . 7  |-  <Q  C_  ( Q.  X.  Q. )
3634, 35sotri 4771 . . . . . 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 403 . . . . 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 270 . . . . . . 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 498 . . . . . . 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 6769 . . . . . . 7  |-  ( (
<. ( 1st `  ( F `  b )
) ,  ( 2nd `  ( F `  b
) ) >.  e.  P.  /\  a  e.  ( 1st `  ( F `  b
) ) )  -> 
a  e.  Q. )
4138, 39, 40syl2anc 403 . . . . . 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 6696 . . . . . 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 403 . . . . 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 145 . . . 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 472 . . . . . . . . . . 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 472 . . . . . . . . . . 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 6669 . . . . . . . . . . 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 403 . . . . . . . . . 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 472 . . . . . . . . . 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 3828 . . . . . . . . 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 108 . . . . . . . . 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 3832 . . . . . . . 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 497 . . . . . . . . 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 6688 . . . . . . . . 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 1170 . . . . . . . 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 165 . . . . . . 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 476 . . . . . . . . 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 6669 . . . . . . . . . . . . 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 403 . . . . . . . . . . . 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 2117 . . . . . . . . . . 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 472 . . . . . . . . . . 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 2159 . . . . . . . . . 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 6663 . . . . . . . . . . . 12  |-  ( ( t  e.  Q.  /\  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  e.  Q. )
6453, 46, 63syl2anc 403 . . . . . . . . . . 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 476 . . . . . . . . . . 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 6839 . . . . . . . . . . 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 403 . . . . . . . . . 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 145 . . . . . . . . 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 3591 . . . . . . . . . . . . . . . . 17  |-  ( r  =  b  ->  <. r ,  1o >.  =  <. b ,  1o >. )
7069eceq1d 6230 . . . . . . . . . . . . . . . 16  |-  ( r  =  b  ->  [ <. r ,  1o >. ]  ~Q  =  [ <. b ,  1o >. ]  ~Q  )
7170fveq2d 5234 . . . . . . . . . . . . . . 15  |-  ( r  =  b  ->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )
7271oveq2d 5580 . . . . . . . . . . . . . 14  |-  ( r  =  b  ->  (
t  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
)  =  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) )
7372breq2d 3818 . . . . . . . . . . . . 13  |-  ( r  =  b  ->  (
p  <Q  ( t  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <->  p  <Q  ( t  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) ) )
7473abbidv 2200 . . . . . . . . . . . 12  |-  ( r  =  b  ->  { p  |  p  <Q  ( t  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) }  =  { p  |  p  <Q  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) } )
7572breq1d 3816 . . . . . . . . . . . . 13  |-  ( r  =  b  ->  (
( t  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  <Q  q  <->  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q 
q ) )
7675abbidv 2200 . . . . . . . . . . . 12  |-  ( r  =  b  ->  { q  |  ( t  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q }  =  {
q  |  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } )
7774, 76opeq12d 3599 . . . . . . . . . . 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 5230 . . . . . . . . . . 11  |-  ( r  =  b  ->  ( F `  r )  =  ( F `  b ) )
7977, 78breq12d 3819 . . . . . . . . . 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 2710 . . . . . . . . 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 403 . . . . . . . 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 6973 . . . . . . . 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 408 . . . . . . 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 300 . . . . . 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 113 . . . . 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 2468 . . . 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 2486 . 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 2486 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 102    <-> wb 103    = wceq 1285    e. wcel 1434   {cab 2069   A.wral 2353   E.wrex 2354   {crab 2357   <.cop 3420   class class class wbr 3806   -->wf 4949   ` cfv 4953  (class class class)co 5564   1stc1st 5817   2ndc2nd 5818   1oc1o 6079   [cec 6192   N.cnpi 6560    <N clti 6563    ~Q ceq 6567   Q.cnq 6568    +Q cplq 6570   *Qcrq 6572    <Q cltq 6573   P.cnp 6579    +P. cpp 6581    <P cltp 6583
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3914  ax-sep 3917  ax-nul 3925  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-iinf 4358
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2826  df-csb 2919  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-nul 3269  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-int 3658  df-iun 3701  df-br 3807  df-opab 3861  df-mpt 3862  df-tr 3897  df-eprel 4073  df-id 4077  df-po 4080  df-iso 4081  df-iord 4150  df-on 4152  df-suc 4155  df-iom 4361  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fn 4956  df-f 4957  df-f1 4958  df-fo 4959  df-f1o 4960  df-fv 4961  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819  df-2nd 5820  df-recs 5975  df-irdg 6040  df-1o 6086  df-oadd 6090  df-omul 6091  df-er 6194  df-ec 6196  df-qs 6200  df-ni 6592  df-pli 6593  df-mi 6594  df-lti 6595  df-plpq 6632  df-mpq 6633  df-enq 6635  df-nqqs 6636  df-plqqs 6637  df-mqqs 6638  df-1nqqs 6639  df-rq 6640  df-ltnqqs 6641  df-inp 6754  df-iltp 6758
This theorem is referenced by:  caucvgprprlemrnd  6989
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