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Theorem caucvgprprlemell 6926
Description: Lemma for caucvgprpr 6953. Membership in the lower cut of the putative limit. (Contributed by Jim Kingdon, 21-Jan-2021.)
Hypothesis
Ref Expression
caucvgprprlemell.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
caucvgprprlemell  |-  ( X  e.  ( 1st `  L
)  <->  ( X  e. 
Q.  /\  E. b  e.  N.  <. { p  |  p  <Q  ( X  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( X  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P 
( F `  b
) ) )
Distinct variable groups:    F, b    F, l, r    u, F, r    X, b, p    X, l, r, p    u, X, p    X, q, b    q,
l, r    u, q
Allowed substitution hints:    F( q, p)    L( u, r, q, p, b, l)

Proof of Theorem caucvgprprlemell
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 oveq1 5544 . . . . . . . 8  |-  ( l  =  X  ->  (
l  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
)  =  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) )
21breq2d 3799 . . . . . . 7  |-  ( l  =  X  ->  (
p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <->  p  <Q  ( X  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
) ) )
32abbidv 2197 . . . . . 6  |-  ( l  =  X  ->  { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) }  =  { p  |  p  <Q  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } )
41breq1d 3797 . . . . . . 7  |-  ( l  =  X  ->  (
( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  <Q  q  <->  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q ) )
54abbidv 2197 . . . . . 6  |-  ( l  =  X  ->  { q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q }  =  {
q  |  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } )
63, 5opeq12d 3580 . . . . 5  |-  ( l  =  X  ->  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q } >.  =  <. { p  |  p  <Q  ( X  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >. )
76breq1d 3797 . . . 4  |-  ( l  =  X  ->  ( <. { p  |  p 
<Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  r )  <->  <. { p  |  p  <Q  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  r )
) )
87rexbidv 2370 . . 3  |-  ( l  =  X  ->  ( 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 )  <->  E. r  e.  N.  <. { p  |  p  <Q  ( X  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  r )
) )
9 caucvgprprlemell.lim . . . . 5  |-  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 } >. } >.
109fveq2i 5206 . . . 4  |-  ( 1st `  L )  =  ( 1st `  <. { 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 } >. } >. )
11 nqex 6604 . . . . . 6  |-  Q.  e.  _V
1211rabex 3924 . . . . 5  |-  { 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 ) }  e.  _V
1311rabex 3924 . . . . 5  |-  { 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 } >. }  e.  _V
1412, 13op1st 5798 . . . 4  |-  ( 1st `  <. { 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 } >. } >. )  =  { 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 ) }
1510, 14eqtri 2102 . . 3  |-  ( 1st `  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 ) }
168, 15elrab2 2752 . 2  |-  ( X  e.  ( 1st `  L
)  <->  ( X  e. 
Q.  /\  E. r  e.  N.  <. { p  |  p  <Q  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P 
( F `  r
) ) )
17 opeq1 3572 . . . . . . . . . . . 12  |-  ( r  =  a  ->  <. r ,  1o >.  =  <. a ,  1o >. )
1817eceq1d 6201 . . . . . . . . . . 11  |-  ( r  =  a  ->  [ <. r ,  1o >. ]  ~Q  =  [ <. a ,  1o >. ]  ~Q  )
1918fveq2d 5207 . . . . . . . . . 10  |-  ( r  =  a  ->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )
2019oveq2d 5553 . . . . . . . . 9  |-  ( r  =  a  ->  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  =  ( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )
2120breq2d 3799 . . . . . . . 8  |-  ( r  =  a  ->  (
p  <Q  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <->  p  <Q  ( X  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
) ) )
2221abbidv 2197 . . . . . . 7  |-  ( r  =  a  ->  { p  |  p  <Q  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) }  =  { p  |  p  <Q  ( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) } )
2320breq1d 3797 . . . . . . . 8  |-  ( r  =  a  ->  (
( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  <Q  q  <->  ( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
q ) )
2423abbidv 2197 . . . . . . 7  |-  ( r  =  a  ->  { q  |  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q  q }  =  { q  |  ( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q  q } )
2522, 24opeq12d 3580 . . . . . 6  |-  ( r  =  a  ->  <. { p  |  p  <Q  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q  q } >.  =  <. { p  |  p  <Q  ( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q  q } >. )
26 fveq2 5203 . . . . . 6  |-  ( r  =  a  ->  ( F `  r )  =  ( F `  a ) )
2725, 26breq12d 3800 . . . . 5  |-  ( r  =  a  ->  ( <. { p  |  p 
<Q  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  r )  <->  <. { p  |  p  <Q  ( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  a )
) )
2827cbvrexv 2579 . . . 4  |-  ( E. r  e.  N.  <. { p  |  p  <Q  ( X  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  r )  <->  E. a  e.  N.  <. { p  |  p  <Q  ( X  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  a )
)
29 opeq1 3572 . . . . . . . . . . . 12  |-  ( a  =  b  ->  <. a ,  1o >.  =  <. b ,  1o >. )
3029eceq1d 6201 . . . . . . . . . . 11  |-  ( a  =  b  ->  [ <. a ,  1o >. ]  ~Q  =  [ <. b ,  1o >. ]  ~Q  )
3130fveq2d 5207 . . . . . . . . . 10  |-  ( a  =  b  ->  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )
3231oveq2d 5553 . . . . . . . . 9  |-  ( a  =  b  ->  ( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  =  ( X  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) )
3332breq2d 3799 . . . . . . . 8  |-  ( a  =  b  ->  (
p  <Q  ( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <->  p  <Q  ( X  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) ) )
3433abbidv 2197 . . . . . . 7  |-  ( a  =  b  ->  { p  |  p  <Q  ( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) }  =  { p  |  p  <Q  ( X  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) } )
3532breq1d 3797 . . . . . . . 8  |-  ( a  =  b  ->  (
( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  <Q  q  <->  ( X  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q 
q ) )
3635abbidv 2197 . . . . . . 7  |-  ( a  =  b  ->  { q  |  ( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q  q }  =  { q  |  ( X  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q  q } )
3734, 36opeq12d 3580 . . . . . 6  |-  ( a  =  b  ->  <. { p  |  p  <Q  ( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q  q } >.  =  <. { p  |  p  <Q  ( X  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( X  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q  q } >. )
38 fveq2 5203 . . . . . 6  |-  ( a  =  b  ->  ( F `  a )  =  ( F `  b ) )
3937, 38breq12d 3800 . . . . 5  |-  ( a  =  b  ->  ( <. { p  |  p 
<Q  ( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  a )  <->  <. { p  |  p  <Q  ( X  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( X  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )
4039cbvrexv 2579 . . . 4  |-  ( E. a  e.  N.  <. { p  |  p  <Q  ( X  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  a )  <->  E. b  e.  N.  <. { p  |  p  <Q  ( X  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( X  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
)
4128, 40bitri 182 . . 3  |-  ( E. r  e.  N.  <. { p  |  p  <Q  ( X  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  r )  <->  E. b  e.  N.  <. { p  |  p  <Q  ( X  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( X  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
)
4241anbi2i 445 . 2  |-  ( ( X  e.  Q.  /\  E. r  e.  N.  <. { p  |  p  <Q  ( X  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  r )
)  <->  ( X  e. 
Q.  /\  E. b  e.  N.  <. { p  |  p  <Q  ( X  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( X  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P 
( F `  b
) ) )
4316, 42bitri 182 1  |-  ( X  e.  ( 1st `  L
)  <->  ( X  e. 
Q.  /\  E. b  e.  N.  <. { p  |  p  <Q  ( X  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( X  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P 
( F `  b
) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   {cab 2068   E.wrex 2350   {crab 2353   <.cop 3403   class class class wbr 3787   ` cfv 4926  (class class class)co 5537   1stc1st 5790   1oc1o 6052   [cec 6163   N.cnpi 6513    ~Q ceq 6520   Q.cnq 6521    +Q cplq 6523   *Qcrq 6525    <Q cltq 6526    +P. cpp 6534    <P cltp 6536
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3895  ax-sep 3898  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-iinf 4331
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-int 3639  df-iun 3682  df-br 3788  df-opab 3842  df-mpt 3843  df-id 4050  df-iom 4334  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-f1 4931  df-fo 4932  df-f1o 4933  df-fv 4934  df-ov 5540  df-1st 5792  df-ec 6167  df-qs 6171  df-ni 6545  df-nqqs 6589
This theorem is referenced by:  caucvgprprlemopl  6938  caucvgprprlemlol  6939  caucvgprprlemdisj  6943  caucvgprprlemloc  6944
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