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

Theorem caucvgprprlemelu 7189
Description: Lemma for caucvgprpr 7215. Membership in the upper cut of the putative limit. (Contributed by Jim Kingdon, 28-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
caucvgprprlemelu  |-  ( X  e.  ( 2nd `  L
)  <->  ( X  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  X } ,  { q  |  X  <Q  q } >. )
)
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 caucvgprprlemelu
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 breq2 3824 . . . . . . 7  |-  ( u  =  X  ->  (
p  <Q  u  <->  p  <Q  X ) )
21abbidv 2202 . . . . . 6  |-  ( u  =  X  ->  { p  |  p  <Q  u }  =  { p  |  p 
<Q  X } )
3 breq1 3823 . . . . . . 7  |-  ( u  =  X  ->  (
u  <Q  q  <->  X  <Q  q ) )
43abbidv 2202 . . . . . 6  |-  ( u  =  X  ->  { q  |  u  <Q  q }  =  { q  |  X  <Q  q } )
52, 4opeq12d 3613 . . . . 5  |-  ( u  =  X  ->  <. { p  |  p  <Q  u } ,  { q  |  u 
<Q  q } >.  =  <. { p  |  p  <Q  X } ,  { q  |  X  <Q  q } >. )
65breq2d 3832 . . . 4  |-  ( u  =  X  ->  (
( ( 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 } >.  <->  ( ( F `
 r )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  X } ,  { q  |  X  <Q  q } >. )
)
76rexbidv 2377 . . 3  |-  ( u  =  X  ->  ( 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. r  e.  N.  ( ( F `  r )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  X } ,  {
q  |  X  <Q  q } >. ) )
8 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 } >. } >.
98fveq2i 5271 . . . 4  |-  ( 2nd `  L )  =  ( 2nd `  <. { 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 } >. } >. )
10 nqex 6866 . . . . . 6  |-  Q.  e.  _V
1110rabex 3958 . . . . 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
1210rabex 3958 . . . . 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
1311, 12op2nd 5875 . . . 4  |-  ( 2nd `  <. { 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 } >. } >. )  =  { 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 } >. }
149, 13eqtri 2105 . . 3  |-  ( 2nd `  L )  =  {
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 } >. }
157, 14elrab2 2765 . 2  |-  ( X  e.  ( 2nd `  L
)  <->  ( X  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  X } ,  { q  |  X  <Q  q } >. )
)
16 fveq2 5268 . . . . . . 7  |-  ( r  =  a  ->  ( F `  r )  =  ( F `  a ) )
17 opeq1 3605 . . . . . . . . . . . 12  |-  ( r  =  a  ->  <. r ,  1o >.  =  <. a ,  1o >. )
1817eceq1d 6280 . . . . . . . . . . 11  |-  ( r  =  a  ->  [ <. r ,  1o >. ]  ~Q  =  [ <. a ,  1o >. ]  ~Q  )
1918fveq2d 5272 . . . . . . . . . 10  |-  ( r  =  a  ->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )
2019breq2d 3832 . . . . . . . . 9  |-  ( r  =  a  ->  (
p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <->  p  <Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )
2120abbidv 2202 . . . . . . . 8  |-  ( r  =  a  ->  { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) }  =  { p  |  p  <Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  ) } )
2219breq1d 3830 . . . . . . . . 9  |-  ( r  =  a  ->  (
( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  q ) )
2322abbidv 2202 . . . . . . . 8  |-  ( r  =  a  ->  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q }  =  {
q  |  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )  <Q  q } )
2421, 23opeq12d 3613 . . . . . . 7  |-  ( r  =  a  ->  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >.  =  <. { p  |  p  <Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q 
q } >. )
2516, 24oveq12d 5631 . . . . . 6  |-  ( r  =  a  ->  (
( F `  r
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  =  ( ( F `
 a )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  q } >. ) )
2625breq1d 3830 . . . . 5  |-  ( r  =  a  ->  (
( ( F `  r )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  X } ,  {
q  |  X  <Q  q } >.  <->  ( ( F `
 a )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  X } ,  { q  |  X  <Q  q } >. )
)
2726cbvrexv 2587 . . . 4  |-  ( 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  X } ,  {
q  |  X  <Q  q } >.  <->  E. a  e.  N.  ( ( F `  a )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  X } ,  {
q  |  X  <Q  q } >. )
28 fveq2 5268 . . . . . . 7  |-  ( a  =  b  ->  ( F `  a )  =  ( F `  b ) )
29 opeq1 3605 . . . . . . . . . . . 12  |-  ( a  =  b  ->  <. a ,  1o >.  =  <. b ,  1o >. )
3029eceq1d 6280 . . . . . . . . . . 11  |-  ( a  =  b  ->  [ <. a ,  1o >. ]  ~Q  =  [ <. b ,  1o >. ]  ~Q  )
3130fveq2d 5272 . . . . . . . . . 10  |-  ( a  =  b  ->  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )
3231breq2d 3832 . . . . . . . . 9  |-  ( a  =  b  ->  (
p  <Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <->  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) )
3332abbidv 2202 . . . . . . . 8  |-  ( a  =  b  ->  { p  |  p  <Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  ) }  =  { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } )
3431breq1d 3830 . . . . . . . . 9  |-  ( a  =  b  ->  (
( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q ) )
3534abbidv 2202 . . . . . . . 8  |-  ( a  =  b  ->  { q  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q 
q }  =  {
q  |  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )  <Q  q } )
3633, 35opeq12d 3613 . . . . . . 7  |-  ( a  =  b  ->  <. { p  |  p  <Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q 
q } >.  =  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )
3728, 36oveq12d 5631 . . . . . 6  |-  ( a  =  b  ->  (
( F `  a
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q 
q } >. )  =  ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. ) )
3837breq1d 3830 . . . . 5  |-  ( a  =  b  ->  (
( ( F `  a )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  X } ,  {
q  |  X  <Q  q } >.  <->  ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  X } ,  { q  |  X  <Q  q } >. )
)
3938cbvrexv 2587 . . . 4  |-  ( E. a  e.  N.  (
( F `  a
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P 
<. { p  |  p 
<Q  X } ,  {
q  |  X  <Q  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  X } ,  {
q  |  X  <Q  q } >. )
4027, 39bitri 182 . . 3  |-  ( 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  X } ,  {
q  |  X  <Q  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  X } ,  {
q  |  X  <Q  q } >. )
4140anbi2i 445 . 2  |-  ( ( X  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  X } ,  {
q  |  X  <Q  q } >. )  <->  ( X  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  X } ,  { q  |  X  <Q  q } >. )
)
4215, 41bitri 182 1  |-  ( X  e.  ( 2nd `  L
)  <->  ( X  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  X } ,  { q  |  X  <Q  q } >. )
)
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    = wceq 1287    e. wcel 1436   {cab 2071   E.wrex 2356   {crab 2359   <.cop 3434   class class class wbr 3820   ` cfv 4981  (class class class)co 5613   2ndc2nd 5867   1oc1o 6128   [cec 6242   N.cnpi 6775    ~Q ceq 6782   Q.cnq 6783    +Q cplq 6785   *Qcrq 6787    <Q cltq 6788    +P. cpp 6796    <P cltp 6798
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3929  ax-sep 3932  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-iinf 4376
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-int 3672  df-iun 3715  df-br 3821  df-opab 3875  df-mpt 3876  df-id 4094  df-iom 4379  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-res 4423  df-ima 4424  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-f1 4986  df-fo 4987  df-f1o 4988  df-fv 4989  df-ov 5616  df-2nd 5869  df-ec 6246  df-qs 6250  df-ni 6807  df-nqqs 6851
This theorem is referenced by:  caucvgprprlemopu  7202  caucvgprprlemupu  7203  caucvgprprlemdisj  7205  caucvgprprlemloc  7206
  Copyright terms: Public domain W3C validator