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Theorem caucvgprprlemdisj 7322
Description: Lemma for caucvgprpr 7332. The putative limit is disjoint. (Contributed by Jim Kingdon, 21-Dec-2020.)
Hypotheses
Ref Expression
caucvgprpr.f  |-  ( ph  ->  F : N. --> P. )
caucvgprpr.cau  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <P  ( ( F `
 k )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( F `  k
)  <P  ( ( F `
 n )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )
) ) )
caucvgprpr.bnd  |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )
caucvgprpr.lim  |-  L  = 
<. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  r ) } ,  { u  e.  Q.  |  E. r  e.  N.  ( ( F `
 r )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  u } ,  { q  |  u 
<Q  q } >. } >.
Assertion
Ref Expression
caucvgprprlemdisj  |-  ( ph  ->  A. s  e.  Q.  -.  ( s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L ) ) )
Distinct variable groups:    A, m    m, F    k, F, n, l    F, r, l    u, F, r    k, L    k, p, r, s    ph, r,
s    k, q, r, s   
p, l, s, q   
u, p, s, q   
u, n    n, l,
k    u, k
Allowed substitution hints:    ph( u, k, m, n, q, p, l)    A( u, k, n, s, r, q, p, l)    F( s, q, p)    L( u, m, n, s, r, q, p, l)

Proof of Theorem caucvgprprlemdisj
Dummy variables  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 caucvgprpr.lim . . . . . . . . 9  |-  L  = 
<. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  r ) } ,  { u  e.  Q.  |  E. r  e.  N.  ( ( F `
 r )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  u } ,  { q  |  u 
<Q  q } >. } >.
21caucvgprprlemell 7305 . . . . . . . 8  |-  ( s  e.  ( 1st `  L
)  <->  ( s  e. 
Q.  /\  E. a  e.  N.  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P 
( F `  a
) ) )
32simprbi 270 . . . . . . 7  |-  ( s  e.  ( 1st `  L
)  ->  E. a  e.  N.  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P 
( F `  a
) )
41caucvgprprlemelu 7306 . . . . . . . 8  |-  ( s  e.  ( 2nd `  L
)  <->  ( s  e. 
Q.  /\  E. b  e.  N.  ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  s } ,  { q  |  s 
<Q  q } >. )
)
54simprbi 270 . . . . . . 7  |-  ( s  e.  ( 2nd `  L
)  ->  E. b  e.  N.  ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  s } ,  { q  |  s 
<Q  q } >. )
63, 5anim12i 332 . . . . . 6  |-  ( ( s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L
) )  ->  ( E. a  e.  N.  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  a )  /\  E. b  e.  N.  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  s } ,  {
q  |  s  <Q 
q } >. )
)
7 reeanv 2537 . . . . . 6  |-  ( E. a  e.  N.  E. b  e.  N.  ( <. { p  |  p 
<Q  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  a )  /\  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  s } ,  {
q  |  s  <Q 
q } >. )  <->  ( E. a  e.  N.  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  a )  /\  E. b  e.  N.  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  s } ,  {
q  |  s  <Q 
q } >. )
)
86, 7sylibr 133 . . . . 5  |-  ( ( s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L
) )  ->  E. a  e.  N.  E. b  e. 
N.  ( <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
q } >.  <P  ( F `  a )  /\  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  s } ,  {
q  |  s  <Q 
q } >. )
)
98adantl 272 . . . 4  |-  ( (
ph  /\  ( s  e.  ( 1st `  L
)  /\  s  e.  ( 2nd `  L ) ) )  ->  E. a  e.  N.  E. b  e. 
N.  ( <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
q } >.  <P  ( F `  a )  /\  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  s } ,  {
q  |  s  <Q 
q } >. )
)
10 caucvgprpr.f . . . . . . . 8  |-  ( ph  ->  F : N. --> P. )
1110ad2antrr 473 . . . . . . 7  |-  ( ( ( ph  /\  (
s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L
) ) )  /\  ( a  e.  N.  /\  b  e.  N. )
)  ->  F : N.
--> P. )
12 caucvgprpr.cau . . . . . . . 8  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <P  ( ( F `
 k )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( F `  k
)  <P  ( ( F `
 n )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )
) ) )
1312ad2antrr 473 . . . . . . 7  |-  ( ( ( ph  /\  (
s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L
) ) )  /\  ( a  e.  N.  /\  b  e.  N. )
)  ->  A. n  e.  N.  A. k  e. 
N.  ( n  <N  k  ->  ( ( F `
 n )  <P 
( ( F `  k )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( F `  k
)  <P  ( ( F `
 n )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )
) ) )
14 simprl 499 . . . . . . 7  |-  ( ( ( ph  /\  (
s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L
) ) )  /\  ( a  e.  N.  /\  b  e.  N. )
)  ->  a  e.  N. )
15 simprr 500 . . . . . . 7  |-  ( ( ( ph  /\  (
s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L
) ) )  /\  ( a  e.  N.  /\  b  e.  N. )
)  ->  b  e.  N. )
161caucvgprprlemell 7305 . . . . . . . . . 10  |-  ( s  e.  ( 1st `  L
)  <->  ( s  e. 
Q.  /\  E. r  e.  N.  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P 
( F `  r
) ) )
1716simplbi 269 . . . . . . . . 9  |-  ( s  e.  ( 1st `  L
)  ->  s  e.  Q. )
1817ad2antrl 475 . . . . . . . 8  |-  ( (
ph  /\  ( s  e.  ( 1st `  L
)  /\  s  e.  ( 2nd `  L ) ) )  ->  s  e.  Q. )
1918adantr 271 . . . . . . 7  |-  ( ( ( ph  /\  (
s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L
) ) )  /\  ( a  e.  N.  /\  b  e.  N. )
)  ->  s  e.  Q. )
2011, 13, 14, 15, 19caucvgprprlemnkj 7312 . . . . . 6  |-  ( ( ( ph  /\  (
s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L
) ) )  /\  ( a  e.  N.  /\  b  e.  N. )
)  ->  -.  ( <. { p  |  p 
<Q  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  a )  /\  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  s } ,  {
q  |  s  <Q 
q } >. )
)
2120pm2.21d 585 . . . . 5  |-  ( ( ( ph  /\  (
s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L
) ) )  /\  ( a  e.  N.  /\  b  e.  N. )
)  ->  ( ( <. { p  |  p 
<Q  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  a )  /\  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  s } ,  {
q  |  s  <Q 
q } >. )  -> F.  ) )
2221rexlimdvva 2497 . . . 4  |-  ( (
ph  /\  ( s  e.  ( 1st `  L
)  /\  s  e.  ( 2nd `  L ) ) )  ->  ( E. a  e.  N.  E. b  e.  N.  ( <. { p  |  p 
<Q  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  a )  /\  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  s } ,  {
q  |  s  <Q 
q } >. )  -> F.  ) )
239, 22mpd 13 . . 3  |-  ( (
ph  /\  ( s  e.  ( 1st `  L
)  /\  s  e.  ( 2nd `  L ) ) )  -> F.  )
2423inegd 1309 . 2  |-  ( ph  ->  -.  ( s  e.  ( 1st `  L
)  /\  s  e.  ( 2nd `  L ) ) )
2524ralrimivw 2448 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 1290   F. wfal 1295    e. wcel 1439   {cab 2075   A.wral 2360   E.wrex 2361   {crab 2364   <.cop 3453   class class class wbr 3851   -->wf 5024   ` cfv 5028  (class class class)co 5666   1stc1st 5923   2ndc2nd 5924   1oc1o 6188   [cec 6304   N.cnpi 6892    <N clti 6895    ~Q ceq 6899   Q.cnq 6900    +Q cplq 6902   *Qcrq 6904    <Q cltq 6905   P.cnp 6911    +P. cpp 6913    <P cltp 6915
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-eprel 4125  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-irdg 6149  df-1o 6195  df-2o 6196  df-oadd 6199  df-omul 6200  df-er 6306  df-ec 6308  df-qs 6312  df-ni 6924  df-pli 6925  df-mi 6926  df-lti 6927  df-plpq 6964  df-mpq 6965  df-enq 6967  df-nqqs 6968  df-plqqs 6969  df-mqqs 6970  df-1nqqs 6971  df-rq 6972  df-ltnqqs 6973  df-enq0 7044  df-nq0 7045  df-0nq0 7046  df-plq0 7047  df-mq0 7048  df-inp 7086  df-iplp 7088  df-iltp 7090
This theorem is referenced by:  caucvgprprlemcl  7324
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