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Theorem cauappcvgprlemopl 6618
Description: Lemma for cauappcvgpr 6634. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 4-Aug-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
cauappcvgprlemopl  s  1st `  L  r  Q.  s  <Q 
r  r  1st `  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 cauappcvgprlemopl
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 oveq1 5462 . . . . . . 7  l  s 
l  +Q  q  s  +Q  q
21breq1d 3765 . . . . . 6  l  s  l  +Q  q  <Q  F `  q  s  +Q  q  <Q  F `  q
32rexbidv 2321 . . . . 5  l  s  q  Q.  l  +Q  q  <Q  F `  q  q  Q.  s  +Q  q  <Q  F `  q
4 cauappcvgpr.lim . . . . . . 7  L 
<. { l  Q.  |  q  Q.  l  +Q  q  <Q  F `  q } ,  {  Q.  |  q  Q.  F `  q  +Q  q  <Q  } >.
54fveq2i 5124 . . . . . 6  1st `  L  1st `  <. { l  Q.  |  q  Q. 
l  +Q  q 
<Q  F `  q } ,  {  Q.  |  q  Q.  F `
 q  +Q  q  <Q  } >.
6 nqex 6347 . . . . . . . 8  Q.  _V
76rabex 3892 . . . . . . 7  { l  Q.  |  q  Q. 
l  +Q  q 
<Q  F `  q }  _V
86rabex 3892 . . . . . . 7  {  Q.  |  q  Q.  F `
 q  +Q  q  <Q  }  _V
97, 8op1st 5715 . . . . . 6  1st `  <. { l 
Q.  |  q  Q.  l  +Q  q  <Q  F `  q } ,  {  Q.  |  q  Q.  F `
 q  +Q  q  <Q  } >.  { l  Q.  |  q  Q. 
l  +Q  q 
<Q  F `  q }
105, 9eqtri 2057 . . . . 5  1st `  L  {
l  Q.  |  q  Q.  l  +Q  q  <Q  F `  q }
113, 10elrab2 2694 . . . 4  s  1st `  L  s  Q.  q  Q.  s  +Q  q  <Q  F `  q
1211simprbi 260 . . 3  s  1st `  L  q 
Q.  s  +Q  q  <Q  F `  q
1312adantl 262 . 2  s  1st `  L  q  Q.  s  +Q  q  <Q  F `  q
14 simprr 484 . . . 4  s  1st `  L 
q  Q.  s  +Q  q  <Q  F `  q  s  +Q  q  <Q  F `  q
15 ltbtwnnqq 6398 . . . 4  s  +Q  q 
<Q  F `  q  t  Q.  s  +Q  q  <Q  t  t  <Q  F `  q
1614, 15sylib 127 . . 3  s  1st `  L 
q  Q.  s  +Q  q  <Q  F `  q  t  Q.  s  +Q  q  <Q  t  t  <Q  F `  q
17 simplrl 487 . . . . . . . 8  s  1st `  L  q  Q.  s  +Q  q  <Q  F `  q  t  Q.  s  +Q  q  <Q  t  t  <Q  F `
 q  q  Q.
1811simplbi 259 . . . . . . . . 9  s  1st `  L  s  Q.
1918ad3antlr 462 . . . . . . . 8  s  1st `  L  q  Q.  s  +Q  q  <Q  F `  q  t  Q.  s  +Q  q  <Q  t  t  <Q  F `
 q  s  Q.
20 ltaddnq 6390 . . . . . . . 8  q  Q.  s  Q.  q  <Q  q  +Q  s
2117, 19, 20syl2anc 391 . . . . . . 7  s  1st `  L  q  Q.  s  +Q  q  <Q  F `  q  t  Q.  s  +Q  q  <Q  t  t  <Q  F `
 q  q  <Q  q  +Q  s
22 addcomnqg 6365 . . . . . . . 8  q  Q.  s  Q.  q  +Q  s  s  +Q  q
2317, 19, 22syl2anc 391 . . . . . . 7  s  1st `  L  q  Q.  s  +Q  q  <Q  F `  q  t  Q.  s  +Q  q  <Q  t  t  <Q  F `
 q 
q  +Q  s  s  +Q  q
2421, 23breqtrd 3779 . . . . . 6  s  1st `  L  q  Q.  s  +Q  q  <Q  F `  q  t  Q.  s  +Q  q  <Q  t  t  <Q  F `
 q  q  <Q  s  +Q  q
25 simprrl 491 . . . . . 6  s  1st `  L  q  Q.  s  +Q  q  <Q  F `  q  t  Q.  s  +Q  q  <Q  t  t  <Q  F `
 q 
s  +Q  q 
<Q  t
26 ltsonq 6382 . . . . . . 7  <Q  Or  Q.
27 ltrelnq 6349 . . . . . . 7  <Q  C_  Q.  X.  Q.
2826, 27sotri 4663 . . . . . 6  q  <Q  s  +Q  q 
s  +Q  q 
<Q  t  q  <Q  t
2924, 25, 28syl2anc 391 . . . . 5  s  1st `  L  q  Q.  s  +Q  q  <Q  F `  q  t  Q.  s  +Q  q  <Q  t  t  <Q  F `
 q  q  <Q  t
30 simprl 483 . . . . . 6  s  1st `  L  q  Q.  s  +Q  q  <Q  F `  q  t  Q.  s  +Q  q  <Q  t  t  <Q  F `
 q  t  Q.
31 ltexnqq 6391 . . . . . 6  q  Q.  t  Q.  q  <Q  t  r  Q.  q  +Q  r  t
3217, 30, 31syl2anc 391 . . . . 5  s  1st `  L  q  Q.  s  +Q  q  <Q  F `  q  t  Q.  s  +Q  q  <Q  t  t  <Q  F `
 q 
q  <Q  t  r  Q.  q  +Q  r  t
3329, 32mpbid 135 . . . 4  s  1st `  L  q  Q.  s  +Q  q  <Q  F `  q  t  Q.  s  +Q  q  <Q  t  t  <Q  F `
 q  r  Q. 
q  +Q  r  t
3425ad2antrr 457 . . . . . . . . . 10  s  1st `  L  q  Q. 
s  +Q  q 
<Q  F `  q 
t  Q.  s  +Q  q  <Q  t  t  <Q  F `  q  r  Q.  q  +Q  r  t  s  +Q  q  <Q  t
3519ad2antrr 457 . . . . . . . . . . . 12  s  1st `  L  q  Q. 
s  +Q  q 
<Q  F `  q 
t  Q.  s  +Q  q  <Q  t  t  <Q  F `  q  r  Q.  q  +Q  r  t  s  Q.
3617ad2antrr 457 . . . . . . . . . . . 12  s  1st `  L  q  Q. 
s  +Q  q 
<Q  F `  q 
t  Q.  s  +Q  q  <Q  t  t  <Q  F `  q  r  Q.  q  +Q  r  t  q  Q.
37 addcomnqg 6365 . . . . . . . . . . . 12  s  Q.  q  Q.  s  +Q  q  q  +Q  s
3835, 36, 37syl2anc 391 . . . . . . . . . . 11  s  1st `  L  q  Q. 
s  +Q  q 
<Q  F `  q 
t  Q.  s  +Q  q  <Q  t  t  <Q  F `  q  r  Q.  q  +Q  r  t  s  +Q  q  q  +Q  s
3938breq1d 3765 . . . . . . . . . 10  s  1st `  L  q  Q. 
s  +Q  q 
<Q  F `  q 
t  Q.  s  +Q  q  <Q  t  t  <Q  F `  q  r  Q.  q  +Q  r  t  s  +Q  q  <Q  t  q  +Q  s  <Q  t
4034, 39mpbid 135 . . . . . . . . 9  s  1st `  L  q  Q. 
s  +Q  q 
<Q  F `  q 
t  Q.  s  +Q  q  <Q  t  t  <Q  F `  q  r  Q.  q  +Q  r  t  q  +Q  s  <Q  t
41 simpr 103 . . . . . . . . 9  s  1st `  L  q  Q. 
s  +Q  q 
<Q  F `  q 
t  Q.  s  +Q  q  <Q  t  t  <Q  F `  q  r  Q.  q  +Q  r  t  q  +Q  r  t
4240, 41breqtrrd 3781 . . . . . . . 8  s  1st `  L  q  Q. 
s  +Q  q 
<Q  F `  q 
t  Q.  s  +Q  q  <Q  t  t  <Q  F `  q  r  Q.  q  +Q  r  t  q  +Q  s  <Q  q  +Q  r
43 simplr 482 . . . . . . . . 9  s  1st `  L  q  Q. 
s  +Q  q 
<Q  F `  q 
t  Q.  s  +Q  q  <Q  t  t  <Q  F `  q  r  Q.  q  +Q  r  t  r  Q.
44 ltanqg 6384 . . . . . . . . 9  s  Q.  r  Q.  q  Q. 
s  <Q  r  q  +Q  s  <Q  q  +Q  r
4535, 43, 36, 44syl3anc 1134 . . . . . . . 8  s  1st `  L  q  Q. 
s  +Q  q 
<Q  F `  q 
t  Q.  s  +Q  q  <Q  t  t  <Q  F `  q  r  Q.  q  +Q  r  t  s  <Q  r  q  +Q  s  <Q  q  +Q  r
4642, 45mpbird 156 . . . . . . 7  s  1st `  L  q  Q. 
s  +Q  q 
<Q  F `  q 
t  Q.  s  +Q  q  <Q  t  t  <Q  F `  q  r  Q.  q  +Q  r  t  s  <Q  r
47 simprrr 492 . . . . . . . . . . 11  s  1st `  L  q  Q.  s  +Q  q  <Q  F `  q  t  Q.  s  +Q  q  <Q  t  t  <Q  F `
 q  t  <Q  F `  q
4847ad2antrr 457 . . . . . . . . . 10  s  1st `  L  q  Q. 
s  +Q  q 
<Q  F `  q 
t  Q.  s  +Q  q  <Q  t  t  <Q  F `  q  r  Q.  q  +Q  r  t  t  <Q  F `  q
49 addcomnqg 6365 . . . . . . . . . . . . 13  q  Q.  r  Q.  q  +Q  r  r  +Q  q
5036, 43, 49syl2anc 391 . . . . . . . . . . . 12  s  1st `  L  q  Q. 
s  +Q  q 
<Q  F `  q 
t  Q.  s  +Q  q  <Q  t  t  <Q  F `  q  r  Q.  q  +Q  r  t  q  +Q  r  r  +Q  q
5150, 41eqtr3d 2071 . . . . . . . . . . 11  s  1st `  L  q  Q. 
s  +Q  q 
<Q  F `  q 
t  Q.  s  +Q  q  <Q  t  t  <Q  F `  q  r  Q.  q  +Q  r  t  r  +Q  q  t
5251breq1d 3765 . . . . . . . . . 10  s  1st `  L  q  Q. 
s  +Q  q 
<Q  F `  q 
t  Q.  s  +Q  q  <Q  t  t  <Q  F `  q  r  Q.  q  +Q  r  t  r  +Q  q  <Q  F `  q  t  <Q  F `  q
5348, 52mpbird 156 . . . . . . . . 9  s  1st `  L  q  Q. 
s  +Q  q 
<Q  F `  q 
t  Q.  s  +Q  q  <Q  t  t  <Q  F `  q  r  Q.  q  +Q  r  t  r  +Q  q  <Q  F `  q
54 rspe 2364 . . . . . . . . 9  q  Q.  r  +Q  q  <Q  F `  q  q  Q.  r  +Q  q  <Q  F `  q
5536, 53, 54syl2anc 391 . . . . . . . 8  s  1st `  L  q  Q. 
s  +Q  q 
<Q  F `  q 
t  Q.  s  +Q  q  <Q  t  t  <Q  F `  q  r  Q.  q  +Q  r  t  q  Q.  r  +Q  q  <Q  F `  q
56 oveq1 5462 . . . . . . . . . . 11  l  r 
l  +Q  q  r  +Q  q
5756breq1d 3765 . . . . . . . . . 10  l  r  l  +Q  q  <Q  F `  q  r  +Q  q  <Q  F `  q
5857rexbidv 2321 . . . . . . . . 9  l  r  q  Q.  l  +Q  q  <Q  F `  q  q  Q.  r  +Q  q  <Q  F `  q
5958, 10elrab2 2694 . . . . . . . 8  r  1st `  L  r  Q.  q  Q.  r  +Q  q  <Q  F `  q
6043, 55, 59sylanbrc 394 . . . . . . 7  s  1st `  L  q  Q. 
s  +Q  q 
<Q  F `  q 
t  Q.  s  +Q  q  <Q  t  t  <Q  F `  q  r  Q.  q  +Q  r  t  r  1st `  L
6146, 60jca 290 . . . . . 6  s  1st `  L  q  Q. 
s  +Q  q 
<Q  F `  q 
t  Q.  s  +Q  q  <Q  t  t  <Q  F `  q  r  Q.  q  +Q  r  t  s  <Q  r  r  1st `  L
6261ex 108 . . . . 5  s  1st `  L  q  Q.  s  +Q  q  <Q  F `  q  t  Q.  s  +Q  q  <Q  t  t  <Q  F `
 q  r  Q.  q  +Q  r  t  s  <Q  r  r  1st `  L
6362reximdva 2415 . . . 4  s  1st `  L  q  Q.  s  +Q  q  <Q  F `  q  t  Q.  s  +Q  q  <Q  t  t  <Q  F `
 q  r  Q.  q  +Q  r  t  r  Q. 
s  <Q  r  r  1st `  L
6433, 63mpd 13 . . 3  s  1st `  L  q  Q.  s  +Q  q  <Q  F `  q  t  Q.  s  +Q  q  <Q  t  t  <Q  F `
 q  r  Q. 
s  <Q  r  r  1st `  L
6516, 64rexlimddv 2431 . 2  s  1st `  L 
q  Q.  s  +Q  q  <Q  F `  q  r  Q.  s  <Q  r  r  1st `  L
6613, 65rexlimddv 2431 1  s  1st `  L  r  Q.  s  <Q 
r  r  1st `  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   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:  cauappcvgprlemrnd  6622
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