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Theorem caucvgprlemdisj 6830
Description: Lemma for caucvgpr 6838. 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 5547 . . . . . . . . . . . 12  |-  ( l  =  s  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) )
21breq1d 3802 . . . . . . . . . . 11  |-  ( l  =  s  ->  (
( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
32rexbidv 2344 . . . . . . . . . 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 5209 . . . . . . . . . . 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 6519 . . . . . . . . . . . . 13  |-  Q.  e.  _V
76rabex 3929 . . . . . . . . . . . 12  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  _V
86rabex 3929 . . . . . . . . . . . 12  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  e.  _V
97, 8op1st 5801 . . . . . . . . . . 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 2076 . . . . . . . . . 10  |-  ( 1st `  L )  =  {
l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }
113, 10elrab2 2723 . . . . . . . . 9  |-  ( s  e.  ( 1st `  L
)  <->  ( s  e. 
Q.  /\  E. j  e.  N.  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
1211simprbi 264 . . . . . . . 8  |-  ( s  e.  ( 1st `  L
)  ->  E. j  e.  N.  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) )
13 opeq1 3577 . . . . . . . . . . . . 13  |-  ( j  =  k  ->  <. j ,  1o >.  =  <. k ,  1o >. )
1413eceq1d 6173 . . . . . . . . . . . 12  |-  ( j  =  k  ->  [ <. j ,  1o >. ]  ~Q  =  [ <. k ,  1o >. ]  ~Q  )
1514fveq2d 5210 . . . . . . . . . . 11  |-  ( j  =  k  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. k ,  1o >. ]  ~Q  ) )
1615oveq2d 5556 . . . . . . . . . 10  |-  ( j  =  k  ->  (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( *Q `  [ <. k ,  1o >. ]  ~Q  ) ) )
17 fveq2 5206 . . . . . . . . . 10  |-  ( j  =  k  ->  ( F `  j )  =  ( F `  k ) )
1816, 17breq12d 3805 . . . . . . . . 9  |-  ( j  =  k  ->  (
( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( s  +Q  ( *Q `  [ <. k ,  1o >. ]  ~Q  ) )  <Q 
( F `  k
) ) )
1918cbvrexv 2551 . . . . . . . 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 131 . . . . . . 7  |-  ( s  e.  ( 1st `  L
)  ->  E. k  e.  N.  ( s  +Q  ( *Q `  [ <. k ,  1o >. ]  ~Q  ) )  <Q 
( F `  k
) )
21 breq2 3796 . . . . . . . . . 10  |-  ( u  =  s  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
s ) )
2221rexbidv 2344 . . . . . . . . 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 5209 . . . . . . . . . 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 5802 . . . . . . . . . 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 2076 . . . . . . . . 9  |-  ( 2nd `  L )  =  {
u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u }
2622, 25elrab2 2723 . . . . . . . 8  |-  ( s  e.  ( 2nd `  L
)  <->  ( s  e. 
Q.  /\  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
s ) )
2726simprbi 264 . . . . . . 7  |-  ( s  e.  ( 2nd `  L
)  ->  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
s )
2820, 27anim12i 325 . . . . . 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 2496 . . . . . 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 141 . . . . 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 266 . . . 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 465 . . . . . . 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 465 . . . . . . 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 491 . . . . . . 7  |-  ( ( ( ph  /\  (
s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L
) ) )  /\  ( k  e.  N.  /\  j  e.  N. )
)  ->  k  e.  N. )
37 simprr 492 . . . . . . 7  |-  ( ( ( ph  /\  (
s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L
) ) )  /\  ( k  e.  N.  /\  j  e.  N. )
)  ->  j  e.  N. )
3811simplbi 263 . . . . . . . . 9  |-  ( s  e.  ( 1st `  L
)  ->  s  e.  Q. )
3938ad2antrl 467 . . . . . . . 8  |-  ( (
ph  /\  ( s  e.  ( 1st `  L
)  /\  s  e.  ( 2nd `  L ) ) )  ->  s  e.  Q. )
4039adantr 265 . . . . . . 7  |-  ( ( ( ph  /\  (
s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L
) ) )  /\  ( k  e.  N.  /\  j  e.  N. )
)  ->  s  e.  Q. )
4133, 35, 36, 37, 40caucvgprlemnkj 6822 . . . . . 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 559 . . . . 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 2457 . . . 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 1279 . 2  |-  ( ph  ->  -.  ( s  e.  ( 1st `  L
)  /\  s  e.  ( 2nd `  L ) ) )
4645ralrimivw 2410 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 101    = wceq 1259   F. wfal 1264    e. wcel 1409   A.wral 2323   E.wrex 2324   {crab 2327   <.cop 3406   class class class wbr 3792   -->wf 4926   ` cfv 4930  (class class class)co 5540   1stc1st 5793   2ndc2nd 5794   1oc1o 6025   [cec 6135   N.cnpi 6428    <N clti 6431    ~Q ceq 6435   Q.cnq 6436    +Q cplq 6438   *Qcrq 6440    <Q cltq 6441
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509
This theorem is referenced by:  caucvgprlemcl  6832
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