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Theorem caucvgprlem1 6931
Description: Lemma for caucvgpr 6934. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 3-Oct-2020.)
Hypotheses
Ref Expression
caucvgpr.f  |-  ( ph  ->  F : N. --> Q. )
caucvgpr.cau  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <Q  ( ( F `
 k )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  /\  ( F `  k ) 
<Q  ( ( F `  n )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  )
) ) ) )
caucvgpr.bnd  |-  ( ph  ->  A. j  e.  N.  A  <Q  ( F `  j ) )
caucvgpr.lim  |-  L  = 
<. { l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) } ,  { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u } >.
caucvgprlemlim.q  |-  ( ph  ->  Q  e.  Q. )
caucvgprlemlim.jk  |-  ( ph  ->  J  <N  K )
caucvgprlemlim.jkq  |-  ( ph  ->  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  Q )
Assertion
Ref Expression
caucvgprlem1  |-  ( ph  -> 
<. { l  |  l 
<Q  ( F `  K
) } ,  {
u  |  ( F `
 K )  <Q  u } >.  <P  ( L  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )
)
Distinct variable groups:    A, j    j, F, l, u    j, K, l, u    Q, j, l, u    Q, k   
j, L, k    u, j    k, F, n    j,
k
Allowed substitution hints:    ph( u, j, k, n, l)    A( u, k, n, l)    Q( n)    J( u, j, k, n, l)    K( k, n)    L( u, n, l)

Proof of Theorem caucvgprlem1
StepHypRef Expression
1 caucvgprlemlim.jk . . . . . 6  |-  ( ph  ->  J  <N  K )
2 ltrelpi 6576 . . . . . . 7  |-  <N  C_  ( N.  X.  N. )
32brel 4418 . . . . . 6  |-  ( J 
<N  K  ->  ( J  e.  N.  /\  K  e.  N. ) )
41, 3syl 14 . . . . 5  |-  ( ph  ->  ( J  e.  N.  /\  K  e.  N. )
)
54simprd 112 . . . 4  |-  ( ph  ->  K  e.  N. )
6 caucvgprlemlim.jkq . . . . . 6  |-  ( ph  ->  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  Q )
71, 6caucvgprlemk 6917 . . . . 5  |-  ( ph  ->  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  Q )
8 caucvgpr.f . . . . . 6  |-  ( ph  ->  F : N. --> Q. )
98, 5ffvelrnd 5335 . . . . 5  |-  ( ph  ->  ( F `  K
)  e.  Q. )
10 ltanqi 6654 . . . . 5  |-  ( ( ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  Q  /\  ( F `  K )  e.  Q. )  -> 
( ( F `  K )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 K )  +Q  Q ) )
117, 9, 10syl2anc 403 . . . 4  |-  ( ph  ->  ( ( F `  K )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 K )  +Q  Q ) )
12 opeq1 3578 . . . . . . . . 9  |-  ( j  =  K  ->  <. j ,  1o >.  =  <. K ,  1o >. )
1312eceq1d 6208 . . . . . . . 8  |-  ( j  =  K  ->  [ <. j ,  1o >. ]  ~Q  =  [ <. K ,  1o >. ]  ~Q  )
1413fveq2d 5213 . . . . . . 7  |-  ( j  =  K  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )
1514oveq2d 5559 . . . . . 6  |-  ( j  =  K  ->  (
( F `  K
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( ( F `  K )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) )
16 fveq2 5209 . . . . . . 7  |-  ( j  =  K  ->  ( F `  j )  =  ( F `  K ) )
1716oveq1d 5558 . . . . . 6  |-  ( j  =  K  ->  (
( F `  j
)  +Q  Q )  =  ( ( F `
 K )  +Q  Q ) )
1815, 17breq12d 3806 . . . . 5  |-  ( j  =  K  ->  (
( ( F `  K )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 j )  +Q  Q )  <->  ( ( F `  K )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  K )  +Q  Q
) ) )
1918rspcev 2702 . . . 4  |-  ( ( K  e.  N.  /\  ( ( F `  K )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 K )  +Q  Q ) )  ->  E. j  e.  N.  ( ( F `  K )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 j )  +Q  Q ) )
205, 11, 19syl2anc 403 . . 3  |-  ( ph  ->  E. j  e.  N.  ( ( F `  K )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 j )  +Q  Q ) )
21 oveq1 5550 . . . . . . . 8  |-  ( l  =  ( F `  K )  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( ( F `  K )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) )
2221breq1d 3803 . . . . . . 7  |-  ( l  =  ( F `  K )  ->  (
( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 j )  +Q  Q )  <->  ( ( F `  K )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  j )  +Q  Q
) ) )
2322rexbidv 2370 . . . . . 6  |-  ( l  =  ( F `  K )  ->  ( E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 j )  +Q  Q )  <->  E. j  e.  N.  ( ( F `
 K )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  j )  +Q  Q
) ) )
2423elrab3 2751 . . . . 5  |-  ( ( F `  K )  e.  Q.  ->  (
( F `  K
)  e.  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 j )  +Q  Q ) }  <->  E. j  e.  N.  ( ( F `
 K )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  j )  +Q  Q
) ) )
259, 24syl 14 . . . 4  |-  ( ph  ->  ( ( F `  K )  e.  {
l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 j )  +Q  Q ) }  <->  E. j  e.  N.  ( ( F `
 K )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  j )  +Q  Q
) ) )
26 caucvgpr.cau . . . . . 6  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <Q  ( ( F `
 k )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  /\  ( F `  k ) 
<Q  ( ( F `  n )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  )
) ) ) )
27 caucvgpr.bnd . . . . . 6  |-  ( ph  ->  A. j  e.  N.  A  <Q  ( F `  j ) )
28 caucvgpr.lim . . . . . 6  |-  L  = 
<. { l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) } ,  { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u } >.
29 caucvgprlemlim.q . . . . . 6  |-  ( ph  ->  Q  e.  Q. )
308, 26, 27, 28, 29caucvgprlemladdrl 6930 . . . . 5  |-  ( ph  ->  { l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 j )  +Q  Q ) }  C_  ( 1st `  ( L  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )
) )
3130sseld 2999 . . . 4  |-  ( ph  ->  ( ( F `  K )  e.  {
l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 j )  +Q  Q ) }  ->  ( F `  K )  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )
) ) )
3225, 31sylbird 168 . . 3  |-  ( ph  ->  ( E. j  e. 
N.  ( ( F `
 K )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( ( F `  j )  +Q  Q
)  ->  ( F `  K )  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )
) ) )
3320, 32mpd 13 . 2  |-  ( ph  ->  ( F `  K
)  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. ) ) )
348, 26, 27, 28caucvgprlemcl 6928 . . . 4  |-  ( ph  ->  L  e.  P. )
35 nqprlu 6799 . . . . 5  |-  ( Q  e.  Q.  ->  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >.  e.  P. )
3629, 35syl 14 . . . 4  |-  ( ph  -> 
<. { l  |  l 
<Q  Q } ,  {
u  |  Q  <Q  u } >.  e.  P. )
37 addclpr 6789 . . . 4  |-  ( ( L  e.  P.  /\  <. { l  |  l 
<Q  Q } ,  {
u  |  Q  <Q  u } >.  e.  P. )  ->  ( L  +P.  <. { l  |  l 
<Q  Q } ,  {
u  |  Q  <Q  u } >. )  e.  P. )
3834, 36, 37syl2anc 403 . . 3  |-  ( ph  ->  ( L  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )  e.  P. )
39 nqprl 6803 . . 3  |-  ( ( ( F `  K
)  e.  Q.  /\  ( L  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )  e.  P. )  ->  (
( F `  K
)  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. ) )  <->  <. { l  |  l  <Q  ( F `  K ) } ,  { u  |  ( F `  K )  <Q  u } >.  <P  ( L  +P.  <. { l  |  l 
<Q  Q } ,  {
u  |  Q  <Q  u } >. ) ) )
409, 38, 39syl2anc 403 . 2  |-  ( ph  ->  ( ( F `  K )  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )
)  <->  <. { l  |  l  <Q  ( F `  K ) } ,  { u  |  ( F `  K )  <Q  u } >.  <P  ( L  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )
) )
4133, 40mpbid 145 1  |-  ( ph  -> 
<. { l  |  l 
<Q  ( F `  K
) } ,  {
u  |  ( F `
 K )  <Q  u } >.  <P  ( L  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   {cab 2068   A.wral 2349   E.wrex 2350   {crab 2353   <.cop 3409   class class class wbr 3793   -->wf 4928   ` cfv 4932  (class class class)co 5543   1stc1st 5796   1oc1o 6058   [cec 6170   N.cnpi 6524    <N clti 6527    ~Q ceq 6531   Q.cnq 6532    +Q cplq 6534   *Qcrq 6536    <Q cltq 6537   P.cnp 6543    +P. cpp 6545    <P cltp 6547
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 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
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 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-2o 6066  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-plpq 6596  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-plqqs 6601  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605  df-enq0 6676  df-nq0 6677  df-0nq0 6678  df-plq0 6679  df-mq0 6680  df-inp 6718  df-iplp 6720  df-iltp 6722
This theorem is referenced by:  caucvgprlemlim  6933
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