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

Theorem cauappcvgprlemlim 6817
Description: Lemma for cauappcvgpr 6818. The putative limit is a limit. (Contributed by Jim Kingdon, 20-Jun-2020.)
Hypotheses
Ref Expression
cauappcvgpr.f  |-  ( ph  ->  F : Q. --> Q. )
cauappcvgpr.app  |-  ( ph  ->  A. p  e.  Q.  A. q  e.  Q.  (
( F `  p
)  <Q  ( ( F `
 q )  +Q  ( p  +Q  q
) )  /\  ( F `  q )  <Q  ( ( F `  p )  +Q  (
p  +Q  q ) ) ) )
cauappcvgpr.bnd  |-  ( ph  ->  A. p  e.  Q.  A  <Q  ( F `  p ) )
cauappcvgpr.lim  |-  L  = 
<. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( F `  q ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  q
)  <Q  u } >.
Assertion
Ref Expression
cauappcvgprlemlim  |-  ( ph  ->  A. q  e.  Q.  A. r  e.  Q.  ( <. { l  |  l 
<Q  ( F `  q
) } ,  {
u  |  ( F `
 q )  <Q  u } >.  <P  ( L  +P.  <. { l  |  l  <Q  ( q  +Q  r ) } ,  { u  |  (
q  +Q  r ) 
<Q  u } >. )  /\  L  <P  <. { l  |  l  <Q  (
( F `  q
)  +Q  ( q  +Q  r ) ) } ,  { u  |  ( ( F `
 q )  +Q  ( q  +Q  r
) )  <Q  u } >. ) )
Distinct variable groups:    A, p    L, p, q    ph, p, q    F, l, p, q, r, u    L, r
Allowed substitution hints:    ph( u, r, l)    A( u, r, q, l)    L( u, l)

Proof of Theorem cauappcvgprlemlim
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cauappcvgpr.f . . . . . 6  |-  ( ph  ->  F : Q. --> Q. )
21adantr 265 . . . . 5  |-  ( (
ph  /\  ( x  e.  Q.  /\  y  e. 
Q. ) )  ->  F : Q. --> Q. )
3 cauappcvgpr.app . . . . . 6  |-  ( ph  ->  A. p  e.  Q.  A. q  e.  Q.  (
( F `  p
)  <Q  ( ( F `
 q )  +Q  ( p  +Q  q
) )  /\  ( F `  q )  <Q  ( ( F `  p )  +Q  (
p  +Q  q ) ) ) )
43adantr 265 . . . . 5  |-  ( (
ph  /\  ( x  e.  Q.  /\  y  e. 
Q. ) )  ->  A. p  e.  Q.  A. q  e.  Q.  (
( F `  p
)  <Q  ( ( F `
 q )  +Q  ( p  +Q  q
) )  /\  ( F `  q )  <Q  ( ( F `  p )  +Q  (
p  +Q  q ) ) ) )
5 cauappcvgpr.bnd . . . . . 6  |-  ( ph  ->  A. p  e.  Q.  A  <Q  ( F `  p ) )
65adantr 265 . . . . 5  |-  ( (
ph  /\  ( x  e.  Q.  /\  y  e. 
Q. ) )  ->  A. p  e.  Q.  A  <Q  ( F `  p ) )
7 cauappcvgpr.lim . . . . 5  |-  L  = 
<. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( F `  q ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  q
)  <Q  u } >.
8 simprl 491 . . . . 5  |-  ( (
ph  /\  ( x  e.  Q.  /\  y  e. 
Q. ) )  ->  x  e.  Q. )
9 simprr 492 . . . . 5  |-  ( (
ph  /\  ( x  e.  Q.  /\  y  e. 
Q. ) )  -> 
y  e.  Q. )
102, 4, 6, 7, 8, 9cauappcvgprlem1 6815 . . . 4  |-  ( (
ph  /\  ( x  e.  Q.  /\  y  e. 
Q. ) )  ->  <. { l  |  l 
<Q  ( F `  x
) } ,  {
u  |  ( F `
 x )  <Q  u } >.  <P  ( L  +P.  <. { l  |  l  <Q  ( x  +Q  y ) } ,  { u  |  (
x  +Q  y ) 
<Q  u } >. )
)
112, 4, 6, 7, 8, 9cauappcvgprlem2 6816 . . . 4  |-  ( (
ph  /\  ( x  e.  Q.  /\  y  e. 
Q. ) )  ->  L  <P  <. { l  |  l  <Q  ( ( F `  x )  +Q  ( x  +Q  y
) ) } ,  { u  |  (
( F `  x
)  +Q  ( x  +Q  y ) ) 
<Q  u } >. )
1210, 11jca 294 . . 3  |-  ( (
ph  /\  ( x  e.  Q.  /\  y  e. 
Q. ) )  -> 
( <. { l  |  l  <Q  ( F `  x ) } ,  { u  |  ( F `  x )  <Q  u } >.  <P  ( L  +P.  <. { l  |  l  <Q  ( x  +Q  y ) } ,  { u  |  (
x  +Q  y ) 
<Q  u } >. )  /\  L  <P  <. { l  |  l  <Q  (
( F `  x
)  +Q  ( x  +Q  y ) ) } ,  { u  |  ( ( F `
 x )  +Q  ( x  +Q  y
) )  <Q  u } >. ) )
1312ralrimivva 2418 . 2  |-  ( ph  ->  A. x  e.  Q.  A. y  e.  Q.  ( <. { l  |  l 
<Q  ( F `  x
) } ,  {
u  |  ( F `
 x )  <Q  u } >.  <P  ( L  +P.  <. { l  |  l  <Q  ( x  +Q  y ) } ,  { u  |  (
x  +Q  y ) 
<Q  u } >. )  /\  L  <P  <. { l  |  l  <Q  (
( F `  x
)  +Q  ( x  +Q  y ) ) } ,  { u  |  ( ( F `
 x )  +Q  ( x  +Q  y
) )  <Q  u } >. ) )
14 fveq2 5206 . . . . . . . 8  |-  ( x  =  q  ->  ( F `  x )  =  ( F `  q ) )
1514breq2d 3804 . . . . . . 7  |-  ( x  =  q  ->  (
l  <Q  ( F `  x )  <->  l  <Q  ( F `  q ) ) )
1615abbidv 2171 . . . . . 6  |-  ( x  =  q  ->  { l  |  l  <Q  ( F `  x ) }  =  { l  |  l  <Q  ( F `
 q ) } )
1714breq1d 3802 . . . . . . 7  |-  ( x  =  q  ->  (
( F `  x
)  <Q  u  <->  ( F `  q )  <Q  u
) )
1817abbidv 2171 . . . . . 6  |-  ( x  =  q  ->  { u  |  ( F `  x )  <Q  u }  =  { u  |  ( F `  q )  <Q  u } )
1916, 18opeq12d 3585 . . . . 5  |-  ( x  =  q  ->  <. { l  |  l  <Q  ( F `  x ) } ,  { u  |  ( F `  x )  <Q  u } >.  =  <. { l  |  l  <Q  ( F `  q ) } ,  { u  |  ( F `  q )  <Q  u } >. )
20 oveq1 5547 . . . . . . . . 9  |-  ( x  =  q  ->  (
x  +Q  y )  =  ( q  +Q  y ) )
2120breq2d 3804 . . . . . . . 8  |-  ( x  =  q  ->  (
l  <Q  ( x  +Q  y )  <->  l  <Q  ( q  +Q  y ) ) )
2221abbidv 2171 . . . . . . 7  |-  ( x  =  q  ->  { l  |  l  <Q  (
x  +Q  y ) }  =  { l  |  l  <Q  (
q  +Q  y ) } )
2320breq1d 3802 . . . . . . . 8  |-  ( x  =  q  ->  (
( x  +Q  y
)  <Q  u  <->  ( q  +Q  y )  <Q  u
) )
2423abbidv 2171 . . . . . . 7  |-  ( x  =  q  ->  { u  |  ( x  +Q  y )  <Q  u }  =  { u  |  ( q  +Q  y )  <Q  u } )
2522, 24opeq12d 3585 . . . . . 6  |-  ( x  =  q  ->  <. { l  |  l  <Q  (
x  +Q  y ) } ,  { u  |  ( x  +Q  y )  <Q  u } >.  =  <. { l  |  l  <Q  (
q  +Q  y ) } ,  { u  |  ( q  +Q  y )  <Q  u } >. )
2625oveq2d 5556 . . . . 5  |-  ( x  =  q  ->  ( L  +P.  <. { l  |  l  <Q  ( x  +Q  y ) } ,  { u  |  (
x  +Q  y ) 
<Q  u } >. )  =  ( L  +P.  <. { l  |  l 
<Q  ( q  +Q  y
) } ,  {
u  |  ( q  +Q  y )  <Q  u } >. ) )
2719, 26breq12d 3805 . . . 4  |-  ( x  =  q  ->  ( <. { l  |  l 
<Q  ( F `  x
) } ,  {
u  |  ( F `
 x )  <Q  u } >.  <P  ( L  +P.  <. { l  |  l  <Q  ( x  +Q  y ) } ,  { u  |  (
x  +Q  y ) 
<Q  u } >. )  <->  <. { l  |  l 
<Q  ( F `  q
) } ,  {
u  |  ( F `
 q )  <Q  u } >.  <P  ( L  +P.  <. { l  |  l  <Q  ( q  +Q  y ) } ,  { u  |  (
q  +Q  y ) 
<Q  u } >. )
) )
2814, 20oveq12d 5558 . . . . . . . 8  |-  ( x  =  q  ->  (
( F `  x
)  +Q  ( x  +Q  y ) )  =  ( ( F `
 q )  +Q  ( q  +Q  y
) ) )
2928breq2d 3804 . . . . . . 7  |-  ( x  =  q  ->  (
l  <Q  ( ( F `
 x )  +Q  ( x  +Q  y
) )  <->  l  <Q  ( ( F `  q
)  +Q  ( q  +Q  y ) ) ) )
3029abbidv 2171 . . . . . 6  |-  ( x  =  q  ->  { l  |  l  <Q  (
( F `  x
)  +Q  ( x  +Q  y ) ) }  =  { l  |  l  <Q  (
( F `  q
)  +Q  ( q  +Q  y ) ) } )
3128breq1d 3802 . . . . . . 7  |-  ( x  =  q  ->  (
( ( F `  x )  +Q  (
x  +Q  y ) )  <Q  u  <->  ( ( F `  q )  +Q  ( q  +Q  y
) )  <Q  u
) )
3231abbidv 2171 . . . . . 6  |-  ( x  =  q  ->  { u  |  ( ( F `
 x )  +Q  ( x  +Q  y
) )  <Q  u }  =  { u  |  ( ( F `
 q )  +Q  ( q  +Q  y
) )  <Q  u } )
3330, 32opeq12d 3585 . . . . 5  |-  ( x  =  q  ->  <. { l  |  l  <Q  (
( F `  x
)  +Q  ( x  +Q  y ) ) } ,  { u  |  ( ( F `
 x )  +Q  ( x  +Q  y
) )  <Q  u } >.  =  <. { l  |  l  <Q  (
( F `  q
)  +Q  ( q  +Q  y ) ) } ,  { u  |  ( ( F `
 q )  +Q  ( q  +Q  y
) )  <Q  u } >. )
3433breq2d 3804 . . . 4  |-  ( x  =  q  ->  ( L  <P  <. { l  |  l  <Q  ( ( F `  x )  +Q  ( x  +Q  y
) ) } ,  { u  |  (
( F `  x
)  +Q  ( x  +Q  y ) ) 
<Q  u } >.  <->  L  <P  <. { l  |  l 
<Q  ( ( F `  q )  +Q  (
q  +Q  y ) ) } ,  {
u  |  ( ( F `  q )  +Q  ( q  +Q  y ) )  <Q  u } >. ) )
3527, 34anbi12d 450 . . 3  |-  ( x  =  q  ->  (
( <. { l  |  l  <Q  ( F `  x ) } ,  { u  |  ( F `  x )  <Q  u } >.  <P  ( L  +P.  <. { l  |  l  <Q  ( x  +Q  y ) } ,  { u  |  (
x  +Q  y ) 
<Q  u } >. )  /\  L  <P  <. { l  |  l  <Q  (
( F `  x
)  +Q  ( x  +Q  y ) ) } ,  { u  |  ( ( F `
 x )  +Q  ( x  +Q  y
) )  <Q  u } >. )  <->  ( <. { l  |  l  <Q 
( F `  q
) } ,  {
u  |  ( F `
 q )  <Q  u } >.  <P  ( L  +P.  <. { l  |  l  <Q  ( q  +Q  y ) } ,  { u  |  (
q  +Q  y ) 
<Q  u } >. )  /\  L  <P  <. { l  |  l  <Q  (
( F `  q
)  +Q  ( q  +Q  y ) ) } ,  { u  |  ( ( F `
 q )  +Q  ( q  +Q  y
) )  <Q  u } >. ) ) )
36 oveq2 5548 . . . . . . . . 9  |-  ( y  =  r  ->  (
q  +Q  y )  =  ( q  +Q  r ) )
3736breq2d 3804 . . . . . . . 8  |-  ( y  =  r  ->  (
l  <Q  ( q  +Q  y )  <->  l  <Q  ( q  +Q  r ) ) )
3837abbidv 2171 . . . . . . 7  |-  ( y  =  r  ->  { l  |  l  <Q  (
q  +Q  y ) }  =  { l  |  l  <Q  (
q  +Q  r ) } )
3936breq1d 3802 . . . . . . . 8  |-  ( y  =  r  ->  (
( q  +Q  y
)  <Q  u  <->  ( q  +Q  r )  <Q  u
) )
4039abbidv 2171 . . . . . . 7  |-  ( y  =  r  ->  { u  |  ( q  +Q  y )  <Q  u }  =  { u  |  ( q  +Q  r )  <Q  u } )
4138, 40opeq12d 3585 . . . . . 6  |-  ( y  =  r  ->  <. { l  |  l  <Q  (
q  +Q  y ) } ,  { u  |  ( q  +Q  y )  <Q  u } >.  =  <. { l  |  l  <Q  (
q  +Q  r ) } ,  { u  |  ( q  +Q  r )  <Q  u } >. )
4241oveq2d 5556 . . . . 5  |-  ( y  =  r  ->  ( L  +P.  <. { l  |  l  <Q  ( q  +Q  y ) } ,  { u  |  (
q  +Q  y ) 
<Q  u } >. )  =  ( L  +P.  <. { l  |  l 
<Q  ( q  +Q  r
) } ,  {
u  |  ( q  +Q  r )  <Q  u } >. ) )
4342breq2d 3804 . . . 4  |-  ( y  =  r  ->  ( <. { l  |  l 
<Q  ( F `  q
) } ,  {
u  |  ( F `
 q )  <Q  u } >.  <P  ( L  +P.  <. { l  |  l  <Q  ( q  +Q  y ) } ,  { u  |  (
q  +Q  y ) 
<Q  u } >. )  <->  <. { l  |  l 
<Q  ( F `  q
) } ,  {
u  |  ( F `
 q )  <Q  u } >.  <P  ( L  +P.  <. { l  |  l  <Q  ( q  +Q  r ) } ,  { u  |  (
q  +Q  r ) 
<Q  u } >. )
) )
4436oveq2d 5556 . . . . . . . 8  |-  ( y  =  r  ->  (
( F `  q
)  +Q  ( q  +Q  y ) )  =  ( ( F `
 q )  +Q  ( q  +Q  r
) ) )
4544breq2d 3804 . . . . . . 7  |-  ( y  =  r  ->  (
l  <Q  ( ( F `
 q )  +Q  ( q  +Q  y
) )  <->  l  <Q  ( ( F `  q
)  +Q  ( q  +Q  r ) ) ) )
4645abbidv 2171 . . . . . 6  |-  ( y  =  r  ->  { l  |  l  <Q  (
( F `  q
)  +Q  ( q  +Q  y ) ) }  =  { l  |  l  <Q  (
( F `  q
)  +Q  ( q  +Q  r ) ) } )
4744breq1d 3802 . . . . . . 7  |-  ( y  =  r  ->  (
( ( F `  q )  +Q  (
q  +Q  y ) )  <Q  u  <->  ( ( F `  q )  +Q  ( q  +Q  r
) )  <Q  u
) )
4847abbidv 2171 . . . . . 6  |-  ( y  =  r  ->  { u  |  ( ( F `
 q )  +Q  ( q  +Q  y
) )  <Q  u }  =  { u  |  ( ( F `
 q )  +Q  ( q  +Q  r
) )  <Q  u } )
4946, 48opeq12d 3585 . . . . 5  |-  ( y  =  r  ->  <. { l  |  l  <Q  (
( F `  q
)  +Q  ( q  +Q  y ) ) } ,  { u  |  ( ( F `
 q )  +Q  ( q  +Q  y
) )  <Q  u } >.  =  <. { l  |  l  <Q  (
( F `  q
)  +Q  ( q  +Q  r ) ) } ,  { u  |  ( ( F `
 q )  +Q  ( q  +Q  r
) )  <Q  u } >. )
5049breq2d 3804 . . . 4  |-  ( y  =  r  ->  ( L  <P  <. { l  |  l  <Q  ( ( F `  q )  +Q  ( q  +Q  y
) ) } ,  { u  |  (
( F `  q
)  +Q  ( q  +Q  y ) ) 
<Q  u } >.  <->  L  <P  <. { l  |  l 
<Q  ( ( F `  q )  +Q  (
q  +Q  r ) ) } ,  {
u  |  ( ( F `  q )  +Q  ( q  +Q  r ) )  <Q  u } >. ) )
5143, 50anbi12d 450 . . 3  |-  ( y  =  r  ->  (
( <. { l  |  l  <Q  ( F `  q ) } ,  { u  |  ( F `  q )  <Q  u } >.  <P  ( L  +P.  <. { l  |  l  <Q  ( q  +Q  y ) } ,  { u  |  (
q  +Q  y ) 
<Q  u } >. )  /\  L  <P  <. { l  |  l  <Q  (
( F `  q
)  +Q  ( q  +Q  y ) ) } ,  { u  |  ( ( F `
 q )  +Q  ( q  +Q  y
) )  <Q  u } >. )  <->  ( <. { l  |  l  <Q 
( F `  q
) } ,  {
u  |  ( F `
 q )  <Q  u } >.  <P  ( L  +P.  <. { l  |  l  <Q  ( q  +Q  r ) } ,  { u  |  (
q  +Q  r ) 
<Q  u } >. )  /\  L  <P  <. { l  |  l  <Q  (
( F `  q
)  +Q  ( q  +Q  r ) ) } ,  { u  |  ( ( F `
 q )  +Q  ( q  +Q  r
) )  <Q  u } >. ) ) )
5235, 51cbvral2v 2558 . 2  |-  ( A. x  e.  Q.  A. y  e.  Q.  ( <. { l  |  l  <Q  ( F `  x ) } ,  { u  |  ( F `  x )  <Q  u } >.  <P  ( L  +P.  <. { l  |  l 
<Q  ( x  +Q  y
) } ,  {
u  |  ( x  +Q  y )  <Q  u } >. )  /\  L  <P 
<. { l  |  l 
<Q  ( ( F `  x )  +Q  (
x  +Q  y ) ) } ,  {
u  |  ( ( F `  x )  +Q  ( x  +Q  y ) )  <Q  u } >. )  <->  A. q  e.  Q.  A. r  e. 
Q.  ( <. { l  |  l  <Q  ( F `  q ) } ,  { u  |  ( F `  q )  <Q  u } >.  <P  ( L  +P.  <. { l  |  l 
<Q  ( q  +Q  r
) } ,  {
u  |  ( q  +Q  r )  <Q  u } >. )  /\  L  <P 
<. { l  |  l 
<Q  ( ( F `  q )  +Q  (
q  +Q  r ) ) } ,  {
u  |  ( ( F `  q )  +Q  ( q  +Q  r ) )  <Q  u } >. ) )
5313, 52sylib 131 1  |-  ( ph  ->  A. q  e.  Q.  A. r  e.  Q.  ( <. { l  |  l 
<Q  ( F `  q
) } ,  {
u  |  ( F `
 q )  <Q  u } >.  <P  ( L  +P.  <. { l  |  l  <Q  ( q  +Q  r ) } ,  { u  |  (
q  +Q  r ) 
<Q  u } >. )  /\  L  <P  <. { l  |  l  <Q  (
( F `  q
)  +Q  ( q  +Q  r ) ) } ,  { u  |  ( ( F `
 q )  +Q  ( q  +Q  r
) )  <Q  u } >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    = wceq 1259    e. wcel 1409   {cab 2042   A.wral 2323   E.wrex 2324   {crab 2327   <.cop 3406   class class class wbr 3792   -->wf 4926   ` cfv 4930  (class class class)co 5540   Q.cnq 6436    +Q cplq 6438    <Q cltq 6441    +P. cpp 6449    <P cltp 6451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-iplp 6624  df-iltp 6626
This theorem is referenced by:  cauappcvgpr  6818
  Copyright terms: Public domain W3C validator