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

Theorem caucvgprlem1 7217
Description: Lemma for caucvgpr 7220. 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 6862 . . . . . . 7  |-  <N  C_  ( N.  X.  N. )
32brel 4478 . . . . . 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 7203 . . . . 5  |-  ( ph  ->  ( *Q `  [ <. K ,  1o >. ]  ~Q  )  <Q  Q )
8 caucvgpr.f . . . . . 6  |-  ( ph  ->  F : N. --> Q. )
98, 5ffvelrnd 5419 . . . . 5  |-  ( ph  ->  ( F `  K
)  e.  Q. )
10 ltanqi 6940 . . . . 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 3617 . . . . . . . . 9  |-  ( j  =  K  ->  <. j ,  1o >.  =  <. K ,  1o >. )
1312eceq1d 6308 . . . . . . . 8  |-  ( j  =  K  ->  [ <. j ,  1o >. ]  ~Q  =  [ <. K ,  1o >. ]  ~Q  )
1413fveq2d 5293 . . . . . . 7  |-  ( j  =  K  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) )
1514oveq2d 5650 . . . . . 6  |-  ( j  =  K  ->  (
( F `  K
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( ( F `  K )  +Q  ( *Q `  [ <. K ,  1o >. ]  ~Q  ) ) )
16 fveq2 5289 . . . . . . 7  |-  ( j  =  K  ->  ( F `  j )  =  ( F `  K ) )
1716oveq1d 5649 . . . . . 6  |-  ( j  =  K  ->  (
( F `  j
)  +Q  Q )  =  ( ( F `
 K )  +Q  Q ) )
1815, 17breq12d 3850 . . . . 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 2722 . . . 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 5641 . . . . . . . 8  |-  ( l  =  ( F `  K )  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( ( F `  K )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) )
2221breq1d 3847 . . . . . . 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 2381 . . . . . 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 2770 . . . . 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 7216 . . . . 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 3022 . . . 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 7214 . . . 4  |-  ( ph  ->  L  e.  P. )
35 nqprlu 7085 . . . . 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 7075 . . . 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 7089 . . 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 1289    e. wcel 1438   {cab 2074   A.wral 2359   E.wrex 2360   {crab 2363   <.cop 3444   class class class wbr 3837   -->wf 4998   ` cfv 5002  (class class class)co 5634   1stc1st 5891   1oc1o 6156   [cec 6270   N.cnpi 6810    <N clti 6813    ~Q ceq 6817   Q.cnq 6818    +Q cplq 6820   *Qcrq 6822    <Q cltq 6823   P.cnp 6829    +P. cpp 6831    <P cltp 6833
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 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
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 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-eprel 4107  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-1o 6163  df-2o 6164  df-oadd 6167  df-omul 6168  df-er 6272  df-ec 6274  df-qs 6278  df-ni 6842  df-pli 6843  df-mi 6844  df-lti 6845  df-plpq 6882  df-mpq 6883  df-enq 6885  df-nqqs 6886  df-plqqs 6887  df-mqqs 6888  df-1nqqs 6889  df-rq 6890  df-ltnqqs 6891  df-enq0 6962  df-nq0 6963  df-0nq0 6964  df-plq0 6965  df-mq0 6966  df-inp 7004  df-iplp 7006  df-iltp 7008
This theorem is referenced by:  caucvgprlemlim  7219
  Copyright terms: Public domain W3C validator