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

Theorem caucvgprlemdisj 6753
Description: Lemma for caucvgpr 6761. The putative limit is disjoint. (Contributed by Jim Kingdon, 27-Sep-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 } >.
Assertion
Ref Expression
caucvgprlemdisj  |-  ( ph  ->  A. s  e.  Q.  -.  ( s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L ) ) )
Distinct variable groups:    A, j    j, F, k    F, l, j   
u, F, j    n, F    j, L, k    ph, j,
s, k    s, l    u, s    k, n
Allowed substitution hints:    ph( u, n, l)    A( u, k, n, s, l)    F( s)    L( u, n, s, l)

Proof of Theorem caucvgprlemdisj
StepHypRef Expression
1 oveq1 5506 . . . . . . . . . . . 12  |-  ( l  =  s  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) )
21breq1d 3771 . . . . . . . . . . 11  |-  ( l  =  s  ->  (
( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
32rexbidv 2324 . . . . . . . . . 10  |-  ( l  =  s  ->  ( E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  E. j  e.  N.  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
4 caucvgpr.lim . . . . . . . . . . . 12  |-  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 } >.
54fveq2i 5168 . . . . . . . . . . 11  |-  ( 1st `  L )  =  ( 1st `  <. { 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 } >. )
6 nqex 6442 . . . . . . . . . . . . 13  |-  Q.  e.  _V
76rabex 3898 . . . . . . . . . . . 12  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  _V
86rabex 3898 . . . . . . . . . . . 12  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  e.  _V
97, 8op1st 5760 . . . . . . . . . . 11  |-  ( 1st `  <. { 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 } >. )  =  { l  e. 
Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) }
105, 9eqtri 2060 . . . . . . . . . 10  |-  ( 1st `  L )  =  {
l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }
113, 10elrab2 2697 . . . . . . . . 9  |-  ( s  e.  ( 1st `  L
)  <->  ( s  e. 
Q.  /\  E. j  e.  N.  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
1211simprbi 260 . . . . . . . 8  |-  ( s  e.  ( 1st `  L
)  ->  E. j  e.  N.  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) )
13 opeq1 3546 . . . . . . . . . . . . 13  |-  ( j  =  k  ->  <. j ,  1o >.  =  <. k ,  1o >. )
1413eceq1d 6129 . . . . . . . . . . . 12  |-  ( j  =  k  ->  [ <. j ,  1o >. ]  ~Q  =  [ <. k ,  1o >. ]  ~Q  )
1514fveq2d 5169 . . . . . . . . . . 11  |-  ( j  =  k  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. k ,  1o >. ]  ~Q  ) )
1615oveq2d 5515 . . . . . . . . . 10  |-  ( j  =  k  ->  (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( *Q `  [ <. k ,  1o >. ]  ~Q  ) ) )
17 fveq2 5165 . . . . . . . . . 10  |-  ( j  =  k  ->  ( F `  j )  =  ( F `  k ) )
1816, 17breq12d 3774 . . . . . . . . 9  |-  ( j  =  k  ->  (
( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( s  +Q  ( *Q `  [ <. k ,  1o >. ]  ~Q  ) )  <Q 
( F `  k
) ) )
1918cbvrexv 2531 . . . . . . . 8  |-  ( E. j  e.  N.  (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  E. k  e.  N.  ( s  +Q  ( *Q `  [ <. k ,  1o >. ]  ~Q  ) )  <Q 
( F `  k
) )
2012, 19sylib 127 . . . . . . 7  |-  ( s  e.  ( 1st `  L
)  ->  E. k  e.  N.  ( s  +Q  ( *Q `  [ <. k ,  1o >. ]  ~Q  ) )  <Q 
( F `  k
) )
21 breq2 3765 . . . . . . . . . 10  |-  ( u  =  s  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
s ) )
2221rexbidv 2324 . . . . . . . . 9  |-  ( u  =  s  ->  ( E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
s ) )
234fveq2i 5168 . . . . . . . . . 10  |-  ( 2nd `  L )  =  ( 2nd `  <. { 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 } >. )
247, 8op2nd 5761 . . . . . . . . . 10  |-  ( 2nd `  <. { 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 } >. )  =  { u  e. 
Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }
2523, 24eqtri 2060 . . . . . . . . 9  |-  ( 2nd `  L )  =  {
u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u }
2622, 25elrab2 2697 . . . . . . . 8  |-  ( s  e.  ( 2nd `  L
)  <->  ( s  e. 
Q.  /\  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
s ) )
2726simprbi 260 . . . . . . 7  |-  ( s  e.  ( 2nd `  L
)  ->  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
s )
2820, 27anim12i 321 . . . . . 6  |-  ( ( s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L
) )  ->  ( E. k  e.  N.  ( s  +Q  ( *Q `  [ <. k ,  1o >. ]  ~Q  )
)  <Q  ( F `  k )  /\  E. j  e.  N.  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  s ) )
29 reeanv 2476 . . . . . 6  |-  ( E. k  e.  N.  E. j  e.  N.  (
( s  +Q  ( *Q `  [ <. k ,  1o >. ]  ~Q  )
)  <Q  ( F `  k )  /\  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  s )  <->  ( E. k  e.  N.  (
s  +Q  ( *Q
`  [ <. k ,  1o >. ]  ~Q  )
)  <Q  ( F `  k )  /\  E. j  e.  N.  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  s ) )
3028, 29sylibr 137 . . . . 5  |-  ( ( s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L
) )  ->  E. k  e.  N.  E. j  e. 
N.  ( ( s  +Q  ( *Q `  [ <. k ,  1o >. ]  ~Q  ) ) 
<Q  ( F `  k
)  /\  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
s ) )
3130adantl 262 . . . 4  |-  ( (
ph  /\  ( s  e.  ( 1st `  L
)  /\  s  e.  ( 2nd `  L ) ) )  ->  E. k  e.  N.  E. j  e. 
N.  ( ( s  +Q  ( *Q `  [ <. k ,  1o >. ]  ~Q  ) ) 
<Q  ( F `  k
)  /\  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
s ) )
32 caucvgpr.f . . . . . . . 8  |-  ( ph  ->  F : N. --> Q. )
3332ad2antrr 457 . . . . . . 7  |-  ( ( ( ph  /\  (
s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L
) ) )  /\  ( k  e.  N.  /\  j  e.  N. )
)  ->  F : N.
--> Q. )
34 caucvgpr.cau . . . . . . . 8  |-  ( 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  )
) ) ) )
3534ad2antrr 457 . . . . . . 7  |-  ( ( ( ph  /\  (
s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L
) ) )  /\  ( k  e.  N.  /\  j  e.  N. )
)  ->  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  )
) ) ) )
36 simprl 483 . . . . . . 7  |-  ( ( ( ph  /\  (
s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L
) ) )  /\  ( k  e.  N.  /\  j  e.  N. )
)  ->  k  e.  N. )
37 simprr 484 . . . . . . 7  |-  ( ( ( ph  /\  (
s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L
) ) )  /\  ( k  e.  N.  /\  j  e.  N. )
)  ->  j  e.  N. )
3811simplbi 259 . . . . . . . . 9  |-  ( s  e.  ( 1st `  L
)  ->  s  e.  Q. )
3938ad2antrl 459 . . . . . . . 8  |-  ( (
ph  /\  ( s  e.  ( 1st `  L
)  /\  s  e.  ( 2nd `  L ) ) )  ->  s  e.  Q. )
4039adantr 261 . . . . . . 7  |-  ( ( ( ph  /\  (
s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L
) ) )  /\  ( k  e.  N.  /\  j  e.  N. )
)  ->  s  e.  Q. )
4133, 35, 36, 37, 40caucvgprlemnkj 6745 . . . . . 6  |-  ( ( ( ph  /\  (
s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L
) ) )  /\  ( k  e.  N.  /\  j  e.  N. )
)  ->  -.  (
( s  +Q  ( *Q `  [ <. k ,  1o >. ]  ~Q  )
)  <Q  ( F `  k )  /\  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  s ) )
4241pm2.21d 549 . . . . 5  |-  ( ( ( ph  /\  (
s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L
) ) )  /\  ( k  e.  N.  /\  j  e.  N. )
)  ->  ( (
( s  +Q  ( *Q `  [ <. k ,  1o >. ]  ~Q  )
)  <Q  ( F `  k )  /\  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  s )  -> F.  ) )
4342rexlimdvva 2437 . . . 4  |-  ( (
ph  /\  ( s  e.  ( 1st `  L
)  /\  s  e.  ( 2nd `  L ) ) )  ->  ( E. k  e.  N.  E. j  e.  N.  (
( s  +Q  ( *Q `  [ <. k ,  1o >. ]  ~Q  )
)  <Q  ( F `  k )  /\  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  s )  -> F.  ) )
4431, 43mpd 13 . . 3  |-  ( (
ph  /\  ( s  e.  ( 1st `  L
)  /\  s  e.  ( 2nd `  L ) ) )  -> F.  )
4544inegd 1263 . 2  |-  ( ph  ->  -.  ( s  e.  ( 1st `  L
)  /\  s  e.  ( 2nd `  L ) ) )
4645ralrimivw 2390 1  |-  ( ph  ->  A. s  e.  Q.  -.  ( s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 97    = wceq 1243   F. wfal 1248    e. wcel 1393   A.wral 2303   E.wrex 2304   {crab 2307   <.cop 3375   class class class wbr 3761   -->wf 4885   ` cfv 4889  (class class class)co 5499   1stc1st 5752   2ndc2nd 5753   1oc1o 5981   [cec 6091   N.cnpi 6351    <N clti 6354    ~Q ceq 6358   Q.cnq 6359    +Q cplq 6361   *Qcrq 6363    <Q cltq 6364
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3869  ax-sep 3872  ax-nul 3880  ax-pow 3924  ax-pr 3941  ax-un 4166  ax-setind 4256  ax-iinf 4298
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2308  df-rex 2309  df-reu 2310  df-rab 2312  df-v 2556  df-sbc 2762  df-csb 2850  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-int 3613  df-iun 3656  df-br 3762  df-opab 3816  df-mpt 3817  df-tr 3852  df-eprel 4023  df-id 4027  df-po 4030  df-iso 4031  df-iord 4099  df-on 4101  df-suc 4104  df-iom 4301  df-xp 4338  df-rel 4339  df-cnv 4340  df-co 4341  df-dm 4342  df-rn 4343  df-res 4344  df-ima 4345  df-iota 4854  df-fun 4891  df-fn 4892  df-f 4893  df-f1 4894  df-fo 4895  df-f1o 4896  df-fv 4897  df-ov 5502  df-oprab 5503  df-mpt2 5504  df-1st 5754  df-2nd 5755  df-recs 5907  df-irdg 5944  df-1o 5988  df-oadd 5992  df-omul 5993  df-er 6093  df-ec 6095  df-qs 6099  df-ni 6383  df-pli 6384  df-mi 6385  df-lti 6386  df-plpq 6423  df-mpq 6424  df-enq 6426  df-nqqs 6427  df-plqqs 6428  df-mqqs 6429  df-1nqqs 6430  df-rq 6431  df-ltnqqs 6432
This theorem is referenced by:  caucvgprlemcl  6755
  Copyright terms: Public domain W3C validator