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Theorem cauappcvgprlemrnd 6622
Description: Lemma for cauappcvgpr 6634. The putative limit is rounded. (Contributed by Jim Kingdon, 18-Jul-2020.)
Hypotheses
Ref Expression
cauappcvgpr.f  F : Q. --> Q.
cauappcvgpr.app  p  Q.  q  Q.  F `  p 
<Q  F `  q  +Q  p  +Q  q  F `  q  <Q  F `
 p  +Q  p  +Q  q
cauappcvgpr.bnd  p  Q.  <Q  F `  p
cauappcvgpr.lim  L 
<. { l  Q.  |  q  Q.  l  +Q  q  <Q  F `  q } ,  {  Q.  |  q  Q.  F `  q  +Q  q  <Q  } >.
Assertion
Ref Expression
cauappcvgprlemrnd  s 
Q.  s  1st `  L  r  Q.  s  <Q  r  r  1st `  L  r  Q.  r  2nd `  L  s  Q.  s  <Q 
r  s  2nd `  L
Distinct variable groups:   , p    L, p, q   , p, q    L, r, s   , s, p    F, l,, p, q, r, s   , r,
s
Allowed substitution hints:   (, l)   (, r, q, l)    L(, l)

Proof of Theorem cauappcvgprlemrnd
StepHypRef Expression
1 cauappcvgpr.f . . . . . 6  F : Q. --> Q.
2 cauappcvgpr.app . . . . . 6  p  Q.  q  Q.  F `  p 
<Q  F `  q  +Q  p  +Q  q  F `  q  <Q  F `
 p  +Q  p  +Q  q
3 cauappcvgpr.bnd . . . . . 6  p  Q.  <Q  F `  p
4 cauappcvgpr.lim . . . . . 6  L 
<. { l  Q.  |  q  Q.  l  +Q  q  <Q  F `  q } ,  {  Q.  |  q  Q.  F `  q  +Q  q  <Q  } >.
51, 2, 3, 4cauappcvgprlemopl 6618 . . . . 5  s  1st `  L  r  Q.  s  <Q 
r  r  1st `  L
65ex 108 . . . 4  s  1st `  L  r  Q.  s  <Q  r  r  1st `  L
71, 2, 3, 4cauappcvgprlemlol 6619 . . . . . 6  s  <Q  r  r  1st `  L  s  1st `  L
873expib 1106 . . . . 5  s  <Q 
r  r  1st `  L  s  1st `  L
98rexlimdvw 2430 . . . 4  r 
Q.  s  <Q 
r  r  1st `  L  s  1st `  L
106, 9impbid 120 . . 3  s  1st `  L  r  Q.  s  <Q  r  r  1st `  L
1110ralrimivw 2387 . 2  s  Q.  s  1st `  L  r  Q.  s  <Q 
r  r  1st `  L
121, 2, 3, 4cauappcvgprlemopu 6620 . . . . 5  r  2nd `  L  s  Q.  s  <Q 
r  s  2nd `  L
1312ex 108 . . . 4  r  2nd `  L  s  Q.  s  <Q  r  s  2nd `  L
141, 2, 3, 4cauappcvgprlemupu 6621 . . . . . 6  s  <Q  r  s  2nd `  L  r  2nd `  L
15143expib 1106 . . . . 5  s  <Q 
r  s  2nd `  L  r  2nd `  L
1615rexlimdvw 2430 . . . 4  s 
Q.  s  <Q 
r  s  2nd `  L  r  2nd `  L
1713, 16impbid 120 . . 3  r  2nd `  L  s  Q.  s  <Q  r  s  2nd `  L
1817ralrimivw 2387 . 2  r  Q.  r  2nd `  L  s  Q.  s  <Q 
r  s  2nd `  L
1911, 18jca 290 1  s 
Q.  s  1st `  L  r  Q.  s  <Q  r  r  1st `  L  r  Q.  r  2nd `  L  s  Q.  s  <Q 
r  s  2nd `  L
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242   wcel 1390  wral 2300  wrex 2301   {crab 2304   <.cop 3370   class class class wbr 3755   -->wf 4841   ` cfv 4845  (class class class)co 5455   1stc1st 5707   2ndc2nd 5708   Q.cnq 6264    +Q cplq 6266    <Q cltq 6269
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337
This theorem is referenced by:  cauappcvgprlemcl  6625
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