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Theorem cauappcvgprlemlim 7218
Description: Lemma for cauappcvgpr 7219. 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 270 . . . . 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 270 . . . . 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 270 . . . . 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 498 . . . . 5  |-  ( (
ph  /\  ( x  e.  Q.  /\  y  e. 
Q. ) )  ->  x  e.  Q. )
9 simprr 499 . . . . 5  |-  ( (
ph  /\  ( x  e.  Q.  /\  y  e. 
Q. ) )  -> 
y  e.  Q. )
102, 4, 6, 7, 8, 9cauappcvgprlem1 7216 . . . 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 7217 . . . 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 300 . . 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 2455 . 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 5305 . . . . . . . 8  |-  ( x  =  q  ->  ( F `  x )  =  ( F `  q ) )
1514breq2d 3857 . . . . . . 7  |-  ( x  =  q  ->  (
l  <Q  ( F `  x )  <->  l  <Q  ( F `  q ) ) )
1615abbidv 2205 . . . . . 6  |-  ( x  =  q  ->  { l  |  l  <Q  ( F `  x ) }  =  { l  |  l  <Q  ( F `
 q ) } )
1714breq1d 3855 . . . . . . 7  |-  ( x  =  q  ->  (
( F `  x
)  <Q  u  <->  ( F `  q )  <Q  u
) )
1817abbidv 2205 . . . . . 6  |-  ( x  =  q  ->  { u  |  ( F `  x )  <Q  u }  =  { u  |  ( F `  q )  <Q  u } )
1916, 18opeq12d 3630 . . . . 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 5659 . . . . . . . . 9  |-  ( x  =  q  ->  (
x  +Q  y )  =  ( q  +Q  y ) )
2120breq2d 3857 . . . . . . . 8  |-  ( x  =  q  ->  (
l  <Q  ( x  +Q  y )  <->  l  <Q  ( q  +Q  y ) ) )
2221abbidv 2205 . . . . . . 7  |-  ( x  =  q  ->  { l  |  l  <Q  (
x  +Q  y ) }  =  { l  |  l  <Q  (
q  +Q  y ) } )
2320breq1d 3855 . . . . . . . 8  |-  ( x  =  q  ->  (
( x  +Q  y
)  <Q  u  <->  ( q  +Q  y )  <Q  u
) )
2423abbidv 2205 . . . . . . 7  |-  ( x  =  q  ->  { u  |  ( x  +Q  y )  <Q  u }  =  { u  |  ( q  +Q  y )  <Q  u } )
2522, 24opeq12d 3630 . . . . . 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 5668 . . . . 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 3858 . . . 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 5670 . . . . . . . 8  |-  ( x  =  q  ->  (
( F `  x
)  +Q  ( x  +Q  y ) )  =  ( ( F `
 q )  +Q  ( q  +Q  y
) ) )
2928breq2d 3857 . . . . . . 7  |-  ( x  =  q  ->  (
l  <Q  ( ( F `
 x )  +Q  ( x  +Q  y
) )  <->  l  <Q  ( ( F `  q
)  +Q  ( q  +Q  y ) ) ) )
3029abbidv 2205 . . . . . 6  |-  ( x  =  q  ->  { l  |  l  <Q  (
( F `  x
)  +Q  ( x  +Q  y ) ) }  =  { l  |  l  <Q  (
( F `  q
)  +Q  ( q  +Q  y ) ) } )
3128breq1d 3855 . . . . . . 7  |-  ( x  =  q  ->  (
( ( F `  x )  +Q  (
x  +Q  y ) )  <Q  u  <->  ( ( F `  q )  +Q  ( q  +Q  y
) )  <Q  u
) )
3231abbidv 2205 . . . . . 6  |-  ( x  =  q  ->  { u  |  ( ( F `
 x )  +Q  ( x  +Q  y
) )  <Q  u }  =  { u  |  ( ( F `
 q )  +Q  ( q  +Q  y
) )  <Q  u } )
3330, 32opeq12d 3630 . . . . 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 3857 . . . 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 457 . . 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 5660 . . . . . . . . 9  |-  ( y  =  r  ->  (
q  +Q  y )  =  ( q  +Q  r ) )
3736breq2d 3857 . . . . . . . 8  |-  ( y  =  r  ->  (
l  <Q  ( q  +Q  y )  <->  l  <Q  ( q  +Q  r ) ) )
3837abbidv 2205 . . . . . . 7  |-  ( y  =  r  ->  { l  |  l  <Q  (
q  +Q  y ) }  =  { l  |  l  <Q  (
q  +Q  r ) } )
3936breq1d 3855 . . . . . . . 8  |-  ( y  =  r  ->  (
( q  +Q  y
)  <Q  u  <->  ( q  +Q  r )  <Q  u
) )
4039abbidv 2205 . . . . . . 7  |-  ( y  =  r  ->  { u  |  ( q  +Q  y )  <Q  u }  =  { u  |  ( q  +Q  r )  <Q  u } )
4138, 40opeq12d 3630 . . . . . 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 5668 . . . . 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 3857 . . . 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 5668 . . . . . . . 8  |-  ( y  =  r  ->  (
( F `  q
)  +Q  ( q  +Q  y ) )  =  ( ( F `
 q )  +Q  ( q  +Q  r
) ) )
4544breq2d 3857 . . . . . . 7  |-  ( y  =  r  ->  (
l  <Q  ( ( F `
 q )  +Q  ( q  +Q  y
) )  <->  l  <Q  ( ( F `  q
)  +Q  ( q  +Q  r ) ) ) )
4645abbidv 2205 . . . . . 6  |-  ( y  =  r  ->  { l  |  l  <Q  (
( F `  q
)  +Q  ( q  +Q  y ) ) }  =  { l  |  l  <Q  (
( F `  q
)  +Q  ( q  +Q  r ) ) } )
4744breq1d 3855 . . . . . . 7  |-  ( y  =  r  ->  (
( ( F `  q )  +Q  (
q  +Q  y ) )  <Q  u  <->  ( ( F `  q )  +Q  ( q  +Q  r
) )  <Q  u
) )
4847abbidv 2205 . . . . . 6  |-  ( y  =  r  ->  { u  |  ( ( F `
 q )  +Q  ( q  +Q  y
) )  <Q  u }  =  { u  |  ( ( F `
 q )  +Q  ( q  +Q  r
) )  <Q  u } )
4946, 48opeq12d 3630 . . . . 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 3857 . . . 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 457 . . 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 2598 . 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 120 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 102    = wceq 1289    e. wcel 1438   {cab 2074   A.wral 2359   E.wrex 2360   {crab 2363   <.cop 3449   class class class wbr 3845   -->wf 5011   ` cfv 5015  (class class class)co 5652   Q.cnq 6837    +Q cplq 6839    <Q cltq 6842    +P. cpp 6850    <P cltp 6852
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-eprel 4116  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-1o 6181  df-2o 6182  df-oadd 6185  df-omul 6186  df-er 6290  df-ec 6292  df-qs 6296  df-ni 6861  df-pli 6862  df-mi 6863  df-lti 6864  df-plpq 6901  df-mpq 6902  df-enq 6904  df-nqqs 6905  df-plqqs 6906  df-mqqs 6907  df-1nqqs 6908  df-rq 6909  df-ltnqqs 6910  df-enq0 6981  df-nq0 6982  df-0nq0 6983  df-plq0 6984  df-mq0 6985  df-inp 7023  df-iplp 7025  df-iltp 7027
This theorem is referenced by:  cauappcvgpr  7219
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