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Theorem cauappcvgprlem2 7480
Description: Lemma for cauappcvgpr 7482. 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 7227 . . . . 5  |-  ( ( Q  e.  Q.  /\  R  e.  Q. )  ->  Q  <Q  ( Q  +Q  R ) )
41, 2, 3syl2anc 408 . . . 4  |-  ( ph  ->  Q  <Q  ( Q  +Q  R ) )
5 cauappcvgpr.f . . . . 5  |-  ( ph  ->  F : Q. --> Q. )
65, 1ffvelrnd 5556 . . . 4  |-  ( ph  ->  ( F `  Q
)  e.  Q. )
7 ltanqi 7222 . . . 4  |-  ( ( Q  <Q  ( Q  +Q  R )  /\  ( F `  Q )  e.  Q. )  ->  (
( F `  Q
)  +Q  Q ) 
<Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) )
84, 6, 7syl2anc 408 . . 3  |-  ( ph  ->  ( ( F `  Q )  +Q  Q
)  <Q  ( ( F `
 Q )  +Q  ( Q  +Q  R
) ) )
9 ltbtwnnqq 7235 . . 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 520 . . . 4  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( ( ( F `  Q
)  +Q  Q ) 
<Q  x  /\  x  <Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) ) ) )  ->  x  e.  Q. )
121adantr 274 . . . . . 6  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( ( ( F `  Q
)  +Q  Q ) 
<Q  x  /\  x  <Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) ) ) )  ->  Q  e.  Q. )
13 simprrl 528 . . . . . 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 5421 . . . . . . . . 9  |-  ( q  =  Q  ->  ( F `  q )  =  ( F `  Q ) )
15 id 19 . . . . . . . . 9  |-  ( q  =  Q  ->  q  =  Q )
1614, 15oveq12d 5792 . . . . . . . 8  |-  ( q  =  Q  ->  (
( F `  q
)  +Q  q )  =  ( ( F `
 Q )  +Q  Q ) )
1716breq1d 3939 . . . . . . 7  |-  ( q  =  Q  ->  (
( ( F `  q )  +Q  q
)  <Q  x  <->  ( ( F `  Q )  +Q  Q )  <Q  x
) )
1817rspcev 2789 . . . . . 6  |-  ( ( Q  e.  Q.  /\  ( ( F `  Q )  +Q  Q
)  <Q  x )  ->  E. q  e.  Q.  ( ( F `  q )  +Q  q
)  <Q  x )
1912, 13, 18syl2anc 408 . . . . 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 3933 . . . . . . 7  |-  ( u  =  x  ->  (
( ( F `  q )  +Q  q
)  <Q  u  <->  ( ( F `  q )  +Q  q )  <Q  x
) )
2120rexbidv 2438 . . . . . 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 5424 . . . . . . 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 7183 . . . . . . . . 9  |-  Q.  e.  _V
2524rabex 4072 . . . . . . . 8  |-  { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( F `  q
) }  e.  _V
2624rabex 4072 . . . . . . . 8  |-  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  u }  e.  _V
2725, 26op2nd 6045 . . . . . . 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 2160 . . . . . 6  |-  ( 2nd `  L )  =  {
u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  q
)  <Q  u }
2921, 28elrab2 2843 . . . . 5  |-  ( x  e.  ( 2nd `  L
)  <->  ( x  e. 
Q.  /\  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  x
) )
3011, 19, 29sylanbrc 413 . . . 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 529 . . . . . 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 2689 . . . . . . 7  |-  x  e. 
_V
33 breq1 3932 . . . . . . 7  |-  ( l  =  x  ->  (
l  <Q  ( ( F `
 Q )  +Q  ( Q  +Q  R
) )  <->  x  <Q  ( ( F `  Q
)  +Q  ( Q  +Q  R ) ) ) )
3432, 33elab 2828 . . . . . 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 7369 . . . . . 6  |-  { l  |  l  <Q  (
( F `  Q
)  +Q  ( Q  +Q  R ) ) }  e.  _V
37 gtnqex 7370 . . . . . 6  |-  { u  |  ( ( F `
 Q )  +Q  ( Q  +Q  R
) )  <Q  u }  e.  _V
3836, 37op1st 6044 . . . . 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 2233 . . . 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 2481 . . . 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 1214 . . 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 7473 . . . . 5  |-  ( ph  ->  L  e.  P. )
4544adantr 274 . . . 4  |-  ( (
ph  /\  ( x  e.  Q.  /\  ( ( ( F `  Q
)  +Q  Q ) 
<Q  x  /\  x  <Q  ( ( F `  Q )  +Q  ( Q  +Q  R ) ) ) ) )  ->  L  e.  P. )
46 addclnq 7195 . . . . . . . 8  |-  ( ( Q  e.  Q.  /\  R  e.  Q. )  ->  ( Q  +Q  R
)  e.  Q. )
471, 2, 46syl2anc 408 . . . . . . 7  |-  ( ph  ->  ( Q  +Q  R
)  e.  Q. )
48 addclnq 7195 . . . . . . 7  |-  ( ( ( F `  Q
)  e.  Q.  /\  ( Q  +Q  R
)  e.  Q. )  ->  ( ( F `  Q )  +Q  ( Q  +Q  R ) )  e.  Q. )
496, 47, 48syl2anc 408 . . . . . 6  |-  ( ph  ->  ( ( F `  Q )  +Q  ( Q  +Q  R ) )  e.  Q. )
50 nqprlu 7367 . . . . . 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 274 . . . 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 7326 . . . 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 408 . . 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 2554 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 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   1stc1st 6036   2ndc2nd 6037   Q.cnq 7100    +Q cplq 7102    <Q cltq 7105   P.cnp 7111    <P cltp 7115
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-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7124  df-pli 7125  df-mi 7126  df-lti 7127  df-plpq 7164  df-mpq 7165  df-enq 7167  df-nqqs 7168  df-plqqs 7169  df-mqqs 7170  df-1nqqs 7171  df-rq 7172  df-ltnqqs 7173  df-inp 7286  df-iltp 7290
This theorem is referenced by:  cauappcvgprlemlim  7481
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