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Theorem caucvgprprlemell 7745
Description: Lemma for caucvgprpr 7772. 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 5925 . . . . . . . 8  |-  ( l  =  X  ->  (
l  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
)  =  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) )
21breq2d 4041 . . . . . . 7  |-  ( l  =  X  ->  (
p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <->  p  <Q  ( X  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
) ) )
32abbidv 2311 . . . . . 6  |-  ( l  =  X  ->  { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) }  =  { p  |  p  <Q  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } )
41breq1d 4039 . . . . . . 7  |-  ( l  =  X  ->  (
( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  <Q  q  <->  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q ) )
54abbidv 2311 . . . . . 6  |-  ( l  =  X  ->  { q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q }  =  {
q  |  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } )
63, 5opeq12d 3812 . . . . 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 4039 . . . 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 2495 . . 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 5557 . . . 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 7423 . . . . . 6  |-  Q.  e.  _V
1211rabex 4173 . . . . 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 4173 . . . . 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 6199 . . . 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 2214 . . 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 2919 . 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 3804 . . . . . . . . . . . 12  |-  ( r  =  a  ->  <. r ,  1o >.  =  <. a ,  1o >. )
1817eceq1d 6623 . . . . . . . . . . 11  |-  ( r  =  a  ->  [ <. r ,  1o >. ]  ~Q  =  [ <. a ,  1o >. ]  ~Q  )
1918fveq2d 5558 . . . . . . . . . 10  |-  ( r  =  a  ->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )
2019oveq2d 5934 . . . . . . . . 9  |-  ( r  =  a  ->  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  =  ( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )
2120breq2d 4041 . . . . . . . 8  |-  ( r  =  a  ->  (
p  <Q  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <->  p  <Q  ( X  +Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )
) ) )
2221abbidv 2311 . . . . . . 7  |-  ( r  =  a  ->  { p  |  p  <Q  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) }  =  { p  |  p  <Q  ( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) } )
2320breq1d 4039 . . . . . . . 8  |-  ( r  =  a  ->  (
( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  <Q  q  <->  ( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
q ) )
2423abbidv 2311 . . . . . . 7  |-  ( r  =  a  ->  { q  |  ( X  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q  q }  =  { q  |  ( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q  q } )
2522, 24opeq12d 3812 . . . . . 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 5554 . . . . . 6  |-  ( r  =  a  ->  ( F `  r )  =  ( F `  a ) )
2725, 26breq12d 4042 . . . . 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 2727 . . . 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 3804 . . . . . . . . . . . 12  |-  ( a  =  b  ->  <. a ,  1o >.  =  <. b ,  1o >. )
3029eceq1d 6623 . . . . . . . . . . 11  |-  ( a  =  b  ->  [ <. a ,  1o >. ]  ~Q  =  [ <. b ,  1o >. ]  ~Q  )
3130fveq2d 5558 . . . . . . . . . 10  |-  ( a  =  b  ->  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )
3231oveq2d 5934 . . . . . . . . 9  |-  ( a  =  b  ->  ( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  =  ( X  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) )
3332breq2d 4041 . . . . . . . 8  |-  ( a  =  b  ->  (
p  <Q  ( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <->  p  <Q  ( X  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) ) )
3433abbidv 2311 . . . . . . 7  |-  ( a  =  b  ->  { p  |  p  <Q  ( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) }  =  { p  |  p  <Q  ( X  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) } )
3532breq1d 4039 . . . . . . . 8  |-  ( a  =  b  ->  (
( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  <Q  q  <->  ( X  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q 
q ) )
3635abbidv 2311 . . . . . . 7  |-  ( a  =  b  ->  { q  |  ( X  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q  q }  =  { q  |  ( X  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q  q } )
3734, 36opeq12d 3812 . . . . . 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 5554 . . . . . 6  |-  ( a  =  b  ->  ( F `  a )  =  ( F `  b ) )
3937, 38breq12d 4042 . . . . 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 2727 . . . 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 184 . . 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 457 . 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 184 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 104    <-> wb 105    = wceq 1364    e. wcel 2164   {cab 2179   E.wrex 2473   {crab 2476   <.cop 3621   class class class wbr 4029   ` cfv 5254  (class class class)co 5918   1stc1st 6191   1oc1o 6462   [cec 6585   N.cnpi 7332    ~Q ceq 7339   Q.cnq 7340    +Q cplq 7342   *Qcrq 7344    <Q cltq 7345    +P. cpp 7353    <P cltp 7355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-iinf 4620
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-1st 6193  df-ec 6589  df-qs 6593  df-ni 7364  df-nqqs 7408
This theorem is referenced by:  caucvgprprlemopl  7757  caucvgprprlemlol  7758  caucvgprprlemdisj  7762  caucvgprprlemloc  7763
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