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Theorem caucvgprprlemopl 7254
Description: Lemma for caucvgprpr 7269. 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 7242 . . . 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 5435 . . . . 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 7032 . . . . 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 7242 . . . . . . . . 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 6979 . . . . . . . . 9  |-  ( b  e.  N.  ->  [ <. b ,  1o >. ]  ~Q  e.  Q. )
16 recclnq 6949 . . . . . . . . 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 6932 . . . . . . 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 7108 . . . . . 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 7045 . . . 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 6964 . . . . . . . 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 6938 . . . . . . . 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 3869 . . . . . 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 6955 . . . . . . 7  |-  <Q  Or  Q.
35 ltrelnq 6922 . . . . . . 7  |-  <Q  C_  ( Q.  X.  Q. )
3634, 35sotri 4827 . . . . . 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 7038 . . . . . . 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 6965 . . . . . 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 6938 . . . . . . . . . . 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 3867 . . . . . . . . 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 3871 . . . . . . . 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 6957 . . . . . . . . 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 1174 . . . . . . . 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 6938 . . . . . . . . . . . . 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 2122 . . . . . . . . . . 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 2164 . . . . . . . . . 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 6932 . . . . . . . . . . . 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 7108 . . . . . . . . . . 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 3622 . . . . . . . . . . . . . . . . 17  |-  ( r  =  b  ->  <. r ,  1o >.  =  <. b ,  1o >. )
7069eceq1d 6326 . . . . . . . . . . . . . . . 16  |-  ( r  =  b  ->  [ <. r ,  1o >. ]  ~Q  =  [ <. b ,  1o >. ]  ~Q  )
7170fveq2d 5309 . . . . . . . . . . . . . . 15  |-  ( r  =  b  ->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )
7271oveq2d 5668 . . . . . . . . . . . . . 14  |-  ( r  =  b  ->  (
t  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
)  =  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) )
7372breq2d 3857 . . . . . . . . . . . . 13  |-  ( r  =  b  ->  (
p  <Q  ( t  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <->  p  <Q  ( t  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) ) )
7473abbidv 2205 . . . . . . . . . . . 12  |-  ( r  =  b  ->  { p  |  p  <Q  ( t  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) }  =  { p  |  p  <Q  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) } )
7572breq1d 3855 . . . . . . . . . . . . 13  |-  ( r  =  b  ->  (
( t  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  <Q  q  <->  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q 
q ) )
7675abbidv 2205 . . . . . . . . . . . 12  |-  ( r  =  b  ->  { q  |  ( t  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q }  =  {
q  |  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } )
7774, 76opeq12d 3630 . . . . . . . . . . 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 5305 . . . . . . . . . . 11  |-  ( r  =  b  ->  ( F `  r )  =  ( F `  b ) )
7977, 78breq12d 3858 . . . . . . . . . 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 2722 . . . . . . . . 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 7242 . . . . . . . 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 2475 . . . 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 2493 . 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 2493 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 1289    e. wcel 1438   {cab 2074   A.wral 2359   E.wrex 2360   {crab 2363   <.cop 3449   class class class wbr 3845   -->wf 5011   ` cfv 5015  (class class class)co 5652   1stc1st 5909   2ndc2nd 5910   1oc1o 6174   [cec 6288   N.cnpi 6829    <N clti 6832    ~Q ceq 6836   Q.cnq 6837    +Q cplq 6839   *Qcrq 6841    <Q cltq 6842   P.cnp 6848    +P. cpp 6850    <P cltp 6852
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-eprel 4116  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-1o 6181  df-oadd 6185  df-omul 6186  df-er 6290  df-ec 6292  df-qs 6296  df-ni 6861  df-pli 6862  df-mi 6863  df-lti 6864  df-plpq 6901  df-mpq 6902  df-enq 6904  df-nqqs 6905  df-plqqs 6906  df-mqqs 6907  df-1nqqs 6908  df-rq 6909  df-ltnqqs 6910  df-inp 7023  df-iltp 7027
This theorem is referenced by:  caucvgprprlemrnd  7258
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