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Theorem caucvgprprlemcl 7524
Description: Lemma for caucvgprpr 7532. The putative limit is a positive real. (Contributed by Jim Kingdon, 21-Nov-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
caucvgprprlemcl  |-  ( ph  ->  L  e.  P. )
Distinct variable groups:    A, m    m, F    A, r    F, l, u, r, k    n, F, k    k, L    ph, r    u, l, p, q, r   
m, r    k, p, q, r    u, n, l, k
Allowed substitution hints:    ph( u, k, m, n, q, p, l)    A( u, k, n, q, p, l)    F( q, p)    L( u, m, n, r, q, p, l)

Proof of Theorem caucvgprprlemcl
Dummy variables  s  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 caucvgprpr.f . . . 4  |-  ( ph  ->  F : N. --> P. )
2 caucvgprpr.cau . . . 4  |-  ( 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 } >. )
) ) )
3 caucvgprpr.bnd . . . 4  |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )
4 caucvgprpr.lim . . . 4  |-  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 } >. } >.
51, 2, 3, 4caucvgprprlemm 7516 . . 3  |-  ( ph  ->  ( E. s  e. 
Q.  s  e.  ( 1st `  L )  /\  E. t  e. 
Q.  t  e.  ( 2nd `  L ) ) )
6 ssrab2 3182 . . . . . 6  |-  { 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 ) }  C_  Q.
7 nqex 7183 . . . . . . 7  |-  Q.  e.  _V
87elpw2 4082 . . . . . 6  |-  ( { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  r ) }  e.  ~P Q.  <->  { 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 ) }  C_  Q. )
96, 8mpbir 145 . . . . 5  |-  { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q } >.  <P  ( F `  r ) }  e.  ~P Q.
10 ssrab2 3182 . . . . . 6  |-  { 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 } >. }  C_  Q.
117elpw2 4082 . . . . . 6  |-  ( { u  e.  Q.  |  E. r  e.  N.  ( ( F `  r )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  u } ,  {
q  |  u  <Q  q } >. }  e.  ~P Q. 
<->  { u  e.  Q.  |  E. r  e.  N.  ( ( F `  r )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  u } ,  {
q  |  u  <Q  q } >. }  C_  Q. )
1210, 11mpbir 145 . . . . 5  |-  { u  e.  Q.  |  E. r  e.  N.  ( ( F `
 r )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  u } ,  { q  |  u 
<Q  q } >. }  e.  ~P Q.
13 opelxpi 4571 . . . . 5  |-  ( ( { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  r ) }  e.  ~P Q.  /\ 
{ u  e.  Q.  |  E. r  e.  N.  ( ( F `  r )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  u } ,  {
q  |  u  <Q  q } >. }  e.  ~P Q. )  ->  <. { 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 } >. } >.  e.  ( ~P Q.  X.  ~P Q. ) )
149, 12, 13mp2an 422 . . . 4  |-  <. { 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 } >. } >.  e.  ( ~P Q.  X.  ~P Q. )
154, 14eqeltri 2212 . . 3  |-  L  e.  ( ~P Q.  X.  ~P Q. )
165, 15jctil 310 . 2  |-  ( ph  ->  ( L  e.  ( ~P Q.  X.  ~P Q. )  /\  ( E. s  e.  Q.  s  e.  ( 1st `  L )  /\  E. t  e.  Q.  t  e.  ( 2nd `  L
) ) ) )
171, 2, 3, 4caucvgprprlemrnd 7521 . . 3  |-  ( ph  ->  ( A. s  e. 
Q.  ( s  e.  ( 1st `  L
)  <->  E. t  e.  Q.  ( s  <Q  t  /\  t  e.  ( 1st `  L ) ) )  /\  A. t  e.  Q.  ( t  e.  ( 2nd `  L
)  <->  E. s  e.  Q.  ( s  <Q  t  /\  s  e.  ( 2nd `  L ) ) ) ) )
181, 2, 3, 4caucvgprprlemdisj 7522 . . 3  |-  ( ph  ->  A. s  e.  Q.  -.  ( s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L ) ) )
191, 2, 3, 4caucvgprprlemloc 7523 . . 3  |-  ( ph  ->  A. s  e.  Q.  A. t  e.  Q.  (
s  <Q  t  ->  (
s  e.  ( 1st `  L )  \/  t  e.  ( 2nd `  L
) ) ) )
2017, 18, 193jca 1161 . 2  |-  ( ph  ->  ( ( A. s  e.  Q.  ( s  e.  ( 1st `  L
)  <->  E. t  e.  Q.  ( s  <Q  t  /\  t  e.  ( 1st `  L ) ) )  /\  A. t  e.  Q.  ( t  e.  ( 2nd `  L
)  <->  E. s  e.  Q.  ( s  <Q  t  /\  s  e.  ( 2nd `  L ) ) ) )  /\  A. s  e.  Q.  -.  (
s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L
) )  /\  A. s  e.  Q.  A. t  e.  Q.  ( s  <Q 
t  ->  ( s  e.  ( 1st `  L
)  \/  t  e.  ( 2nd `  L
) ) ) ) )
21 elnp1st2nd 7296 . 2  |-  ( L  e.  P.  <->  ( ( L  e.  ( ~P Q.  X.  ~P Q. )  /\  ( E. s  e. 
Q.  s  e.  ( 1st `  L )  /\  E. t  e. 
Q.  t  e.  ( 2nd `  L ) ) )  /\  (
( A. s  e. 
Q.  ( s  e.  ( 1st `  L
)  <->  E. t  e.  Q.  ( s  <Q  t  /\  t  e.  ( 1st `  L ) ) )  /\  A. t  e.  Q.  ( t  e.  ( 2nd `  L
)  <->  E. s  e.  Q.  ( s  <Q  t  /\  s  e.  ( 2nd `  L ) ) ) )  /\  A. s  e.  Q.  -.  (
s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L
) )  /\  A. s  e.  Q.  A. t  e.  Q.  ( s  <Q 
t  ->  ( s  e.  ( 1st `  L
)  \/  t  e.  ( 2nd `  L
) ) ) ) ) )
2216, 20, 21sylanbrc 413 1  |-  ( ph  ->  L  e.  P. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 697    /\ w3a 962    = wceq 1331    e. wcel 1480   {cab 2125   A.wral 2416   E.wrex 2417   {crab 2420    C_ wss 3071   ~Pcpw 3510   <.cop 3530   class class class wbr 3929    X. cxp 4537   -->wf 5119   ` cfv 5123  (class class class)co 5774   1stc1st 6036   2ndc2nd 6037   1oc1o 6306   [cec 6427   N.cnpi 7092    <N clti 7095    ~Q ceq 7099   Q.cnq 7100    +Q cplq 7102   *Qcrq 7104    <Q cltq 7105   P.cnp 7111    +P. cpp 7113    <P cltp 7115
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 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
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 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7124  df-pli 7125  df-mi 7126  df-lti 7127  df-plpq 7164  df-mpq 7165  df-enq 7167  df-nqqs 7168  df-plqqs 7169  df-mqqs 7170  df-1nqqs 7171  df-rq 7172  df-ltnqqs 7173  df-enq0 7244  df-nq0 7245  df-0nq0 7246  df-plq0 7247  df-mq0 7248  df-inp 7286  df-iplp 7288  df-iltp 7290
This theorem is referenced by:  caucvgprprlemclphr  7525  caucvgprprlemaddq  7528  caucvgprprlem2  7530  caucvgprpr  7532
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