ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  caucvgprlem1 Unicode version

Theorem caucvgprlem1 7451
Description: Lemma for caucvgpr 7454. 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 7096 . . . . . . 7  |-  <N  C_  ( N.  X.  N. )
32brel 4559 . . . . . 6  |-  ( J 
<N  K  ->  ( J  e.  N.  /\  K  e.  N. ) )
41, 3syl 14 . . . . 5  |-  ( ph  ->  ( J  e.  N.  /\  K  e.  N. )
)
54simprd 113 . . . 4  |-  ( ph  ->  K  e.  N. )
6 caucvgprlemlim.jkq . . . . . 6  |-  ( ph  ->  ( *Q `  [ <. J ,  1o >. ]  ~Q  )  <Q  Q )
71, 6caucvgprlemk 7437 . . . . 5  |-  ( ph  ->  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  Q )
8 caucvgpr.f . . . . . 6  |-  ( ph  ->  F : N. --> Q. )
98, 5ffvelrnd 5522 . . . . 5  |-  ( ph  ->  ( F `  K
)  e.  Q. )
10 ltanqi 7174 . . . . 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 406 . . . 4  |-  ( ph  ->  ( ( F `  K )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 K )  +Q  Q ) )
12 opeq1 3673 . . . . . . . . 9  |-  ( j  =  K  ->  <. j ,  1o >.  =  <. K ,  1o >. )
1312eceq1d 6431 . . . . . . . 8  |-  ( j  =  K  ->  [ <. j ,  1o >. ]  ~Q  =  [ <. K ,  1o >. ]  ~Q  )
1413fveq2d 5391 . . . . . . 7  |-  ( j  =  K  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )
1514oveq2d 5756 . . . . . 6  |-  ( j  =  K  ->  (
( F `  K
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( ( F `  K )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) )
16 fveq2 5387 . . . . . . 7  |-  ( j  =  K  ->  ( F `  j )  =  ( F `  K ) )
1716oveq1d 5755 . . . . . 6  |-  ( j  =  K  ->  (
( F `  j
)  +Q  Q )  =  ( ( F `
 K )  +Q  Q ) )
1815, 17breq12d 3910 . . . . 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 2761 . . . 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 406 . . 3  |-  ( ph  ->  E. j  e.  N.  ( ( F `  K )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( ( F `
 j )  +Q  Q ) )
21 oveq1 5747 . . . . . . . 8  |-  ( l  =  ( F `  K )  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( ( F `  K )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) )
2221breq1d 3907 . . . . . . 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 2413 . . . . . 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 2812 . . . . 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 7450 . . . . 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 3064 . . . 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 169 . . 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 7448 . . . 4  |-  ( ph  ->  L  e.  P. )
35 nqprlu 7319 . . . . 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 7309 . . . 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 406 . . 3  |-  ( ph  ->  ( L  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )  e.  P. )
39 nqprl 7323 . . 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 406 . 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 146 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 103    <-> wb 104    = wceq 1314    e. wcel 1463   {cab 2101   A.wral 2391   E.wrex 2392   {crab 2395   <.cop 3498   class class class wbr 3897   -->wf 5087   ` cfv 5091  (class class class)co 5740   1stc1st 6002   1oc1o 6272   [cec 6393   N.cnpi 7044    <N clti 7047    ~Q ceq 7051   Q.cnq 7052    +Q cplq 7054   *Qcrq 7056    <Q cltq 7057   P.cnp 7063    +P. cpp 7065    <P cltp 7067
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-2o 6280  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-pli 7077  df-mi 7078  df-lti 7079  df-plpq 7116  df-mpq 7117  df-enq 7119  df-nqqs 7120  df-plqqs 7121  df-mqqs 7122  df-1nqqs 7123  df-rq 7124  df-ltnqqs 7125  df-enq0 7196  df-nq0 7197  df-0nq0 7198  df-plq0 7199  df-mq0 7200  df-inp 7238  df-iplp 7240  df-iltp 7242
This theorem is referenced by:  caucvgprlemlim  7453
  Copyright terms: Public domain W3C validator