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

Theorem cauappcvgpr 7477
Description: A Cauchy approximation has a limit. A Cauchy approximation, here  F, is similar to a Cauchy sequence but is indexed by the desired tolerance (that is, how close together terms needs to be) rather than by natural numbers. This is basically Theorem 11.2.12 of [HoTT], p. (varies) with a few differences such as that we are proving the existence of a limit without anything about how fast it converges (that is, mere existence instead of existence, in HoTT terms), and that the codomain of  F is  Q. rather than  P.. We also specify that every term needs to be larger than a fraction  A, to avoid the case where we have positive terms which "converge" to zero (which is not a positive real).

This proof (including its lemmas) is similar to the proofs of caucvgpr 7497 and caucvgprpr 7527 but is somewhat simpler, so reading this one first may help understanding the other two.

(Contributed by Jim Kingdon, 19-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 ) )
Assertion
Ref Expression
cauappcvgpr  |-  ( ph  ->  E. y  e.  P.  A. q  e.  Q.  A. r  e.  Q.  ( <. { l  |  l 
<Q  ( F `  q
) } ,  {
u  |  ( F `
 q )  <Q  u } >.  <P  ( y  +P.  <. { l  |  l  <Q  ( q  +Q  r ) } ,  { u  |  (
q  +Q  r ) 
<Q  u } >. )  /\  y  <P  <. { l  |  l  <Q  (
( F `  q
)  +Q  ( q  +Q  r ) ) } ,  { u  |  ( ( F `
 q )  +Q  ( q  +Q  r
) )  <Q  u } >. ) )
Distinct variable groups:    A, p    F, q, y, r, u    F, p, l, q    y, l, r    u, q, y, r    u, p, r, q, l    ph, q, p
Allowed substitution hints:    ph( y, u, r, l)    A( y, u, r, q, l)

Proof of Theorem cauappcvgpr
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 cauappcvgpr.f . . 3  |-  ( ph  ->  F : Q. --> Q. )
2 cauappcvgpr.app . . 3  |-  ( 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 ) ) ) )
3 cauappcvgpr.bnd . . 3  |-  ( ph  ->  A. p  e.  Q.  A  <Q  ( F `  p ) )
4 oveq2 5782 . . . . . . . 8  |-  ( z  =  q  ->  (
l  +Q  z )  =  ( l  +Q  q ) )
5 fveq2 5421 . . . . . . . 8  |-  ( z  =  q  ->  ( F `  z )  =  ( F `  q ) )
64, 5breq12d 3942 . . . . . . 7  |-  ( z  =  q  ->  (
( l  +Q  z
)  <Q  ( F `  z )  <->  ( l  +Q  q )  <Q  ( F `  q )
) )
76cbvrexv 2655 . . . . . 6  |-  ( E. z  e.  Q.  (
l  +Q  z ) 
<Q  ( F `  z
)  <->  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( F `  q ) )
87a1i 9 . . . . 5  |-  ( l  e.  Q.  ->  ( E. z  e.  Q.  ( l  +Q  z
)  <Q  ( F `  z )  <->  E. q  e.  Q.  ( l  +Q  q )  <Q  ( F `  q )
) )
98rabbiia 2671 . . . 4  |-  { l  e.  Q.  |  E. z  e.  Q.  (
l  +Q  z ) 
<Q  ( F `  z
) }  =  {
l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( F `  q ) }
10 id 19 . . . . . . . . 9  |-  ( z  =  q  ->  z  =  q )
115, 10oveq12d 5792 . . . . . . . 8  |-  ( z  =  q  ->  (
( F `  z
)  +Q  z )  =  ( ( F `
 q )  +Q  q ) )
1211breq1d 3939 . . . . . . 7  |-  ( z  =  q  ->  (
( ( F `  z )  +Q  z
)  <Q  u  <->  ( ( F `  q )  +Q  q )  <Q  u
) )
1312cbvrexv 2655 . . . . . 6  |-  ( E. z  e.  Q.  (
( F `  z
)  +Q  z ) 
<Q  u  <->  E. q  e.  Q.  ( ( F `  q )  +Q  q
)  <Q  u )
1413a1i 9 . . . . 5  |-  ( u  e.  Q.  ->  ( E. z  e.  Q.  ( ( F `  z )  +Q  z
)  <Q  u  <->  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  u
) )
1514rabbiia 2671 . . . 4  |-  { u  e.  Q.  |  E. z  e.  Q.  ( ( F `
 z )  +Q  z )  <Q  u }  =  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  u }
169, 15opeq12i 3710 . . 3  |-  <. { l  e.  Q.  |  E. z  e.  Q.  (
l  +Q  z ) 
<Q  ( F `  z
) } ,  {
u  e.  Q.  |  E. z  e.  Q.  ( ( F `  z )  +Q  z
)  <Q  u } >.  = 
<. { 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 } >.
171, 2, 3, 16cauappcvgprlemcl 7468 . 2  |-  ( ph  -> 
<. { l  e.  Q.  |  E. z  e.  Q.  ( l  +Q  z
)  <Q  ( F `  z ) } ,  { u  e.  Q.  |  E. z  e.  Q.  ( ( F `  z )  +Q  z
)  <Q  u } >.  e. 
P. )
181, 2, 3, 16cauappcvgprlemlim 7476 . 2  |-  ( ph  ->  A. q  e.  Q.  A. r  e.  Q.  ( <. { l  |  l 
<Q  ( F `  q
) } ,  {
u  |  ( F `
 q )  <Q  u } >.  <P  ( <. { l  e.  Q.  |  E. z  e.  Q.  ( l  +Q  z
)  <Q  ( F `  z ) } ,  { u  e.  Q.  |  E. z  e.  Q.  ( ( F `  z )  +Q  z
)  <Q  u } >.  +P. 
<. { l  |  l 
<Q  ( q  +Q  r
) } ,  {
u  |  ( q  +Q  r )  <Q  u } >. )  /\  <. { l  e.  Q.  |  E. z  e.  Q.  ( l  +Q  z
)  <Q  ( F `  z ) } ,  { u  e.  Q.  |  E. z  e.  Q.  ( ( F `  z )  +Q  z
)  <Q  u } >.  <P  <. { l  |  l 
<Q  ( ( F `  q )  +Q  (
q  +Q  r ) ) } ,  {
u  |  ( ( F `  q )  +Q  ( q  +Q  r ) )  <Q  u } >. ) )
19 oveq1 5781 . . . . . 6  |-  ( y  =  <. { l  e. 
Q.  |  E. z  e.  Q.  ( l  +Q  z )  <Q  ( F `  z ) } ,  { u  e.  Q.  |  E. z  e.  Q.  ( ( F `
 z )  +Q  z )  <Q  u } >.  ->  ( y  +P.  <. { l  |  l  <Q  ( q  +Q  r ) } ,  { u  |  (
q  +Q  r ) 
<Q  u } >. )  =  ( <. { l  e.  Q.  |  E. z  e.  Q.  (
l  +Q  z ) 
<Q  ( F `  z
) } ,  {
u  e.  Q.  |  E. z  e.  Q.  ( ( F `  z )  +Q  z
)  <Q  u } >.  +P. 
<. { l  |  l 
<Q  ( q  +Q  r
) } ,  {
u  |  ( q  +Q  r )  <Q  u } >. ) )
2019breq2d 3941 . . . . 5  |-  ( y  =  <. { l  e. 
Q.  |  E. z  e.  Q.  ( l  +Q  z )  <Q  ( F `  z ) } ,  { u  e.  Q.  |  E. z  e.  Q.  ( ( F `
 z )  +Q  z )  <Q  u } >.  ->  ( <. { l  |  l  <Q 
( F `  q
) } ,  {
u  |  ( F `
 q )  <Q  u } >.  <P  ( y  +P.  <. { l  |  l  <Q  ( q  +Q  r ) } ,  { u  |  (
q  +Q  r ) 
<Q  u } >. )  <->  <. { l  |  l 
<Q  ( F `  q
) } ,  {
u  |  ( F `
 q )  <Q  u } >.  <P  ( <. { l  e.  Q.  |  E. z  e.  Q.  ( l  +Q  z
)  <Q  ( F `  z ) } ,  { u  e.  Q.  |  E. z  e.  Q.  ( ( F `  z )  +Q  z
)  <Q  u } >.  +P. 
<. { l  |  l 
<Q  ( q  +Q  r
) } ,  {
u  |  ( q  +Q  r )  <Q  u } >. ) ) )
21 breq1 3932 . . . . 5  |-  ( y  =  <. { l  e. 
Q.  |  E. z  e.  Q.  ( l  +Q  z )  <Q  ( F `  z ) } ,  { u  e.  Q.  |  E. z  e.  Q.  ( ( F `
 z )  +Q  z )  <Q  u } >.  ->  ( y  <P 
<. { l  |  l 
<Q  ( ( F `  q )  +Q  (
q  +Q  r ) ) } ,  {
u  |  ( ( F `  q )  +Q  ( q  +Q  r ) )  <Q  u } >.  <->  <. { l  e. 
Q.  |  E. z  e.  Q.  ( l  +Q  z )  <Q  ( F `  z ) } ,  { u  e.  Q.  |  E. z  e.  Q.  ( ( F `
 z )  +Q  z )  <Q  u } >.  <P  <. { l  |  l  <Q  ( ( F `  q )  +Q  ( q  +Q  r
) ) } ,  { u  |  (
( F `  q
)  +Q  ( q  +Q  r ) ) 
<Q  u } >. )
)
2220, 21anbi12d 464 . . . 4  |-  ( y  =  <. { l  e. 
Q.  |  E. z  e.  Q.  ( l  +Q  z )  <Q  ( F `  z ) } ,  { u  e.  Q.  |  E. z  e.  Q.  ( ( F `
 z )  +Q  z )  <Q  u } >.  ->  ( ( <. { l  |  l 
<Q  ( F `  q
) } ,  {
u  |  ( F `
 q )  <Q  u } >.  <P  ( y  +P.  <. { l  |  l  <Q  ( q  +Q  r ) } ,  { u  |  (
q  +Q  r ) 
<Q  u } >. )  /\  y  <P  <. { l  |  l  <Q  (
( F `  q
)  +Q  ( q  +Q  r ) ) } ,  { u  |  ( ( F `
 q )  +Q  ( q  +Q  r
) )  <Q  u } >. )  <->  ( <. { l  |  l  <Q 
( F `  q
) } ,  {
u  |  ( F `
 q )  <Q  u } >.  <P  ( <. { l  e.  Q.  |  E. z  e.  Q.  ( l  +Q  z
)  <Q  ( F `  z ) } ,  { u  e.  Q.  |  E. z  e.  Q.  ( ( F `  z )  +Q  z
)  <Q  u } >.  +P. 
<. { l  |  l 
<Q  ( q  +Q  r
) } ,  {
u  |  ( q  +Q  r )  <Q  u } >. )  /\  <. { l  e.  Q.  |  E. z  e.  Q.  ( l  +Q  z
)  <Q  ( F `  z ) } ,  { u  e.  Q.  |  E. z  e.  Q.  ( ( F `  z )  +Q  z
)  <Q  u } >.  <P  <. { l  |  l 
<Q  ( ( F `  q )  +Q  (
q  +Q  r ) ) } ,  {
u  |  ( ( F `  q )  +Q  ( q  +Q  r ) )  <Q  u } >. ) ) )
23222ralbidv 2459 . . 3  |-  ( y  =  <. { l  e. 
Q.  |  E. z  e.  Q.  ( l  +Q  z )  <Q  ( F `  z ) } ,  { u  e.  Q.  |  E. z  e.  Q.  ( ( F `
 z )  +Q  z )  <Q  u } >.  ->  ( A. q  e.  Q.  A. r  e.  Q.  ( <. { l  |  l  <Q  ( F `  q ) } ,  { u  |  ( F `  q )  <Q  u } >.  <P  ( y  +P. 
<. { l  |  l 
<Q  ( q  +Q  r
) } ,  {
u  |  ( q  +Q  r )  <Q  u } >. )  /\  y  <P 
<. { l  |  l 
<Q  ( ( F `  q )  +Q  (
q  +Q  r ) ) } ,  {
u  |  ( ( F `  q )  +Q  ( q  +Q  r ) )  <Q  u } >. )  <->  A. q  e.  Q.  A. r  e. 
Q.  ( <. { l  |  l  <Q  ( F `  q ) } ,  { u  |  ( F `  q )  <Q  u } >.  <P  ( <. { l  e.  Q.  |  E. z  e.  Q.  (
l  +Q  z ) 
<Q  ( F `  z
) } ,  {
u  e.  Q.  |  E. z  e.  Q.  ( ( F `  z )  +Q  z
)  <Q  u } >.  +P. 
<. { l  |  l 
<Q  ( q  +Q  r
) } ,  {
u  |  ( q  +Q  r )  <Q  u } >. )  /\  <. { l  e.  Q.  |  E. z  e.  Q.  ( l  +Q  z
)  <Q  ( F `  z ) } ,  { u  e.  Q.  |  E. z  e.  Q.  ( ( F `  z )  +Q  z
)  <Q  u } >.  <P  <. { l  |  l 
<Q  ( ( F `  q )  +Q  (
q  +Q  r ) ) } ,  {
u  |  ( ( F `  q )  +Q  ( q  +Q  r ) )  <Q  u } >. ) ) )
2423rspcev 2789 . 2  |-  ( (
<. { l  e.  Q.  |  E. z  e.  Q.  ( l  +Q  z
)  <Q  ( F `  z ) } ,  { u  e.  Q.  |  E. z  e.  Q.  ( ( F `  z )  +Q  z
)  <Q  u } >.  e. 
P.  /\  A. q  e.  Q.  A. r  e. 
Q.  ( <. { l  |  l  <Q  ( F `  q ) } ,  { u  |  ( F `  q )  <Q  u } >.  <P  ( <. { l  e.  Q.  |  E. z  e.  Q.  (
l  +Q  z ) 
<Q  ( F `  z
) } ,  {
u  e.  Q.  |  E. z  e.  Q.  ( ( F `  z )  +Q  z
)  <Q  u } >.  +P. 
<. { l  |  l 
<Q  ( q  +Q  r
) } ,  {
u  |  ( q  +Q  r )  <Q  u } >. )  /\  <. { l  e.  Q.  |  E. z  e.  Q.  ( l  +Q  z
)  <Q  ( F `  z ) } ,  { u  e.  Q.  |  E. z  e.  Q.  ( ( F `  z )  +Q  z
)  <Q  u } >.  <P  <. { l  |  l 
<Q  ( ( F `  q )  +Q  (
q  +Q  r ) ) } ,  {
u  |  ( ( F `  q )  +Q  ( q  +Q  r ) )  <Q  u } >. ) )  ->  E. y  e.  P.  A. q  e.  Q.  A. r  e.  Q.  ( <. { l  |  l 
<Q  ( F `  q
) } ,  {
u  |  ( F `
 q )  <Q  u } >.  <P  ( y  +P.  <. { l  |  l  <Q  ( q  +Q  r ) } ,  { u  |  (
q  +Q  r ) 
<Q  u } >. )  /\  y  <P  <. { l  |  l  <Q  (
( F `  q
)  +Q  ( q  +Q  r ) ) } ,  { u  |  ( ( F `
 q )  +Q  ( q  +Q  r
) )  <Q  u } >. ) )
2517, 18, 24syl2anc 408 1  |-  ( ph  ->  E. y  e.  P.  A. q  e.  Q.  A. r  e.  Q.  ( <. { l  |  l 
<Q  ( F `  q
) } ,  {
u  |  ( F `
 q )  <Q  u } >.  <P  ( y  +P.  <. { l  |  l  <Q  ( q  +Q  r ) } ,  { u  |  (
q  +Q  r ) 
<Q  u } >. )  /\  y  <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    <-> wb 104    = wceq 1331    e. wcel 1480   {cab 2125   A.wral 2416   E.wrex 2417   {crab 2420   <.cop 3530   class class class wbr 3929   -->wf 5119   ` cfv 5123  (class class class)co 5774   Q.cnq 7095    +Q cplq 7097    <Q cltq 7100   P.cnp 7106    +P. cpp 7108    <P cltp 7110
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7119  df-pli 7120  df-mi 7121  df-lti 7122  df-plpq 7159  df-mpq 7160  df-enq 7162  df-nqqs 7163  df-plqqs 7164  df-mqqs 7165  df-1nqqs 7166  df-rq 7167  df-ltnqqs 7168  df-enq0 7239  df-nq0 7240  df-0nq0 7241  df-plq0 7242  df-mq0 7243  df-inp 7281  df-iplp 7283  df-iltp 7285
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator