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Theorem cauappcvgprlemlim 7581
Description: Lemma for cauappcvgpr 7582. 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 274 . . . . 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 274 . . . . 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 274 . . . . 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 521 . . . . 5  |-  ( (
ph  /\  ( x  e.  Q.  /\  y  e. 
Q. ) )  ->  x  e.  Q. )
9 simprr 522 . . . . 5  |-  ( (
ph  /\  ( x  e.  Q.  /\  y  e. 
Q. ) )  -> 
y  e.  Q. )
102, 4, 6, 7, 8, 9cauappcvgprlem1 7579 . . . 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 7580 . . . 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 304 . . 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 2539 . 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 5468 . . . . . . . 8  |-  ( x  =  q  ->  ( F `  x )  =  ( F `  q ) )
1514breq2d 3977 . . . . . . 7  |-  ( x  =  q  ->  (
l  <Q  ( F `  x )  <->  l  <Q  ( F `  q ) ) )
1615abbidv 2275 . . . . . 6  |-  ( x  =  q  ->  { l  |  l  <Q  ( F `  x ) }  =  { l  |  l  <Q  ( F `
 q ) } )
1714breq1d 3975 . . . . . . 7  |-  ( x  =  q  ->  (
( F `  x
)  <Q  u  <->  ( F `  q )  <Q  u
) )
1817abbidv 2275 . . . . . 6  |-  ( x  =  q  ->  { u  |  ( F `  x )  <Q  u }  =  { u  |  ( F `  q )  <Q  u } )
1916, 18opeq12d 3749 . . . . 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 5831 . . . . . . . . 9  |-  ( x  =  q  ->  (
x  +Q  y )  =  ( q  +Q  y ) )
2120breq2d 3977 . . . . . . . 8  |-  ( x  =  q  ->  (
l  <Q  ( x  +Q  y )  <->  l  <Q  ( q  +Q  y ) ) )
2221abbidv 2275 . . . . . . 7  |-  ( x  =  q  ->  { l  |  l  <Q  (
x  +Q  y ) }  =  { l  |  l  <Q  (
q  +Q  y ) } )
2320breq1d 3975 . . . . . . . 8  |-  ( x  =  q  ->  (
( x  +Q  y
)  <Q  u  <->  ( q  +Q  y )  <Q  u
) )
2423abbidv 2275 . . . . . . 7  |-  ( x  =  q  ->  { u  |  ( x  +Q  y )  <Q  u }  =  { u  |  ( q  +Q  y )  <Q  u } )
2522, 24opeq12d 3749 . . . . . 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 5840 . . . . 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 3978 . . . 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 5842 . . . . . . . 8  |-  ( x  =  q  ->  (
( F `  x
)  +Q  ( x  +Q  y ) )  =  ( ( F `
 q )  +Q  ( q  +Q  y
) ) )
2928breq2d 3977 . . . . . . 7  |-  ( x  =  q  ->  (
l  <Q  ( ( F `
 x )  +Q  ( x  +Q  y
) )  <->  l  <Q  ( ( F `  q
)  +Q  ( q  +Q  y ) ) ) )
3029abbidv 2275 . . . . . 6  |-  ( x  =  q  ->  { l  |  l  <Q  (
( F `  x
)  +Q  ( x  +Q  y ) ) }  =  { l  |  l  <Q  (
( F `  q
)  +Q  ( q  +Q  y ) ) } )
3128breq1d 3975 . . . . . . 7  |-  ( x  =  q  ->  (
( ( F `  x )  +Q  (
x  +Q  y ) )  <Q  u  <->  ( ( F `  q )  +Q  ( q  +Q  y
) )  <Q  u
) )
3231abbidv 2275 . . . . . 6  |-  ( x  =  q  ->  { u  |  ( ( F `
 x )  +Q  ( x  +Q  y
) )  <Q  u }  =  { u  |  ( ( F `
 q )  +Q  ( q  +Q  y
) )  <Q  u } )
3330, 32opeq12d 3749 . . . . 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 3977 . . . 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 465 . . 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 5832 . . . . . . . . 9  |-  ( y  =  r  ->  (
q  +Q  y )  =  ( q  +Q  r ) )
3736breq2d 3977 . . . . . . . 8  |-  ( y  =  r  ->  (
l  <Q  ( q  +Q  y )  <->  l  <Q  ( q  +Q  r ) ) )
3837abbidv 2275 . . . . . . 7  |-  ( y  =  r  ->  { l  |  l  <Q  (
q  +Q  y ) }  =  { l  |  l  <Q  (
q  +Q  r ) } )
3936breq1d 3975 . . . . . . . 8  |-  ( y  =  r  ->  (
( q  +Q  y
)  <Q  u  <->  ( q  +Q  r )  <Q  u
) )
4039abbidv 2275 . . . . . . 7  |-  ( y  =  r  ->  { u  |  ( q  +Q  y )  <Q  u }  =  { u  |  ( q  +Q  r )  <Q  u } )
4138, 40opeq12d 3749 . . . . . 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 5840 . . . . 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 3977 . . . 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 5840 . . . . . . . 8  |-  ( y  =  r  ->  (
( F `  q
)  +Q  ( q  +Q  y ) )  =  ( ( F `
 q )  +Q  ( q  +Q  r
) ) )
4544breq2d 3977 . . . . . . 7  |-  ( y  =  r  ->  (
l  <Q  ( ( F `
 q )  +Q  ( q  +Q  y
) )  <->  l  <Q  ( ( F `  q
)  +Q  ( q  +Q  r ) ) ) )
4645abbidv 2275 . . . . . 6  |-  ( y  =  r  ->  { l  |  l  <Q  (
( F `  q
)  +Q  ( q  +Q  y ) ) }  =  { l  |  l  <Q  (
( F `  q
)  +Q  ( q  +Q  r ) ) } )
4744breq1d 3975 . . . . . . 7  |-  ( y  =  r  ->  (
( ( F `  q )  +Q  (
q  +Q  y ) )  <Q  u  <->  ( ( F `  q )  +Q  ( q  +Q  r
) )  <Q  u
) )
4847abbidv 2275 . . . . . 6  |-  ( y  =  r  ->  { u  |  ( ( F `
 q )  +Q  ( q  +Q  y
) )  <Q  u }  =  { u  |  ( ( F `
 q )  +Q  ( q  +Q  r
) )  <Q  u } )
4946, 48opeq12d 3749 . . . . 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 3977 . . . 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 465 . . 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 2691 . 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 121 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 103    = wceq 1335    e. wcel 2128   {cab 2143   A.wral 2435   E.wrex 2436   {crab 2439   <.cop 3563   class class class wbr 3965   -->wf 5166   ` cfv 5170  (class class class)co 5824   Q.cnq 7200    +Q cplq 7202    <Q cltq 7205    +P. cpp 7213    <P cltp 7215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-eprel 4249  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-irdg 6317  df-1o 6363  df-2o 6364  df-oadd 6367  df-omul 6368  df-er 6480  df-ec 6482  df-qs 6486  df-ni 7224  df-pli 7225  df-mi 7226  df-lti 7227  df-plpq 7264  df-mpq 7265  df-enq 7267  df-nqqs 7268  df-plqqs 7269  df-mqqs 7270  df-1nqqs 7271  df-rq 7272  df-ltnqqs 7273  df-enq0 7344  df-nq0 7345  df-0nq0 7346  df-plq0 7347  df-mq0 7348  df-inp 7386  df-iplp 7388  df-iltp 7390
This theorem is referenced by:  cauappcvgpr  7582
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