ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  caucvgprprlemopl Unicode version

Theorem caucvgprprlemopl 7498
Description: Lemma for caucvgprpr 7513. 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 7486 . . . 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 273 . . 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 275 . 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 479 . . . . . 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 520 . . . . . 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 5549 . . . . 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 7276 . . . . 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 521 . . . . 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 7486 . . . . . . . . 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 272 . . . . . . . 8  |-  ( s  e.  ( 1st `  L
)  ->  s  e.  Q. )
1413ad2antlr 480 . . . . . . 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 7223 . . . . . . . . 9  |-  ( b  e.  N.  ->  [ <. b ,  1o >. ]  ~Q  e.  Q. )
16 recclnq 7193 . . . . . . . . 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 481 . . . . . . 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 7176 . . . . . . 7  |-  ( ( s  e.  Q.  /\  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  e.  Q. )
2014, 18, 19syl2anc 408 . . . . . 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 7352 . . . . . 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 408 . . . . 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 166 . . . 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 7289 . . . 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 408 . . 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 274 . . . . . . . 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 274 . . . . . . . 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 7208 . . . . . . . 8  |-  ( ( ( *Q `  [ <. b ,  1o >. ]  ~Q  )  e.  Q.  /\  s  e.  Q. )  ->  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  (
( *Q `  [ <. b ,  1o >. ]  ~Q  )  +Q  s
) )
2926, 27, 28syl2anc 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 ) )  ->  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
( ( *Q `  [ <. b ,  1o >. ]  ~Q  )  +Q  s ) )
30 addcomnqg 7182 . . . . . . . 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 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 ) )  ->  ( ( *Q
`  [ <. b ,  1o >. ]  ~Q  )  +Q  s )  =  ( s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) )
3229, 31breqtrd 3949 . . . . . 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 521 . . . . . 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 7199 . . . . . . 7  |-  <Q  Or  Q.
35 ltrelnq 7166 . . . . . . 7  |-  <Q  C_  ( Q.  X.  Q. )
3634, 35sotri 4929 . . . . . 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 408 . . . . 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 274 . . . . . . 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 520 . . . . . . 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 7282 . . . . . . 7  |-  ( (
<. ( 1st `  ( F `  b )
) ,  ( 2nd `  ( F `  b
) ) >.  e.  P.  /\  a  e.  ( 1st `  ( F `  b
) ) )  -> 
a  e.  Q. )
4138, 39, 40syl2anc 408 . . . . . 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 7209 . . . . . 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 408 . . . . 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 146 . . . 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 479 . . . . . . . . . . 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 479 . . . . . . . . . . 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 7182 . . . . . . . . . . 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 408 . . . . . . . . . 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 479 . . . . . . . . . 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 3947 . . . . . . . . 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 109 . . . . . . . . 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 3951 . . . . . . . 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 519 . . . . . . . . 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 7201 . . . . . . . . 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 1216 . . . . . . . 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 166 . . . . . . 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 483 . . . . . . . . 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 7182 . . . . . . . . . . . . 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 408 . . . . . . . . . . . 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 2172 . . . . . . . . . . 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 479 . . . . . . . . . . 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 2214 . . . . . . . . . 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 7176 . . . . . . . . . . . 12  |-  ( ( t  e.  Q.  /\  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  e.  Q. )
6453, 46, 63syl2anc 408 . . . . . . . . . . 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 483 . . . . . . . . . . 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 7352 . . . . . . . . . . 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 408 . . . . . . . . . 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 146 . . . . . . . . 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 3700 . . . . . . . . . . . . . . . . 17  |-  ( r  =  b  ->  <. r ,  1o >.  =  <. b ,  1o >. )
7069eceq1d 6458 . . . . . . . . . . . . . . . 16  |-  ( r  =  b  ->  [ <. r ,  1o >. ]  ~Q  =  [ <. b ,  1o >. ]  ~Q  )
7170fveq2d 5418 . . . . . . . . . . . . . . 15  |-  ( r  =  b  ->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )
7271oveq2d 5783 . . . . . . . . . . . . . 14  |-  ( r  =  b  ->  (
t  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
)  =  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) )
7372breq2d 3936 . . . . . . . . . . . . 13  |-  ( r  =  b  ->  (
p  <Q  ( t  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <->  p  <Q  ( t  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) ) )
7473abbidv 2255 . . . . . . . . . . . 12  |-  ( r  =  b  ->  { p  |  p  <Q  ( t  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) }  =  { p  |  p  <Q  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) } )
7572breq1d 3934 . . . . . . . . . . . . 13  |-  ( r  =  b  ->  (
( t  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  <Q  q  <->  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q 
q ) )
7675abbidv 2255 . . . . . . . . . . . 12  |-  ( r  =  b  ->  { q  |  ( t  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q }  =  {
q  |  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } )
7774, 76opeq12d 3708 . . . . . . . . . . 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 5414 . . . . . . . . . . 11  |-  ( r  =  b  ->  ( F `  r )  =  ( F `  b ) )
7977, 78breq12d 3937 . . . . . . . . . 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 2784 . . . . . . . . 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 408 . . . . . . . 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 7486 . . . . . . . 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 413 . . . . . . 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 304 . . . . . 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 114 . . . . 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 2532 . . . 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 2552 . 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 2552 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 103    <-> wb 104    = wceq 1331    e. wcel 1480   {cab 2123   A.wral 2414   E.wrex 2415   {crab 2418   <.cop 3525   class class class wbr 3924   -->wf 5114   ` cfv 5118  (class class class)co 5767   1stc1st 6029   2ndc2nd 6030   1oc1o 6299   [cec 6420   N.cnpi 7073    <N clti 7076    ~Q ceq 7080   Q.cnq 7081    +Q cplq 7083   *Qcrq 7085    <Q cltq 7086   P.cnp 7092    +P. cpp 7094    <P cltp 7096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-pli 7106  df-mi 7107  df-lti 7108  df-plpq 7145  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-plqqs 7150  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154  df-inp 7267  df-iltp 7271
This theorem is referenced by:  caucvgprprlemrnd  7502
  Copyright terms: Public domain W3C validator