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Theorem caucvgprlemdisj 7446
Description: Lemma for caucvgpr 7454. 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 5747 . . . . . . . . . . . 12  |-  ( l  =  s  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) )
21breq1d 3907 . . . . . . . . . . 11  |-  ( l  =  s  ->  (
( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
32rexbidv 2413 . . . . . . . . . 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 5390 . . . . . . . . . . 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 7135 . . . . . . . . . . . . 13  |-  Q.  e.  _V
76rabex 4040 . . . . . . . . . . . 12  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  _V
86rabex 4040 . . . . . . . . . . . 12  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  e.  _V
97, 8op1st 6010 . . . . . . . . . . 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 2136 . . . . . . . . . 10  |-  ( 1st `  L )  =  {
l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }
113, 10elrab2 2814 . . . . . . . . 9  |-  ( s  e.  ( 1st `  L
)  <->  ( s  e. 
Q.  /\  E. j  e.  N.  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
1211simprbi 271 . . . . . . . 8  |-  ( s  e.  ( 1st `  L
)  ->  E. j  e.  N.  ( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) )
13 opeq1 3673 . . . . . . . . . . . . 13  |-  ( j  =  k  ->  <. j ,  1o >.  =  <. k ,  1o >. )
1413eceq1d 6431 . . . . . . . . . . . 12  |-  ( j  =  k  ->  [ <. j ,  1o >. ]  ~Q  =  [ <. k ,  1o >. ]  ~Q  )
1514fveq2d 5391 . . . . . . . . . . 11  |-  ( j  =  k  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. k ,  1o >. ]  ~Q  ) )
1615oveq2d 5756 . . . . . . . . . 10  |-  ( j  =  k  ->  (
s  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( *Q `  [ <. k ,  1o >. ]  ~Q  ) ) )
17 fveq2 5387 . . . . . . . . . 10  |-  ( j  =  k  ->  ( F `  j )  =  ( F `  k ) )
1816, 17breq12d 3910 . . . . . . . . 9  |-  ( j  =  k  ->  (
( s  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( s  +Q  ( *Q `  [ <. k ,  1o >. ]  ~Q  ) )  <Q 
( F `  k
) ) )
1918cbvrexv 2630 . . . . . . . 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 121 . . . . . . 7  |-  ( s  e.  ( 1st `  L
)  ->  E. k  e.  N.  ( s  +Q  ( *Q `  [ <. k ,  1o >. ]  ~Q  ) )  <Q 
( F `  k
) )
21 breq2 3901 . . . . . . . . . 10  |-  ( u  =  s  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
s ) )
2221rexbidv 2413 . . . . . . . . 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 5390 . . . . . . . . . 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 6011 . . . . . . . . . 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 2136 . . . . . . . . 9  |-  ( 2nd `  L )  =  {
u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u }
2622, 25elrab2 2814 . . . . . . . 8  |-  ( s  e.  ( 2nd `  L
)  <->  ( s  e. 
Q.  /\  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
s ) )
2726simprbi 271 . . . . . . 7  |-  ( s  e.  ( 2nd `  L
)  ->  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
s )
2820, 27anim12i 334 . . . . . 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 2575 . . . . . 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 133 . . . . 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 273 . . . 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 477 . . . . . . 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 477 . . . . . . 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 503 . . . . . . 7  |-  ( ( ( ph  /\  (
s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L
) ) )  /\  ( k  e.  N.  /\  j  e.  N. )
)  ->  k  e.  N. )
37 simprr 504 . . . . . . 7  |-  ( ( ( ph  /\  (
s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L
) ) )  /\  ( k  e.  N.  /\  j  e.  N. )
)  ->  j  e.  N. )
3811simplbi 270 . . . . . . . . 9  |-  ( s  e.  ( 1st `  L
)  ->  s  e.  Q. )
3938ad2antrl 479 . . . . . . . 8  |-  ( (
ph  /\  ( s  e.  ( 1st `  L
)  /\  s  e.  ( 2nd `  L ) ) )  ->  s  e.  Q. )
4039adantr 272 . . . . . . 7  |-  ( ( ( ph  /\  (
s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L
) ) )  /\  ( k  e.  N.  /\  j  e.  N. )
)  ->  s  e.  Q. )
4133, 35, 36, 37, 40caucvgprlemnkj 7438 . . . . . 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 591 . . . . 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 2532 . . . 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 1333 . 2  |-  ( ph  ->  -.  ( s  e.  ( 1st `  L
)  /\  s  e.  ( 2nd `  L ) ) )
4645ralrimivw 2481 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 103    = wceq 1314   F. wfal 1319    e. wcel 1463   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   2ndc2nd 6003   1oc1o 6272   [cec 6393   N.cnpi 7044    <N clti 7047    ~Q ceq 7051   Q.cnq 7052    +Q cplq 7054   *Qcrq 7056    <Q cltq 7057
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-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
This theorem is referenced by:  caucvgprlemcl  7448
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