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

Theorem caucvgprprlemelu 7676
Description: Lemma for caucvgprpr 7702. 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 4004 . . . . . . 7  |-  ( u  =  X  ->  (
p  <Q  u  <->  p  <Q  X ) )
21abbidv 2295 . . . . . 6  |-  ( u  =  X  ->  { p  |  p  <Q  u }  =  { p  |  p 
<Q  X } )
3 breq1 4003 . . . . . . 7  |-  ( u  =  X  ->  (
u  <Q  q  <->  X  <Q  q ) )
43abbidv 2295 . . . . . 6  |-  ( u  =  X  ->  { q  |  u  <Q  q }  =  { q  |  X  <Q  q } )
52, 4opeq12d 3784 . . . . 5  |-  ( u  =  X  ->  <. { p  |  p  <Q  u } ,  { q  |  u 
<Q  q } >.  =  <. { p  |  p  <Q  X } ,  { q  |  X  <Q  q } >. )
65breq2d 4012 . . . 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 2478 . . 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 5514 . . . 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 7353 . . . . . 6  |-  Q.  e.  _V
1110rabex 4144 . . . . 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 4144 . . . . 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 6142 . . . 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 2198 . . 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 2896 . 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 5511 . . . . . . 7  |-  ( r  =  a  ->  ( F `  r )  =  ( F `  a ) )
17 opeq1 3776 . . . . . . . . . . . 12  |-  ( r  =  a  ->  <. r ,  1o >.  =  <. a ,  1o >. )
1817eceq1d 6565 . . . . . . . . . . 11  |-  ( r  =  a  ->  [ <. r ,  1o >. ]  ~Q  =  [ <. a ,  1o >. ]  ~Q  )
1918fveq2d 5515 . . . . . . . . . 10  |-  ( r  =  a  ->  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )
2019breq2d 4012 . . . . . . . . 9  |-  ( r  =  a  ->  (
p  <Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <->  p  <Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )
2120abbidv 2295 . . . . . . . 8  |-  ( r  =  a  ->  { p  |  p  <Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  ) }  =  { p  |  p  <Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  ) } )
2219breq1d 4010 . . . . . . . . 9  |-  ( r  =  a  ->  (
( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  q ) )
2322abbidv 2295 . . . . . . . 8  |-  ( r  =  a  ->  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q 
q }  =  {
q  |  ( *Q
`  [ <. a ,  1o >. ]  ~Q  )  <Q  q } )
2421, 23opeq12d 3784 . . . . . . 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 5887 . . . . . 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 4010 . . . . 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 2704 . . . 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 5511 . . . . . . 7  |-  ( a  =  b  ->  ( F `  a )  =  ( F `  b ) )
29 opeq1 3776 . . . . . . . . . . . 12  |-  ( a  =  b  ->  <. a ,  1o >.  =  <. b ,  1o >. )
3029eceq1d 6565 . . . . . . . . . . 11  |-  ( a  =  b  ->  [ <. a ,  1o >. ]  ~Q  =  [ <. b ,  1o >. ]  ~Q  )
3130fveq2d 5515 . . . . . . . . . 10  |-  ( a  =  b  ->  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )
3231breq2d 4012 . . . . . . . . 9  |-  ( a  =  b  ->  (
p  <Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <->  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) )
3332abbidv 2295 . . . . . . . 8  |-  ( a  =  b  ->  { p  |  p  <Q  ( *Q
`  [ <. a ,  1o >. ]  ~Q  ) }  =  { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } )
3431breq1d 4010 . . . . . . . . 9  |-  ( a  =  b  ->  (
( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q  q  <->  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q ) )
3534abbidv 2295 . . . . . . . 8  |-  ( a  =  b  ->  { q  |  ( *Q `  [ <. a ,  1o >. ]  ~Q  )  <Q 
q }  =  {
q  |  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )  <Q  q } )
3633, 35opeq12d 3784 . . . . . . 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 5887 . . . . . 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 4010 . . . . 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 2704 . . . 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 184 . . 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 457 . 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 184 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 104    <-> wb 105    = wceq 1353    e. wcel 2148   {cab 2163   E.wrex 2456   {crab 2459   <.cop 3594   class class class wbr 4000   ` cfv 5212  (class class class)co 5869   2ndc2nd 6134   1oc1o 6404   [cec 6527   N.cnpi 7262    ~Q ceq 7269   Q.cnq 7270    +Q cplq 7272   *Qcrq 7274    <Q cltq 7275    +P. cpp 7283    <P cltp 7285
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-2nd 6136  df-ec 6531  df-qs 6535  df-ni 7294  df-nqqs 7338
This theorem is referenced by:  caucvgprprlemopu  7689  caucvgprprlemupu  7690  caucvgprprlemdisj  7692  caucvgprprlemloc  7693
  Copyright terms: Public domain W3C validator