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Theorem caucvgprlemdisj 6772
Description: Lemma for caucvgpr 6780. 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 5519 . . . . . . . . . . . 12  |-  ( l  =  s  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) )
21breq1d 3774 . . . . . . . . . . 11  |-  ( l  =  s  ->  (
( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
32rexbidv 2327 . . . . . . . . . 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 5181 . . . . . . . . . . 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 6461 . . . . . . . . . . . . 13  |-  Q.  e.  _V
76rabex 3901 . . . . . . . . . . . 12  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  _V
86rabex 3901 . . . . . . . . . . . 12  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  e.  _V
97, 8op1st 5773 . . . . . . . . . . 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 2700 . . . . . . . . 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 3549 . . . . . . . . . . . . 13  |-  ( j  =  k  ->  <. j ,  1o >.  =  <. k ,  1o >. )
1413eceq1d 6142 . . . . . . . . . . . 12  |-  ( j  =  k  ->  [ <. j ,  1o >. ]  ~Q  =  [ <. k ,  1o >. ]  ~Q  )
1514fveq2d 5182 . . . . . . . . . . 11  |-  ( j  =  k  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. k ,  1o >. ]  ~Q  ) )
1615oveq2d 5528 . . . . . . . . . 10  |-  ( j  =  k  ->  (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( *Q `  [ <. k ,  1o >. ]  ~Q  ) ) )
17 fveq2 5178 . . . . . . . . . 10  |-  ( j  =  k  ->  ( F `  j )  =  ( F `  k ) )
1816, 17breq12d 3777 . . . . . . . . 9  |-  ( j  =  k  ->  (
( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( s  +Q  ( *Q `  [ <. k ,  1o >. ]  ~Q  ) )  <Q 
( F `  k
) ) )
1918cbvrexv 2534 . . . . . . . 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 3768 . . . . . . . . . 10  |-  ( u  =  s  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
s ) )
2221rexbidv 2327 . . . . . . . . 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 5181 . . . . . . . . . 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 5774 . . . . . . . . . 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 2700 . . . . . . . 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 2479 . . . . . 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 6764 . . . . . 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 2440 . . . 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 2393 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 2306   E.wrex 2307   {crab 2310   <.cop 3378   class class class wbr 3764   -->wf 4898   ` cfv 4902  (class class class)co 5512   1stc1st 5765   2ndc2nd 5766   1oc1o 5994   [cec 6104   N.cnpi 6370    <N clti 6373    ~Q ceq 6377   Q.cnq 6378    +Q cplq 6380   *Qcrq 6382    <Q cltq 6383
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 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
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 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451
This theorem is referenced by:  caucvgprlemcl  6774
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