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Theorem cauappcvgprlem2 7672
Description: Lemma for cauappcvgpr 7674. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 23-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 } >.
cauappcvgprlem.q  |-  ( ph  ->  Q  e.  Q. )
cauappcvgprlem.r  |-  ( ph  ->  R  e.  Q. )
Assertion
Ref Expression
cauappcvgprlem2  |-  ( ph  ->  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, p, q, l, u    Q, p, q, l, u    R, p, q, l, u
Allowed substitution hints:    ph( u, l)    A( u, q, l)    L( u, l)

Proof of Theorem cauappcvgprlem2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 cauappcvgprlem.q . . . . 5  |-  ( ph  ->  Q  e.  Q. )
2 cauappcvgprlem.r . . . . 5  |-  ( ph  ->  R  e.  Q. )
3 ltaddnq 7419 . . . . 5  |-  ( ( Q  e.  Q.  /\  R  e.  Q. )  ->  Q  <Q  ( Q  +Q  R ) )
41, 2, 3syl2anc 411 . . . 4  |-  ( ph  ->  Q  <Q  ( Q  +Q  R ) )
5 cauappcvgpr.f . . . . 5  |-  ( ph  ->  F : Q. --> Q. )
65, 1ffvelcdmd 5665 . . . 4  |-  ( ph  ->  ( F `  Q
)  e.  Q. )
7 ltanqi 7414 . . . 4  |-  ( ( Q  <Q  ( Q  +Q  R )  /\  ( F `  Q )  e.  Q. )  ->  (
( F `  Q
)  +Q  Q ) 
<Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) )
84, 6, 7syl2anc 411 . . 3  |-  ( ph  ->  ( ( F `  Q )  +Q  Q
)  <Q  ( ( F `
 Q )  +Q  ( Q  +Q  R
) ) )
9 ltbtwnnqq 7427 . . 3  |-  ( ( ( F `  Q
)  +Q  Q ) 
<Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) )  <->  E. x  e.  Q.  ( ( ( F `
 Q )  +Q  Q )  <Q  x  /\  x  <Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) ) )
108, 9sylib 122 . 2  |-  ( ph  ->  E. x  e.  Q.  ( ( ( F `
 Q )  +Q  Q )  <Q  x  /\  x  <Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) ) )
11 simprl 529 . . . 4  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( ( ( F `  Q
)  +Q  Q ) 
<Q  x  /\  x  <Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) ) ) )  ->  x  e.  Q. )
121adantr 276 . . . . . 6  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( ( ( F `  Q
)  +Q  Q ) 
<Q  x  /\  x  <Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) ) ) )  ->  Q  e.  Q. )
13 simprrl 539 . . . . . 6  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( ( ( F `  Q
)  +Q  Q ) 
<Q  x  /\  x  <Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) ) ) )  -> 
( ( F `  Q )  +Q  Q
)  <Q  x )
14 fveq2 5527 . . . . . . . . 9  |-  ( q  =  Q  ->  ( F `  q )  =  ( F `  Q ) )
15 id 19 . . . . . . . . 9  |-  ( q  =  Q  ->  q  =  Q )
1614, 15oveq12d 5906 . . . . . . . 8  |-  ( q  =  Q  ->  (
( F `  q
)  +Q  q )  =  ( ( F `
 Q )  +Q  Q ) )
1716breq1d 4025 . . . . . . 7  |-  ( q  =  Q  ->  (
( ( F `  q )  +Q  q
)  <Q  x  <->  ( ( F `  Q )  +Q  Q )  <Q  x
) )
1817rspcev 2853 . . . . . 6  |-  ( ( Q  e.  Q.  /\  ( ( F `  Q )  +Q  Q
)  <Q  x )  ->  E. q  e.  Q.  ( ( F `  q )  +Q  q
)  <Q  x )
1912, 13, 18syl2anc 411 . . . . 5  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( ( ( F `  Q
)  +Q  Q ) 
<Q  x  /\  x  <Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) ) ) )  ->  E. q  e.  Q.  ( ( F `  q )  +Q  q
)  <Q  x )
20 breq2 4019 . . . . . . 7  |-  ( u  =  x  ->  (
( ( F `  q )  +Q  q
)  <Q  u  <->  ( ( F `  q )  +Q  q )  <Q  x
) )
2120rexbidv 2488 . . . . . 6  |-  ( u  =  x  ->  ( E. q  e.  Q.  ( ( F `  q )  +Q  q
)  <Q  u  <->  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  x
) )
22 cauappcvgpr.lim . . . . . . . 8  |-  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 } >.
2322fveq2i 5530 . . . . . . 7  |-  ( 2nd `  L )  =  ( 2nd `  <. { 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 } >. )
24 nqex 7375 . . . . . . . . 9  |-  Q.  e.  _V
2524rabex 4159 . . . . . . . 8  |-  { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( F `  q
) }  e.  _V
2624rabex 4159 . . . . . . . 8  |-  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  u }  e.  _V
2725, 26op2nd 6161 . . . . . . 7  |-  ( 2nd `  <. { 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 } >. )  =  {
u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  q
)  <Q  u }
2823, 27eqtri 2208 . . . . . 6  |-  ( 2nd `  L )  =  {
u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  q
)  <Q  u }
2921, 28elrab2 2908 . . . . 5  |-  ( x  e.  ( 2nd `  L
)  <->  ( x  e. 
Q.  /\  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  x
) )
3011, 19, 29sylanbrc 417 . . . 4  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( ( ( F `  Q
)  +Q  Q ) 
<Q  x  /\  x  <Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) ) ) )  ->  x  e.  ( 2nd `  L ) )
31 simprrr 540 . . . . . 6  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( ( ( F `  Q
)  +Q  Q ) 
<Q  x  /\  x  <Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) ) ) )  ->  x  <Q  ( ( F `
 Q )  +Q  ( Q  +Q  R
) ) )
32 vex 2752 . . . . . . 7  |-  x  e. 
_V
33 breq1 4018 . . . . . . 7  |-  ( l  =  x  ->  (
l  <Q  ( ( F `
 Q )  +Q  ( Q  +Q  R
) )  <->  x  <Q  ( ( F `  Q
)  +Q  ( Q  +Q  R ) ) ) )
3432, 33elab 2893 . . . . . 6  |-  ( x  e.  { l  |  l  <Q  ( ( F `  Q )  +Q  ( Q  +Q  R
) ) }  <->  x  <Q  ( ( F `  Q
)  +Q  ( Q  +Q  R ) ) )
3531, 34sylibr 134 . . . . 5  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( ( ( F `  Q
)  +Q  Q ) 
<Q  x  /\  x  <Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) ) ) )  ->  x  e.  { l  |  l  <Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) } )
36 ltnqex 7561 . . . . . 6  |-  { l  |  l  <Q  (
( F `  Q
)  +Q  ( Q  +Q  R ) ) }  e.  _V
37 gtnqex 7562 . . . . . 6  |-  { u  |  ( ( F `
 Q )  +Q  ( Q  +Q  R
) )  <Q  u }  e.  _V
3836, 37op1st 6160 . . . . 5  |-  ( 1st `  <. { l  |  l  <Q  ( ( F `  Q )  +Q  ( Q  +Q  R
) ) } ,  { u  |  (
( F `  Q
)  +Q  ( Q  +Q  R ) ) 
<Q  u } >. )  =  { l  |  l 
<Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) }
3935, 38eleqtrrdi 2281 . . . 4  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( ( ( F `  Q
)  +Q  Q ) 
<Q  x  /\  x  <Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) ) ) )  ->  x  e.  ( 1st ` 
<. { l  |  l 
<Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) } ,  { u  |  ( ( F `
 Q )  +Q  ( Q  +Q  R
) )  <Q  u } >. ) )
40 rspe 2536 . . . 4  |-  ( ( x  e.  Q.  /\  ( x  e.  ( 2nd `  L )  /\  x  e.  ( 1st ` 
<. { l  |  l 
<Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) } ,  { u  |  ( ( F `
 Q )  +Q  ( Q  +Q  R
) )  <Q  u } >. ) ) )  ->  E. x  e.  Q.  ( x  e.  ( 2nd `  L )  /\  x  e.  ( 1st ` 
<. { l  |  l 
<Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) } ,  { u  |  ( ( F `
 Q )  +Q  ( Q  +Q  R
) )  <Q  u } >. ) ) )
4111, 30, 39, 40syl12anc 1246 . . 3  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( ( ( F `  Q
)  +Q  Q ) 
<Q  x  /\  x  <Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) ) ) )  ->  E. x  e.  Q.  ( x  e.  ( 2nd `  L )  /\  x  e.  ( 1st ` 
<. { l  |  l 
<Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) } ,  { u  |  ( ( F `
 Q )  +Q  ( Q  +Q  R
) )  <Q  u } >. ) ) )
42 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 ) ) ) )
43 cauappcvgpr.bnd . . . . . 6  |-  ( ph  ->  A. p  e.  Q.  A  <Q  ( F `  p ) )
445, 42, 43, 22cauappcvgprlemcl 7665 . . . . 5  |-  ( ph  ->  L  e.  P. )
4544adantr 276 . . . 4  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( ( ( F `  Q
)  +Q  Q ) 
<Q  x  /\  x  <Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) ) ) )  ->  L  e.  P. )
46 addclnq 7387 . . . . . . . 8  |-  ( ( Q  e.  Q.  /\  R  e.  Q. )  ->  ( Q  +Q  R
)  e.  Q. )
471, 2, 46syl2anc 411 . . . . . . 7  |-  ( ph  ->  ( Q  +Q  R
)  e.  Q. )
48 addclnq 7387 . . . . . . 7  |-  ( ( ( F `  Q
)  e.  Q.  /\  ( Q  +Q  R
)  e.  Q. )  ->  ( ( F `  Q )  +Q  ( Q  +Q  R ) )  e.  Q. )
496, 47, 48syl2anc 411 . . . . . 6  |-  ( ph  ->  ( ( F `  Q )  +Q  ( Q  +Q  R ) )  e.  Q. )
50 nqprlu 7559 . . . . . 6  |-  ( ( ( F `  Q
)  +Q  ( Q  +Q  R ) )  e.  Q.  ->  <. { l  |  l  <Q  (
( F `  Q
)  +Q  ( Q  +Q  R ) ) } ,  { u  |  ( ( F `
 Q )  +Q  ( Q  +Q  R
) )  <Q  u } >.  e.  P. )
5149, 50syl 14 . . . . 5  |-  ( ph  -> 
<. { l  |  l 
<Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) } ,  { u  |  ( ( F `
 Q )  +Q  ( Q  +Q  R
) )  <Q  u } >.  e.  P. )
5251adantr 276 . . . 4  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( ( ( F `  Q
)  +Q  Q ) 
<Q  x  /\  x  <Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) ) ) )  ->  <. { l  |  l 
<Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) } ,  { u  |  ( ( F `
 Q )  +Q  ( Q  +Q  R
) )  <Q  u } >.  e.  P. )
53 ltdfpr 7518 . . . 4  |-  ( ( L  e.  P.  /\  <. { l  |  l 
<Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) } ,  { u  |  ( ( F `
 Q )  +Q  ( Q  +Q  R
) )  <Q  u } >.  e.  P. )  ->  ( L  <P  <. { l  |  l  <Q  (
( F `  Q
)  +Q  ( Q  +Q  R ) ) } ,  { u  |  ( ( F `
 Q )  +Q  ( Q  +Q  R
) )  <Q  u } >. 
<->  E. x  e.  Q.  ( x  e.  ( 2nd `  L )  /\  x  e.  ( 1st ` 
<. { l  |  l 
<Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) } ,  { u  |  ( ( F `
 Q )  +Q  ( Q  +Q  R
) )  <Q  u } >. ) ) ) )
5445, 52, 53syl2anc 411 . . 3  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( ( ( F `  Q
)  +Q  Q ) 
<Q  x  /\  x  <Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) ) ) )  -> 
( L  <P  <. { l  |  l  <Q  (
( F `  Q
)  +Q  ( Q  +Q  R ) ) } ,  { u  |  ( ( F `
 Q )  +Q  ( Q  +Q  R
) )  <Q  u } >. 
<->  E. x  e.  Q.  ( x  e.  ( 2nd `  L )  /\  x  e.  ( 1st ` 
<. { l  |  l 
<Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) } ,  { u  |  ( ( F `
 Q )  +Q  ( Q  +Q  R
) )  <Q  u } >. ) ) ) )
5541, 54mpbird 167 . 2  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( ( ( F `  Q
)  +Q  Q ) 
<Q  x  /\  x  <Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) ) ) )  ->  L  <P  <. { l  |  l  <Q  ( ( F `  Q )  +Q  ( Q  +Q  R
) ) } ,  { u  |  (
( F `  Q
)  +Q  ( Q  +Q  R ) ) 
<Q  u } >. )
5610, 55rexlimddv 2609 1  |-  ( ph  ->  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 104    <-> wb 105    = wceq 1363    e. wcel 2158   {cab 2173   A.wral 2465   E.wrex 2466   {crab 2469   <.cop 3607   class class class wbr 4015   -->wf 5224   ` cfv 5228  (class class class)co 5888   1stc1st 6152   2ndc2nd 6153   Q.cnq 7292    +Q cplq 7294    <Q cltq 7297   P.cnp 7303    <P cltp 7307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-eprel 4301  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-irdg 6384  df-1o 6430  df-oadd 6434  df-omul 6435  df-er 6548  df-ec 6550  df-qs 6554  df-ni 7316  df-pli 7317  df-mi 7318  df-lti 7319  df-plpq 7356  df-mpq 7357  df-enq 7359  df-nqqs 7360  df-plqqs 7361  df-mqqs 7362  df-1nqqs 7363  df-rq 7364  df-ltnqqs 7365  df-inp 7478  df-iltp 7482
This theorem is referenced by:  cauappcvgprlemlim  7673
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