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Theorem caucvgprprlemell 7500
Description: Lemma for caucvgprpr 7527. 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 5781 . . . . . . . 8  |-  ( l  =  X  ->  (
l  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
)  =  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) )
21breq2d 3941 . . . . . . 7  |-  ( l  =  X  ->  (
p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <->  p  <Q  ( X  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
) ) )
32abbidv 2257 . . . . . 6  |-  ( l  =  X  ->  { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) }  =  { p  |  p  <Q  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } )
41breq1d 3939 . . . . . . 7  |-  ( l  =  X  ->  (
( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  <Q  q  <->  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q ) )
54abbidv 2257 . . . . . 6  |-  ( l  =  X  ->  { q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q }  =  {
q  |  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } )
63, 5opeq12d 3713 . . . . 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 3939 . . . 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 2438 . . 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 5424 . . . 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 7178 . . . . . 6  |-  Q.  e.  _V
1211rabex 4072 . . . . 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 4072 . . . . 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 6044 . . . 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 2160 . . 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 2843 . 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 3705 . . . . . . . . . . . 12  |-  ( r  =  a  ->  <. r ,  1o >.  =  <. a ,  1o >. )
1817eceq1d 6465 . . . . . . . . . . 11  |-  ( r  =  a  ->  [ <. r ,  1o >. ]  ~Q  =  [ <. a ,  1o >. ]  ~Q  )
1918fveq2d 5425 . . . . . . . . . 10  |-  ( r  =  a  ->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )
2019oveq2d 5790 . . . . . . . . 9  |-  ( r  =  a  ->  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  =  ( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )
2120breq2d 3941 . . . . . . . 8  |-  ( r  =  a  ->  (
p  <Q  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <->  p  <Q  ( X  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
) ) )
2221abbidv 2257 . . . . . . 7  |-  ( r  =  a  ->  { p  |  p  <Q  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) }  =  { p  |  p  <Q  ( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) } )
2320breq1d 3939 . . . . . . . 8  |-  ( r  =  a  ->  (
( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  <Q  q  <->  ( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
q ) )
2423abbidv 2257 . . . . . . 7  |-  ( r  =  a  ->  { q  |  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q  q }  =  { q  |  ( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q  q } )
2522, 24opeq12d 3713 . . . . . 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 5421 . . . . . 6  |-  ( r  =  a  ->  ( F `  r )  =  ( F `  a ) )
2725, 26breq12d 3942 . . . . 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 2655 . . . 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 3705 . . . . . . . . . . . 12  |-  ( a  =  b  ->  <. a ,  1o >.  =  <. b ,  1o >. )
3029eceq1d 6465 . . . . . . . . . . 11  |-  ( a  =  b  ->  [ <. a ,  1o >. ]  ~Q  =  [ <. b ,  1o >. ]  ~Q  )
3130fveq2d 5425 . . . . . . . . . 10  |-  ( a  =  b  ->  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )
3231oveq2d 5790 . . . . . . . . 9  |-  ( a  =  b  ->  ( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  =  ( X  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) )
3332breq2d 3941 . . . . . . . 8  |-  ( a  =  b  ->  (
p  <Q  ( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <->  p  <Q  ( X  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) ) )
3433abbidv 2257 . . . . . . 7  |-  ( a  =  b  ->  { p  |  p  <Q  ( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) }  =  { p  |  p  <Q  ( X  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) } )
3532breq1d 3939 . . . . . . . 8  |-  ( a  =  b  ->  (
( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  <Q  q  <->  ( X  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q 
q ) )
3635abbidv 2257 . . . . . . 7  |-  ( a  =  b  ->  { q  |  ( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q  q }  =  { q  |  ( X  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q  q } )
3734, 36opeq12d 3713 . . . . . 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 5421 . . . . . 6  |-  ( a  =  b  ->  ( F `  a )  =  ( F `  b ) )
3937, 38breq12d 3942 . . . . 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 2655 . . . 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 183 . . 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 452 . 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 183 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 103    <-> wb 104    = wceq 1331    e. wcel 1480   {cab 2125   E.wrex 2417   {crab 2420   <.cop 3530   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   1stc1st 6036   1oc1o 6306   [cec 6427   N.cnpi 7087    ~Q ceq 7094   Q.cnq 7095    +Q cplq 7097   *Qcrq 7099    <Q cltq 7100    +P. cpp 7108    <P cltp 7110
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-1st 6038  df-ec 6431  df-qs 6435  df-ni 7119  df-nqqs 7163
This theorem is referenced by:  caucvgprprlemopl  7512  caucvgprprlemlol  7513  caucvgprprlemdisj  7517  caucvgprprlemloc  7518
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