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

Theorem caucvgprlemupu 7444
Description: Lemma for caucvgpr 7454. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 20-Oct-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
caucvgprlemupu  |-  ( (
ph  /\  s  <Q  r  /\  s  e.  ( 2nd `  L ) )  ->  r  e.  ( 2nd `  L ) )
Distinct variable groups:    A, j    F, l, r, s    u, F   
j, L, r, s   
j, l, s    ph, j,
r, s    u, j,
r, s
Allowed substitution hints:    ph( u, k, n, l)    A( u, k, n, s, r, l)    F( j, k, n)    L( u, k, n, l)

Proof of Theorem caucvgprlemupu
StepHypRef Expression
1 ltrelnq 7137 . . . . 5  |-  <Q  C_  ( Q.  X.  Q. )
21brel 4559 . . . 4  |-  ( s 
<Q  r  ->  ( s  e.  Q.  /\  r  e.  Q. ) )
32simprd 113 . . 3  |-  ( s 
<Q  r  ->  r  e. 
Q. )
433ad2ant2 986 . 2  |-  ( (
ph  /\  s  <Q  r  /\  s  e.  ( 2nd `  L ) )  ->  r  e.  Q. )
5 breq2 3901 . . . . . . 7  |-  ( u  =  s  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
s ) )
65rexbidv 2413 . . . . . 6  |-  ( 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 ) )
7 caucvgpr.lim . . . . . . . 8  |-  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 } >.
87fveq2i 5390 . . . . . . 7  |-  ( 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 } >. )
9 nqex 7135 . . . . . . . . 9  |-  Q.  e.  _V
109rabex 4040 . . . . . . . 8  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  _V
119rabex 4040 . . . . . . . 8  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  e.  _V
1210, 11op2nd 6011 . . . . . . 7  |-  ( 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 }
138, 12eqtri 2136 . . . . . 6  |-  ( 2nd `  L )  =  {
u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u }
146, 13elrab2 2814 . . . . 5  |-  ( s  e.  ( 2nd `  L
)  <->  ( s  e. 
Q.  /\  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
s ) )
1514simprbi 271 . . . 4  |-  ( s  e.  ( 2nd `  L
)  ->  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
s )
16153ad2ant3 987 . . 3  |-  ( (
ph  /\  s  <Q  r  /\  s  e.  ( 2nd `  L ) )  ->  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
s )
17 ltsonq 7170 . . . . . . 7  |-  <Q  Or  Q.
1817, 1sotri 4902 . . . . . 6  |-  ( ( ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  s  /\  s  <Q  r )  ->  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  r )
1918expcom 115 . . . . 5  |-  ( s 
<Q  r  ->  ( ( ( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  s  ->  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  r ) )
20193ad2ant2 986 . . . 4  |-  ( (
ph  /\  s  <Q  r  /\  s  e.  ( 2nd `  L ) )  ->  ( (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  s  ->  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  r ) )
2120reximdv 2508 . . 3  |-  ( (
ph  /\  s  <Q  r  /\  s  e.  ( 2nd `  L ) )  ->  ( E. j  e.  N.  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  s  ->  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
r ) )
2216, 21mpd 13 . 2  |-  ( (
ph  /\  s  <Q  r  /\  s  e.  ( 2nd `  L ) )  ->  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
r )
23 breq2 3901 . . . 4  |-  ( u  =  r  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
r ) )
2423rexbidv 2413 . . 3  |-  ( u  =  r  ->  ( 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 
r ) )
2524, 13elrab2 2814 . 2  |-  ( r  e.  ( 2nd `  L
)  <->  ( r  e. 
Q.  /\  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
r ) )
264, 22, 25sylanbrc 411 1  |-  ( (
ph  /\  s  <Q  r  /\  s  e.  ( 2nd `  L ) )  ->  r  e.  ( 2nd `  L ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 945    = wceq 1314    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   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-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-mi 7078  df-lti 7079  df-enq 7119  df-nqqs 7120  df-ltnqqs 7125
This theorem is referenced by:  caucvgprlemrnd  7445
  Copyright terms: Public domain W3C validator