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Theorem cauappcvgprlemdisj 6623
Description: Lemma for cauappcvgpr 6634. The putative limit is disjoint. (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
cauappcvgprlemdisj  s  Q.  s  1st `  L  s  2nd `  L
Distinct variable groups:   , p    L, p, q   , p, q    L, s   , s, p    F, l,, p, q, s   , s
Allowed substitution hints:   (, l)   (, q, l)    L(, l)

Proof of Theorem cauappcvgprlemdisj
Dummy variables  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cauappcvgpr.app . . . . . . 7  p  Q.  q  Q.  F `  p 
<Q  F `  q  +Q  p  +Q  q  F `  q  <Q  F `
 p  +Q  p  +Q  q
2 simpl 102 . . . . . . . . 9  F `  p  <Q  F `
 q  +Q  p  +Q  q  F `  q  <Q  F `  p  +Q  p  +Q  q  F `
 p  <Q  F `  q  +Q  p  +Q  q
32ralimi 2378 . . . . . . . 8  q  Q.  F `  p 
<Q  F `  q  +Q  p  +Q  q  F `  q  <Q  F `
 p  +Q  p  +Q  q  q  Q.  F `  p  <Q  F `  q  +Q  p  +Q  q
43ralimi 2378 . . . . . . 7  p  Q.  q  Q.  F `
 p  <Q  F `  q  +Q  p  +Q  q  F `  q 
<Q  F `  p  +Q  p  +Q  q  p  Q.  q 
Q.  F `  p  <Q  F `  q  +Q  p  +Q  q
51, 4syl 14 . . . . . 6  p  Q.  q  Q.  F `  p  <Q  F `  q  +Q  p  +Q  q
65adantr 261 . . . . 5  s  1st `  L  s  2nd `  L  p  Q.  q  Q.  F `  p  <Q  F `  q  +Q  p  +Q  q
7 oveq1 5462 . . . . . . . . . . . . 13  l  s 
l  +Q  q  s  +Q  q
87breq1d 3765 . . . . . . . . . . . 12  l  s  l  +Q  q  <Q  F `  q  s  +Q  q  <Q  F `  q
98rexbidv 2321 . . . . . . . . . . 11  l  s  q  Q.  l  +Q  q  <Q  F `  q  q  Q.  s  +Q  q  <Q  F `  q
10 cauappcvgpr.lim . . . . . . . . . . . . 13  L 
<. { l  Q.  |  q  Q.  l  +Q  q  <Q  F `  q } ,  {  Q.  |  q  Q.  F `  q  +Q  q  <Q  } >.
1110fveq2i 5124 . . . . . . . . . . . 12  1st `  L  1st `  <. { l  Q.  |  q  Q. 
l  +Q  q 
<Q  F `  q } ,  {  Q.  |  q  Q.  F `
 q  +Q  q  <Q  } >.
12 nqex 6347 . . . . . . . . . . . . . 14  Q.  _V
1312rabex 3892 . . . . . . . . . . . . 13  { l  Q.  |  q  Q. 
l  +Q  q 
<Q  F `  q }  _V
1412rabex 3892 . . . . . . . . . . . . 13  {  Q.  |  q  Q.  F `
 q  +Q  q  <Q  }  _V
1513, 14op1st 5715 . . . . . . . . . . . 12  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 }
1611, 15eqtri 2057 . . . . . . . . . . 11  1st `  L  {
l  Q.  |  q  Q.  l  +Q  q  <Q  F `  q }
179, 16elrab2 2694 . . . . . . . . . 10  s  1st `  L  s  Q.  q  Q.  s  +Q  q  <Q  F `  q
1817simprbi 260 . . . . . . . . 9  s  1st `  L  q 
Q.  s  +Q  q  <Q  F `  q
19 oveq2 5463 . . . . . . . . . . 11  q  p 
s  +Q  q  s  +Q  p
20 fveq2 5121 . . . . . . . . . . 11  q  p  F `  q  F `  p
2119, 20breq12d 3768 . . . . . . . . . 10  q  p  s  +Q  q  <Q  F `  q  s  +Q  p  <Q  F `  p
2221cbvrexv 2528 . . . . . . . . 9  q  Q. 
s  +Q  q 
<Q  F `  q  p  Q.  s  +Q  p  <Q  F `  p
2318, 22sylib 127 . . . . . . . 8  s  1st `  L  p 
Q.  s  +Q  p  <Q  F `  p
24 breq2 3759 . . . . . . . . . . 11  s  F `  q  +Q  q  <Q  F `  q  +Q  q  <Q  s
2524rexbidv 2321 . . . . . . . . . 10  s  q  Q.  F `  q  +Q  q  <Q  q  Q.  F `  q  +Q  q  <Q  s
2610fveq2i 5124 . . . . . . . . . . 11  2nd `  L  2nd `  <. { l  Q.  |  q  Q. 
l  +Q  q 
<Q  F `  q } ,  {  Q.  |  q  Q.  F `
 q  +Q  q  <Q  } >.
2713, 14op2nd 5716 . . . . . . . . . . 11  2nd `  <. { l 
Q.  |  q  Q.  l  +Q  q  <Q  F `  q } ,  {  Q.  |  q  Q.  F `
 q  +Q  q  <Q  } >.  {  Q.  |  q  Q.  F `
 q  +Q  q  <Q  }
2826, 27eqtri 2057 . . . . . . . . . 10  2nd `  L  {  Q.  |  q  Q.  F `  q  +Q  q  <Q  }
2925, 28elrab2 2694 . . . . . . . . 9  s  2nd `  L  s  Q.  q  Q.  F `  q  +Q  q  <Q  s
3029simprbi 260 . . . . . . . 8  s  2nd `  L  q 
Q.  F `
 q  +Q  q  <Q  s
3123, 30anim12i 321 . . . . . . 7  s  1st `  L  s  2nd `  L  p  Q.  s  +Q  p  <Q  F `  p  q  Q.  F `
 q  +Q  q  <Q  s
32 reeanv 2473 . . . . . . 7  p  Q.  q  Q.  s  +Q  p  <Q  F `  p  F `  q  +Q  q  <Q  s  p  Q. 
s  +Q  p 
<Q  F `  p  q 
Q.  F `
 q  +Q  q  <Q  s
3331, 32sylibr 137 . . . . . 6  s  1st `  L  s  2nd `  L  p  Q.  q 
Q.  s  +Q  p  <Q  F `  p  F `  q  +Q  q  <Q  s
3433adantl 262 . . . . 5  s  1st `  L  s  2nd `  L  p  Q.  q  Q.  s  +Q  p  <Q  F `  p  F `  q  +Q  q  <Q  s
356, 34r19.29d2r 2449 . . . 4  s  1st `  L  s  2nd `  L  p  Q.  q  Q.  F `
 p  <Q  F `  q  +Q  p  +Q  q  s  +Q  p  <Q  F `  p  F `  q  +Q  q  <Q  s
36 simprl 483 . . . . . . . . . . . 12  F `  p  <Q  F `
 q  +Q  p  +Q  q  s  +Q  p  <Q  F `  p  F `  q  +Q  q  <Q  s 
s  +Q  p 
<Q  F `  p
37 simpl 102 . . . . . . . . . . . 12  F `  p  <Q  F `
 q  +Q  p  +Q  q  s  +Q  p  <Q  F `  p  F `  q  +Q  q  <Q  s  F `  p  <Q  F `  q  +Q  p  +Q  q
3836, 37jca 290 . . . . . . . . . . 11  F `  p  <Q  F `
 q  +Q  p  +Q  q  s  +Q  p  <Q  F `  p  F `  q  +Q  q  <Q  s  s  +Q  p  <Q  F `  p  F `  p  <Q  F `  q  +Q  p  +Q  q
3917simplbi 259 . . . . . . . . . . . . . . 15  s  1st `  L  s  Q.
4039adantr 261 . . . . . . . . . . . . . 14  s  1st `  L  s  2nd `  L  s  Q.
4140ad3antlr 462 . . . . . . . . . . . . 13  s  1st `  L  s  2nd `  L  p  Q.  q  Q.  s  Q.
42 simplr 482 . . . . . . . . . . . . 13  s  1st `  L  s  2nd `  L  p  Q.  q  Q.  p  Q.
43 addclnq 6359 . . . . . . . . . . . . 13  s  Q.  p  Q.  s  +Q  p  Q.
4441, 42, 43syl2anc 391 . . . . . . . . . . . 12  s  1st `  L  s  2nd `  L  p  Q.  q  Q.  s  +Q  p  Q.
45 cauappcvgpr.f . . . . . . . . . . . . . 14  F : Q. --> Q.
4645ad3antrrr 461 . . . . . . . . . . . . 13  s  1st `  L  s  2nd `  L  p  Q.  q  Q.  F : Q. --> Q.
4746, 42ffvelrnd 5246 . . . . . . . . . . . 12  s  1st `  L  s  2nd `  L  p  Q.  q  Q.  F `  p  Q.
48 simpr 103 . . . . . . . . . . . . . 14  s  1st `  L  s  2nd `  L  p  Q.  q  Q.  q  Q.
4946, 48ffvelrnd 5246 . . . . . . . . . . . . 13  s  1st `  L  s  2nd `  L  p  Q.  q  Q.  F `  q  Q.
50 addclnq 6359 . . . . . . . . . . . . . 14  p  Q.  q  Q.  p  +Q  q  Q.
5142, 48, 50syl2anc 391 . . . . . . . . . . . . 13  s  1st `  L  s  2nd `  L  p  Q.  q  Q.  p  +Q  q  Q.
52 addclnq 6359 . . . . . . . . . . . . 13  F `  q  Q.  p  +Q  q 
Q.  F `  q  +Q  p  +Q  q 
Q.
5349, 51, 52syl2anc 391 . . . . . . . . . . . 12  s  1st `  L  s  2nd `  L  p  Q.  q  Q.  F `
 q  +Q  p  +Q  q  Q.
54 ltsonq 6382 . . . . . . . . . . . . 13  <Q  Or  Q.
55 sotr 4046 . . . . . . . . . . . . 13 
<Q  Or  Q.  s  +Q  p  Q.  F `  p  Q.  F `  q  +Q  p  +Q  q 
Q.  s  +Q  p  <Q  F `  p  F `  p  <Q  F `
 q  +Q  p  +Q  q 
s  +Q  p 
<Q  F `  q  +Q  p  +Q  q
5654, 55mpan 400 . . . . . . . . . . . 12  s  +Q  p  Q.  F `  p  Q.  F `  q  +Q  p  +Q  q 
Q.  s  +Q  p  <Q  F `  p  F `  p  <Q  F `  q  +Q  p  +Q  q  s  +Q  p  <Q  F `  q  +Q  p  +Q  q
5744, 47, 53, 56syl3anc 1134 . . . . . . . . . . 11  s  1st `  L  s  2nd `  L  p  Q.  q  Q.  s  +Q  p 
<Q  F `  p  F `  p  <Q  F `  q  +Q  p  +Q  q  s  +Q  p  <Q  F `
 q  +Q  p  +Q  q
5838, 57syl5 28 . . . . . . . . . 10  s  1st `  L  s  2nd `  L  p  Q.  q  Q.  F `  p 
<Q  F `  q  +Q  p  +Q  q  s  +Q  p  <Q  F `  p  F `  q  +Q  q  <Q  s  s  +Q  p  <Q  F `
 q  +Q  p  +Q  q
59 simprr 484 . . . . . . . . . . 11  F `  p  <Q  F `
 q  +Q  p  +Q  q  s  +Q  p  <Q  F `  p  F `  q  +Q  q  <Q  s  F `  q  +Q  q  <Q  s
6059a1i 9 . . . . . . . . . 10  s  1st `  L  s  2nd `  L  p  Q.  q  Q.  F `  p 
<Q  F `  q  +Q  p  +Q  q  s  +Q  p  <Q  F `  p  F `  q  +Q  q  <Q  s  F `  q  +Q  q  <Q  s
6158, 60jcad 291 . . . . . . . . 9  s  1st `  L  s  2nd `  L  p  Q.  q  Q.  F `  p 
<Q  F `  q  +Q  p  +Q  q  s  +Q  p  <Q  F `  p  F `  q  +Q  q  <Q  s  s  +Q  p  <Q  F `  q  +Q  p  +Q  q  F `  q  +Q  q  <Q  s
62 addcomnqg 6365 . . . . . . . . . . . 12  s  Q.  p  Q.  s  +Q  p  p  +Q  s
6341, 42, 62syl2anc 391 . . . . . . . . . . 11  s  1st `  L  s  2nd `  L  p  Q.  q  Q.  s  +Q  p  p  +Q  s
64 addcomnqg 6365 . . . . . . . . . . . . 13  Q.  Q.  +Q  +Q
6564adantl 262 . . . . . . . . . . . 12  s  1st `  L  s  2nd `  L  p 
Q.  q  Q.  Q.  Q.  +Q  +Q
66 addassnqg 6366 . . . . . . . . . . . . 13  Q.  Q.  h  Q.  +Q  +Q  h  +Q  +Q  h
6766adantl 262 . . . . . . . . . . . 12  s  1st `  L  s  2nd `  L  p 
Q.  q  Q.  Q.  Q.  h 
Q.  +Q  +Q  h  +Q  +Q  h
6849, 42, 48, 65, 67caov12d 5624 . . . . . . . . . . 11  s  1st `  L  s  2nd `  L  p  Q.  q  Q.  F `
 q  +Q  p  +Q  q  p  +Q  F `
 q  +Q  q
6963, 68breq12d 3768 . . . . . . . . . 10  s  1st `  L  s  2nd `  L  p  Q.  q  Q.  s  +Q  p  <Q  F `  q  +Q  p  +Q  q  p  +Q  s  <Q  p  +Q  F `  q  +Q  q
7069anbi1d 438 . . . . . . . . 9  s  1st `  L  s  2nd `  L  p  Q.  q  Q.  s  +Q  p 
<Q  F `  q  +Q  p  +Q  q  F `  q  +Q  q  <Q  s  p  +Q  s 
<Q  p  +Q  F `  q  +Q  q  F `  q  +Q  q  <Q  s
7161, 70sylibd 138 . . . . . . . 8  s  1st `  L  s  2nd `  L  p  Q.  q  Q.  F `  p 
<Q  F `  q  +Q  p  +Q  q  s  +Q  p  <Q  F `  p  F `  q  +Q  q  <Q  s  p  +Q  s  <Q  p  +Q  F `
 q  +Q  q  F `  q  +Q  q  <Q  s
72 addclnq 6359 . . . . . . . . . . 11  F `  q  Q.  q  Q.  F `  q  +Q  q 
Q.
7349, 48, 72syl2anc 391 . . . . . . . . . 10  s  1st `  L  s  2nd `  L  p  Q.  q  Q.  F `
 q  +Q  q  Q.
74 ltanqg 6384 . . . . . . . . . 10  s  Q.  F `  q  +Q  q 
Q.  p  Q. 
s  <Q  F `
 q  +Q  q  p  +Q  s  <Q  p  +Q  F `
 q  +Q  q
7541, 73, 42, 74syl3anc 1134 . . . . . . . . 9  s  1st `  L  s  2nd `  L  p  Q.  q  Q.  s  <Q  F `  q  +Q  q  p  +Q  s  <Q  p  +Q  F `  q  +Q  q
7675anbi1d 438 . . . . . . . 8  s  1st `  L  s  2nd `  L  p  Q.  q  Q.  s 
<Q  F `  q  +Q  q  F `
 q  +Q  q  <Q  s  p  +Q  s  <Q  p  +Q  F `
 q  +Q  q  F `  q  +Q  q  <Q  s
7771, 76sylibrd 158 . . . . . . 7  s  1st `  L  s  2nd `  L  p  Q.  q  Q.  F `  p 
<Q  F `  q  +Q  p  +Q  q  s  +Q  p  <Q  F `  p  F `  q  +Q  q  <Q  s  s  <Q  F `  q  +Q  q  F `  q  +Q  q  <Q  s
78 so2nr 4049 . . . . . . . . . 10 
<Q  Or  Q. 
s  Q.  F `  q  +Q  q 
Q.  s  <Q  F `  q  +Q  q  F `  q  +Q  q  <Q  s
7954, 78mpan 400 . . . . . . . . 9  s  Q.  F `  q  +Q  q 
Q.  s  <Q  F `  q  +Q  q  F `  q  +Q  q  <Q  s
8041, 73, 79syl2anc 391 . . . . . . . 8  s  1st `  L  s  2nd `  L  p  Q.  q  Q.  s 
<Q  F `  q  +Q  q  F `
 q  +Q  q  <Q  s
8180pm2.21d 549 . . . . . . 7  s  1st `  L  s  2nd `  L  p  Q.  q  Q.  s 
<Q  F `  q  +Q  q  F `
 q  +Q  q  <Q  s
8277, 81syld 40 . . . . . 6  s  1st `  L  s  2nd `  L  p  Q.  q  Q.  F `  p 
<Q  F `  q  +Q  p  +Q  q  s  +Q  p  <Q  F `  p  F `  q  +Q  q  <Q  s
8382rexlimdva 2427 . . . . 5 
s  1st `  L  s  2nd `  L  p  Q.  q  Q.  F `  p  <Q  F `
 q  +Q  p  +Q  q  s  +Q  p  <Q  F `  p  F `  q  +Q  q  <Q  s
8483rexlimdva 2427 . . . 4  s  1st `  L  s  2nd `  L  p  Q.  q  Q.  F `  p 
<Q  F `  q  +Q  p  +Q  q  s  +Q  p  <Q  F `  p  F `  q  +Q  q  <Q  s
8535, 84mpd 13 . . 3  s  1st `  L  s  2nd `  L
8685inegd 1262 . 2  s  1st `  L  s  2nd `  L
8786ralrimivw 2387 1  s  Q.  s  1st `  L  s  2nd `  L
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97   wb 98   w3a 884   wceq 1242   wfal 1247   wcel 1390  wral 2300  wrex 2301   {crab 2304   <.cop 3370   class class class wbr 3755    Or wor 4023   -->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-bnd 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-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-enq 6331  df-nqqs 6332  df-plqqs 6333  df-ltnqqs 6337
This theorem is referenced by:  cauappcvgprlemcl  6625
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