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Theorem caucvgprlemdisj 6645
Description: Lemma for caucvgpr 6653. The putative limit is disjoint. (Contributed by Jim Kingdon, 27-Sep-2020.)
Hypotheses
Ref Expression
caucvgpr.f (φ𝐹:NQ)
caucvgpr.cau (φ𝑛 N 𝑘 N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))
caucvgpr.bnd (φ𝑗 N A <Q (𝐹𝑗))
caucvgpr.lim 𝐿 = ⟨{𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}⟩
Assertion
Ref Expression
caucvgprlemdisj (φ𝑠 Q ¬ (𝑠 (1st𝐿) 𝑠 (2nd𝐿)))
Distinct variable groups:   A,𝑗   𝑗,𝐹,𝑘   𝐹,𝑙,𝑗   u,𝐹,𝑗   𝑛,𝐹   𝑗,𝐿,𝑘   φ,𝑗,𝑠,𝑘   𝑠,𝑙   u,𝑠   𝑘,𝑛
Allowed substitution hints:   φ(u,𝑛,𝑙)   A(u,𝑘,𝑛,𝑠,𝑙)   𝐹(𝑠)   𝐿(u,𝑛,𝑠,𝑙)

Proof of Theorem caucvgprlemdisj
StepHypRef Expression
1 oveq1 5462 . . . . . . . . . . . 12 (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )))
21breq1d 3765 . . . . . . . . . . 11 (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
32rexbidv 2321 . . . . . . . . . 10 (𝑙 = 𝑠 → (𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ 𝑗 N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
4 caucvgpr.lim . . . . . . . . . . . 12 𝐿 = ⟨{𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}⟩
54fveq2i 5124 . . . . . . . . . . 11 (1st𝐿) = (1st ‘⟨{𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}⟩)
6 nqex 6347 . . . . . . . . . . . . 13 Q V
76rabex 3892 . . . . . . . . . . . 12 {𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)} V
86rabex 3892 . . . . . . . . . . . 12 {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u} V
97, 8op1st 5715 . . . . . . . . . . 11 (1st ‘⟨{𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}⟩) = {𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}
105, 9eqtri 2057 . . . . . . . . . 10 (1st𝐿) = {𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}
113, 10elrab2 2694 . . . . . . . . 9 (𝑠 (1st𝐿) ↔ (𝑠 Q 𝑗 N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
1211simprbi 260 . . . . . . . 8 (𝑠 (1st𝐿) → 𝑗 N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))
13 opeq1 3540 . . . . . . . . . . . . 13 (𝑗 = 𝑘 → ⟨𝑗, 1𝑜⟩ = ⟨𝑘, 1𝑜⟩)
1413eceq1d 6078 . . . . . . . . . . . 12 (𝑗 = 𝑘 → [⟨𝑗, 1𝑜⟩] ~Q = [⟨𝑘, 1𝑜⟩] ~Q )
1514fveq2d 5125 . . . . . . . . . . 11 (𝑗 = 𝑘 → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑘, 1𝑜⟩] ~Q ))
1615oveq2d 5471 . . . . . . . . . 10 (𝑗 = 𝑘 → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑘, 1𝑜⟩] ~Q )))
17 fveq2 5121 . . . . . . . . . 10 (𝑗 = 𝑘 → (𝐹𝑗) = (𝐹𝑘))
1816, 17breq12d 3768 . . . . . . . . 9 (𝑗 = 𝑘 → ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑘, 1𝑜⟩] ~Q )) <Q (𝐹𝑘)))
1918cbvrexv 2528 . . . . . . . 8 (𝑗 N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ 𝑘 N (𝑠 +Q (*Q‘[⟨𝑘, 1𝑜⟩] ~Q )) <Q (𝐹𝑘))
2012, 19sylib 127 . . . . . . 7 (𝑠 (1st𝐿) → 𝑘 N (𝑠 +Q (*Q‘[⟨𝑘, 1𝑜⟩] ~Q )) <Q (𝐹𝑘))
21 breq2 3759 . . . . . . . . . 10 (u = 𝑠 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u ↔ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
2221rexbidv 2321 . . . . . . . . 9 (u = 𝑠 → (𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
234fveq2i 5124 . . . . . . . . . 10 (2nd𝐿) = (2nd ‘⟨{𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}⟩)
247, 8op2nd 5716 . . . . . . . . . 10 (2nd ‘⟨{𝑙 Q𝑗 N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}⟩) = {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}
2523, 24eqtri 2057 . . . . . . . . 9 (2nd𝐿) = {u Q𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q u}
2622, 25elrab2 2694 . . . . . . . 8 (𝑠 (2nd𝐿) ↔ (𝑠 Q 𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
2726simprbi 260 . . . . . . 7 (𝑠 (2nd𝐿) → 𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠)
2820, 27anim12i 321 . . . . . 6 ((𝑠 (1st𝐿) 𝑠 (2nd𝐿)) → (𝑘 N (𝑠 +Q (*Q‘[⟨𝑘, 1𝑜⟩] ~Q )) <Q (𝐹𝑘) 𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
29 reeanv 2473 . . . . . 6 (𝑘 N 𝑗 N ((𝑠 +Q (*Q‘[⟨𝑘, 1𝑜⟩] ~Q )) <Q (𝐹𝑘) ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠) ↔ (𝑘 N (𝑠 +Q (*Q‘[⟨𝑘, 1𝑜⟩] ~Q )) <Q (𝐹𝑘) 𝑗 N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
3028, 29sylibr 137 . . . . 5 ((𝑠 (1st𝐿) 𝑠 (2nd𝐿)) → 𝑘 N 𝑗 N ((𝑠 +Q (*Q‘[⟨𝑘, 1𝑜⟩] ~Q )) <Q (𝐹𝑘) ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
3130adantl 262 . . . 4 ((φ (𝑠 (1st𝐿) 𝑠 (2nd𝐿))) → 𝑘 N 𝑗 N ((𝑠 +Q (*Q‘[⟨𝑘, 1𝑜⟩] ~Q )) <Q (𝐹𝑘) ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
32 caucvgpr.f . . . . . . . 8 (φ𝐹:NQ)
3332ad2antrr 457 . . . . . . 7 (((φ (𝑠 (1st𝐿) 𝑠 (2nd𝐿))) (𝑘 N 𝑗 N)) → 𝐹:NQ)
34 caucvgpr.cau . . . . . . . 8 (φ𝑛 N 𝑘 N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))
3534ad2antrr 457 . . . . . . 7 (((φ (𝑠 (1st𝐿) 𝑠 (2nd𝐿))) (𝑘 N 𝑗 N)) → 𝑛 N 𝑘 N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))
36 simprl 483 . . . . . . 7 (((φ (𝑠 (1st𝐿) 𝑠 (2nd𝐿))) (𝑘 N 𝑗 N)) → 𝑘 N)
37 simprr 484 . . . . . . 7 (((φ (𝑠 (1st𝐿) 𝑠 (2nd𝐿))) (𝑘 N 𝑗 N)) → 𝑗 N)
3811simplbi 259 . . . . . . . . 9 (𝑠 (1st𝐿) → 𝑠 Q)
3938ad2antrl 459 . . . . . . . 8 ((φ (𝑠 (1st𝐿) 𝑠 (2nd𝐿))) → 𝑠 Q)
4039adantr 261 . . . . . . 7 (((φ (𝑠 (1st𝐿) 𝑠 (2nd𝐿))) (𝑘 N 𝑗 N)) → 𝑠 Q)
4133, 35, 36, 37, 40caucvgprlemnkj 6637 . . . . . 6 (((φ (𝑠 (1st𝐿) 𝑠 (2nd𝐿))) (𝑘 N 𝑗 N)) → ¬ ((𝑠 +Q (*Q‘[⟨𝑘, 1𝑜⟩] ~Q )) <Q (𝐹𝑘) ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
4241pm2.21d 549 . . . . 5 (((φ (𝑠 (1st𝐿) 𝑠 (2nd𝐿))) (𝑘 N 𝑗 N)) → (((𝑠 +Q (*Q‘[⟨𝑘, 1𝑜⟩] ~Q )) <Q (𝐹𝑘) ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠) → ⊥ ))
4342rexlimdvva 2434 . . . 4 ((φ (𝑠 (1st𝐿) 𝑠 (2nd𝐿))) → (𝑘 N 𝑗 N ((𝑠 +Q (*Q‘[⟨𝑘, 1𝑜⟩] ~Q )) <Q (𝐹𝑘) ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠) → ⊥ ))
4431, 43mpd 13 . . 3 ((φ (𝑠 (1st𝐿) 𝑠 (2nd𝐿))) → ⊥ )
4544inegd 1262 . 2 (φ → ¬ (𝑠 (1st𝐿) 𝑠 (2nd𝐿)))
4645ralrimivw 2387 1 (φ𝑠 Q ¬ (𝑠 (1st𝐿) 𝑠 (2nd𝐿)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   = wceq 1242  wfal 1247   wcel 1390  wral 2300  wrex 2301  {crab 2304  cop 3370   class class class wbr 3755  wf 4841  cfv 4845  (class class class)co 5455  1st c1st 5707  2nd c2nd 5708  1𝑜c1o 5933  [cec 6040  Ncnpi 6256   <N clti 6259   ~Q ceq 6263  Qcnq 6264   +Q cplq 6266  *Qcrq 6268   <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:  caucvgprlemcl  6647
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