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Theorem cauappcvgprlemlim 7623
Description: Lemma for cauappcvgpr 7624. 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 526 . . . . 5  |-  ( (
ph  /\  ( x  e.  Q.  /\  y  e. 
Q. ) )  ->  x  e.  Q. )
9 simprr 527 . . . . 5  |-  ( (
ph  /\  ( x  e.  Q.  /\  y  e. 
Q. ) )  -> 
y  e.  Q. )
102, 4, 6, 7, 8, 9cauappcvgprlem1 7621 . . . 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 7622 . . . 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 2552 . 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 5496 . . . . . . . 8  |-  ( x  =  q  ->  ( F `  x )  =  ( F `  q ) )
1514breq2d 4001 . . . . . . 7  |-  ( x  =  q  ->  (
l  <Q  ( F `  x )  <->  l  <Q  ( F `  q ) ) )
1615abbidv 2288 . . . . . 6  |-  ( x  =  q  ->  { l  |  l  <Q  ( F `  x ) }  =  { l  |  l  <Q  ( F `
 q ) } )
1714breq1d 3999 . . . . . . 7  |-  ( x  =  q  ->  (
( F `  x
)  <Q  u  <->  ( F `  q )  <Q  u
) )
1817abbidv 2288 . . . . . 6  |-  ( x  =  q  ->  { u  |  ( F `  x )  <Q  u }  =  { u  |  ( F `  q )  <Q  u } )
1916, 18opeq12d 3773 . . . . 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 5860 . . . . . . . . 9  |-  ( x  =  q  ->  (
x  +Q  y )  =  ( q  +Q  y ) )
2120breq2d 4001 . . . . . . . 8  |-  ( x  =  q  ->  (
l  <Q  ( x  +Q  y )  <->  l  <Q  ( q  +Q  y ) ) )
2221abbidv 2288 . . . . . . 7  |-  ( x  =  q  ->  { l  |  l  <Q  (
x  +Q  y ) }  =  { l  |  l  <Q  (
q  +Q  y ) } )
2320breq1d 3999 . . . . . . . 8  |-  ( x  =  q  ->  (
( x  +Q  y
)  <Q  u  <->  ( q  +Q  y )  <Q  u
) )
2423abbidv 2288 . . . . . . 7  |-  ( x  =  q  ->  { u  |  ( x  +Q  y )  <Q  u }  =  { u  |  ( q  +Q  y )  <Q  u } )
2522, 24opeq12d 3773 . . . . . 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 5869 . . . . 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 4002 . . . 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 5871 . . . . . . . 8  |-  ( x  =  q  ->  (
( F `  x
)  +Q  ( x  +Q  y ) )  =  ( ( F `
 q )  +Q  ( q  +Q  y
) ) )
2928breq2d 4001 . . . . . . 7  |-  ( x  =  q  ->  (
l  <Q  ( ( F `
 x )  +Q  ( x  +Q  y
) )  <->  l  <Q  ( ( F `  q
)  +Q  ( q  +Q  y ) ) ) )
3029abbidv 2288 . . . . . 6  |-  ( x  =  q  ->  { l  |  l  <Q  (
( F `  x
)  +Q  ( x  +Q  y ) ) }  =  { l  |  l  <Q  (
( F `  q
)  +Q  ( q  +Q  y ) ) } )
3128breq1d 3999 . . . . . . 7  |-  ( x  =  q  ->  (
( ( F `  x )  +Q  (
x  +Q  y ) )  <Q  u  <->  ( ( F `  q )  +Q  ( q  +Q  y
) )  <Q  u
) )
3231abbidv 2288 . . . . . 6  |-  ( x  =  q  ->  { u  |  ( ( F `
 x )  +Q  ( x  +Q  y
) )  <Q  u }  =  { u  |  ( ( F `
 q )  +Q  ( q  +Q  y
) )  <Q  u } )
3330, 32opeq12d 3773 . . . . 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 4001 . . . 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 470 . . 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 5861 . . . . . . . . 9  |-  ( y  =  r  ->  (
q  +Q  y )  =  ( q  +Q  r ) )
3736breq2d 4001 . . . . . . . 8  |-  ( y  =  r  ->  (
l  <Q  ( q  +Q  y )  <->  l  <Q  ( q  +Q  r ) ) )
3837abbidv 2288 . . . . . . 7  |-  ( y  =  r  ->  { l  |  l  <Q  (
q  +Q  y ) }  =  { l  |  l  <Q  (
q  +Q  r ) } )
3936breq1d 3999 . . . . . . . 8  |-  ( y  =  r  ->  (
( q  +Q  y
)  <Q  u  <->  ( q  +Q  r )  <Q  u
) )
4039abbidv 2288 . . . . . . 7  |-  ( y  =  r  ->  { u  |  ( q  +Q  y )  <Q  u }  =  { u  |  ( q  +Q  r )  <Q  u } )
4138, 40opeq12d 3773 . . . . . 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 5869 . . . . 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 4001 . . . 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 5869 . . . . . . . 8  |-  ( y  =  r  ->  (
( F `  q
)  +Q  ( q  +Q  y ) )  =  ( ( F `
 q )  +Q  ( q  +Q  r
) ) )
4544breq2d 4001 . . . . . . 7  |-  ( y  =  r  ->  (
l  <Q  ( ( F `
 q )  +Q  ( q  +Q  y
) )  <->  l  <Q  ( ( F `  q
)  +Q  ( q  +Q  r ) ) ) )
4645abbidv 2288 . . . . . 6  |-  ( y  =  r  ->  { l  |  l  <Q  (
( F `  q
)  +Q  ( q  +Q  y ) ) }  =  { l  |  l  <Q  (
( F `  q
)  +Q  ( q  +Q  r ) ) } )
4744breq1d 3999 . . . . . . 7  |-  ( y  =  r  ->  (
( ( F `  q )  +Q  (
q  +Q  y ) )  <Q  u  <->  ( ( F `  q )  +Q  ( q  +Q  r
) )  <Q  u
) )
4847abbidv 2288 . . . . . 6  |-  ( y  =  r  ->  { u  |  ( ( F `
 q )  +Q  ( q  +Q  y
) )  <Q  u }  =  { u  |  ( ( F `
 q )  +Q  ( q  +Q  r
) )  <Q  u } )
4946, 48opeq12d 3773 . . . . 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 4001 . . . 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 470 . . 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 2709 . 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 1348    e. wcel 2141   {cab 2156   A.wral 2448   E.wrex 2449   {crab 2452   <.cop 3586   class class class wbr 3989   -->wf 5194   ` cfv 5198  (class class class)co 5853   Q.cnq 7242    +Q cplq 7244    <Q cltq 7247    +P. cpp 7255    <P cltp 7257
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-iplp 7430  df-iltp 7432
This theorem is referenced by:  cauappcvgpr  7624
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