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Theorem cauappcvgprlem2 7634
Description: Lemma for cauappcvgpr 7636. 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 7381 . . . . 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 5644 . . . 4  |-  ( ph  ->  ( F `  Q
)  e.  Q. )
7 ltanqi 7376 . . . 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 7389 . . 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 5507 . . . . . . . . 9  |-  ( q  =  Q  ->  ( F `  q )  =  ( F `  Q ) )
15 id 19 . . . . . . . . 9  |-  ( q  =  Q  ->  q  =  Q )
1614, 15oveq12d 5883 . . . . . . . 8  |-  ( q  =  Q  ->  (
( F `  q
)  +Q  q )  =  ( ( F `
 Q )  +Q  Q ) )
1716breq1d 4008 . . . . . . 7  |-  ( q  =  Q  ->  (
( ( F `  q )  +Q  q
)  <Q  x  <->  ( ( F `  Q )  +Q  Q )  <Q  x
) )
1817rspcev 2839 . . . . . 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 4002 . . . . . . 7  |-  ( u  =  x  ->  (
( ( F `  q )  +Q  q
)  <Q  u  <->  ( ( F `  q )  +Q  q )  <Q  x
) )
2120rexbidv 2476 . . . . . 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 5510 . . . . . . 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 7337 . . . . . . . . 9  |-  Q.  e.  _V
2524rabex 4142 . . . . . . . 8  |-  { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( F `  q
) }  e.  _V
2624rabex 4142 . . . . . . . 8  |-  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  u }  e.  _V
2725, 26op2nd 6138 . . . . . . 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 2196 . . . . . 6  |-  ( 2nd `  L )  =  {
u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  q
)  <Q  u }
2921, 28elrab2 2894 . . . . 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 2738 . . . . . . 7  |-  x  e. 
_V
33 breq1 4001 . . . . . . 7  |-  ( l  =  x  ->  (
l  <Q  ( ( F `
 Q )  +Q  ( Q  +Q  R
) )  <->  x  <Q  ( ( F `  Q
)  +Q  ( Q  +Q  R ) ) ) )
3432, 33elab 2879 . . . . . 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 7523 . . . . . 6  |-  { l  |  l  <Q  (
( F `  Q
)  +Q  ( Q  +Q  R ) ) }  e.  _V
37 gtnqex 7524 . . . . . 6  |-  { u  |  ( ( F `
 Q )  +Q  ( Q  +Q  R
) )  <Q  u }  e.  _V
3836, 37op1st 6137 . . . . 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 2269 . . . 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 2524 . . . 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 1236 . . 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 7627 . . . . 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 7349 . . . . . . . 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 7349 . . . . . . 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 7521 . . . . . 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 7480 . . . 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 2597 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 1353    e. wcel 2146   {cab 2161   A.wral 2453   E.wrex 2454   {crab 2457   <.cop 3592   class class class wbr 3998   -->wf 5204   ` cfv 5208  (class class class)co 5865   1stc1st 6129   2ndc2nd 6130   Q.cnq 7254    +Q cplq 7256    <Q cltq 7259   P.cnp 7265    <P cltp 7269
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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-eprel 4283  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-1o 6407  df-oadd 6411  df-omul 6412  df-er 6525  df-ec 6527  df-qs 6531  df-ni 7278  df-pli 7279  df-mi 7280  df-lti 7281  df-plpq 7318  df-mpq 7319  df-enq 7321  df-nqqs 7322  df-plqqs 7323  df-mqqs 7324  df-1nqqs 7325  df-rq 7326  df-ltnqqs 7327  df-inp 7440  df-iltp 7444
This theorem is referenced by:  cauappcvgprlemlim  7635
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