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Theorem cauappcvgprlem2 7280
Description: Lemma for cauappcvgpr 7282. 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 7027 . . . . 5  |-  ( ( Q  e.  Q.  /\  R  e.  Q. )  ->  Q  <Q  ( Q  +Q  R ) )
41, 2, 3syl2anc 404 . . . 4  |-  ( ph  ->  Q  <Q  ( Q  +Q  R ) )
5 cauappcvgpr.f . . . . 5  |-  ( ph  ->  F : Q. --> Q. )
65, 1ffvelrnd 5449 . . . 4  |-  ( ph  ->  ( F `  Q
)  e.  Q. )
7 ltanqi 7022 . . . 4  |-  ( ( Q  <Q  ( Q  +Q  R )  /\  ( F `  Q )  e.  Q. )  ->  (
( F `  Q
)  +Q  Q ) 
<Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) )
84, 6, 7syl2anc 404 . . 3  |-  ( ph  ->  ( ( F `  Q )  +Q  Q
)  <Q  ( ( F `
 Q )  +Q  ( Q  +Q  R
) ) )
9 ltbtwnnqq 7035 . . 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 121 . 2  |-  ( ph  ->  E. x  e.  Q.  ( ( ( F `
 Q )  +Q  Q )  <Q  x  /\  x  <Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) ) )
11 simprl 499 . . . 4  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( ( ( F `  Q
)  +Q  Q ) 
<Q  x  /\  x  <Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) ) ) )  ->  x  e.  Q. )
121adantr 271 . . . . . 6  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( ( ( F `  Q
)  +Q  Q ) 
<Q  x  /\  x  <Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) ) ) )  ->  Q  e.  Q. )
13 simprrl 507 . . . . . 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 5318 . . . . . . . . 9  |-  ( q  =  Q  ->  ( F `  q )  =  ( F `  Q ) )
15 id 19 . . . . . . . . 9  |-  ( q  =  Q  ->  q  =  Q )
1614, 15oveq12d 5684 . . . . . . . 8  |-  ( q  =  Q  ->  (
( F `  q
)  +Q  q )  =  ( ( F `
 Q )  +Q  Q ) )
1716breq1d 3861 . . . . . . 7  |-  ( q  =  Q  ->  (
( ( F `  q )  +Q  q
)  <Q  x  <->  ( ( F `  Q )  +Q  Q )  <Q  x
) )
1817rspcev 2723 . . . . . 6  |-  ( ( Q  e.  Q.  /\  ( ( F `  Q )  +Q  Q
)  <Q  x )  ->  E. q  e.  Q.  ( ( F `  q )  +Q  q
)  <Q  x )
1912, 13, 18syl2anc 404 . . . . 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 3855 . . . . . . 7  |-  ( u  =  x  ->  (
( ( F `  q )  +Q  q
)  <Q  u  <->  ( ( F `  q )  +Q  q )  <Q  x
) )
2120rexbidv 2382 . . . . . 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 5321 . . . . . . 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 6983 . . . . . . . . 9  |-  Q.  e.  _V
2524rabex 3989 . . . . . . . 8  |-  { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( F `  q
) }  e.  _V
2624rabex 3989 . . . . . . . 8  |-  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  u }  e.  _V
2725, 26op2nd 5932 . . . . . . 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 2109 . . . . . 6  |-  ( 2nd `  L )  =  {
u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  q
)  <Q  u }
2921, 28elrab2 2775 . . . . 5  |-  ( x  e.  ( 2nd `  L
)  <->  ( x  e. 
Q.  /\  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  x
) )
3011, 19, 29sylanbrc 409 . . . 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 508 . . . . . 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 2623 . . . . . . 7  |-  x  e. 
_V
33 breq1 3854 . . . . . . 7  |-  ( l  =  x  ->  (
l  <Q  ( ( F `
 Q )  +Q  ( Q  +Q  R
) )  <->  x  <Q  ( ( F `  Q
)  +Q  ( Q  +Q  R ) ) ) )
3432, 33elab 2761 . . . . . 6  |-  ( x  e.  { l  |  l  <Q  ( ( F `  Q )  +Q  ( Q  +Q  R
) ) }  <->  x  <Q  ( ( F `  Q
)  +Q  ( Q  +Q  R ) ) )
3531, 34sylibr 133 . . . . 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 7169 . . . . . 6  |-  { l  |  l  <Q  (
( F `  Q
)  +Q  ( Q  +Q  R ) ) }  e.  _V
37 gtnqex 7170 . . . . . 6  |-  { u  |  ( ( F `
 Q )  +Q  ( Q  +Q  R
) )  <Q  u }  e.  _V
3836, 37op1st 5931 . . . . 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, 38syl6eleqr 2182 . . . 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 2425 . . . 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 1173 . . 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 7273 . . . . 5  |-  ( ph  ->  L  e.  P. )
4544adantr 271 . . . 4  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( ( ( F `  Q
)  +Q  Q ) 
<Q  x  /\  x  <Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) ) ) )  ->  L  e.  P. )
46 addclnq 6995 . . . . . . . 8  |-  ( ( Q  e.  Q.  /\  R  e.  Q. )  ->  ( Q  +Q  R
)  e.  Q. )
471, 2, 46syl2anc 404 . . . . . . 7  |-  ( ph  ->  ( Q  +Q  R
)  e.  Q. )
48 addclnq 6995 . . . . . . 7  |-  ( ( ( F `  Q
)  e.  Q.  /\  ( Q  +Q  R
)  e.  Q. )  ->  ( ( F `  Q )  +Q  ( Q  +Q  R ) )  e.  Q. )
496, 47, 48syl2anc 404 . . . . . 6  |-  ( ph  ->  ( ( F `  Q )  +Q  ( Q  +Q  R ) )  e.  Q. )
50 nqprlu 7167 . . . . . 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 271 . . . 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 7126 . . . 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 404 . . 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 166 . 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 2494 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 103    <-> wb 104    = wceq 1290    e. wcel 1439   {cab 2075   A.wral 2360   E.wrex 2361   {crab 2364   <.cop 3453   class class class wbr 3851   -->wf 5024   ` cfv 5028  (class class class)co 5666   1stc1st 5923   2ndc2nd 5924   Q.cnq 6900    +Q cplq 6902    <Q cltq 6905   P.cnp 6911    <P cltp 6915
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-eprel 4125  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-irdg 6149  df-1o 6195  df-oadd 6199  df-omul 6200  df-er 6306  df-ec 6308  df-qs 6312  df-ni 6924  df-pli 6925  df-mi 6926  df-lti 6927  df-plpq 6964  df-mpq 6965  df-enq 6967  df-nqqs 6968  df-plqqs 6969  df-mqqs 6970  df-1nqqs 6971  df-rq 6972  df-ltnqqs 6973  df-inp 7086  df-iltp 7090
This theorem is referenced by:  cauappcvgprlemlim  7281
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