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Theorem caucvgprlemdisj 7233
Description: Lemma for caucvgpr 7241. The putative limit is disjoint. (Contributed by Jim Kingdon, 27-Sep-2020.)
Hypotheses
Ref Expression
caucvgpr.f (𝜑𝐹:NQ)
caucvgpr.cau (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))
caucvgpr.bnd (𝜑 → ∀𝑗N 𝐴 <Q (𝐹𝑗))
caucvgpr.lim 𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩
Assertion
Ref Expression
caucvgprlemdisj (𝜑 → ∀𝑠Q ¬ (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 5659 . . . . . . . . . . . 12 (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )))
21breq1d 3855 . . . . . . . . . . 11 (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
32rexbidv 2381 . . . . . . . . . 10 (𝑙 = 𝑠 → (∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
4 caucvgpr.lim . . . . . . . . . . . 12 𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩
54fveq2i 5308 . . . . . . . . . . 11 (1st𝐿) = (1st ‘⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩)
6 nqex 6922 . . . . . . . . . . . . 13 Q ∈ V
76rabex 3983 . . . . . . . . . . . 12 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)} ∈ V
86rabex 3983 . . . . . . . . . . . 12 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢} ∈ V
97, 8op1st 5917 . . . . . . . . . . 11 (1st ‘⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩) = {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}
105, 9eqtri 2108 . . . . . . . . . 10 (1st𝐿) = {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}
113, 10elrab2 2774 . . . . . . . . 9 (𝑠 ∈ (1st𝐿) ↔ (𝑠Q ∧ ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
1211simprbi 269 . . . . . . . 8 (𝑠 ∈ (1st𝐿) → ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))
13 opeq1 3622 . . . . . . . . . . . . 13 (𝑗 = 𝑘 → ⟨𝑗, 1𝑜⟩ = ⟨𝑘, 1𝑜⟩)
1413eceq1d 6328 . . . . . . . . . . . 12 (𝑗 = 𝑘 → [⟨𝑗, 1𝑜⟩] ~Q = [⟨𝑘, 1𝑜⟩] ~Q )
1514fveq2d 5309 . . . . . . . . . . 11 (𝑗 = 𝑘 → (*Q‘[⟨𝑗, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑘, 1𝑜⟩] ~Q ))
1615oveq2d 5668 . . . . . . . . . 10 (𝑗 = 𝑘 → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑘, 1𝑜⟩] ~Q )))
17 fveq2 5305 . . . . . . . . . 10 (𝑗 = 𝑘 → (𝐹𝑗) = (𝐹𝑘))
1816, 17breq12d 3858 . . . . . . . . 9 (𝑗 = 𝑘 → ((𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑘, 1𝑜⟩] ~Q )) <Q (𝐹𝑘)))
1918cbvrexv 2591 . . . . . . . 8 (∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ ∃𝑘N (𝑠 +Q (*Q‘[⟨𝑘, 1𝑜⟩] ~Q )) <Q (𝐹𝑘))
2012, 19sylib 120 . . . . . . 7 (𝑠 ∈ (1st𝐿) → ∃𝑘N (𝑠 +Q (*Q‘[⟨𝑘, 1𝑜⟩] ~Q )) <Q (𝐹𝑘))
21 breq2 3849 . . . . . . . . . 10 (𝑢 = 𝑠 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢 ↔ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
2221rexbidv 2381 . . . . . . . . 9 (𝑢 = 𝑠 → (∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢 ↔ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
234fveq2i 5308 . . . . . . . . . 10 (2nd𝐿) = (2nd ‘⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩)
247, 8op2nd 5918 . . . . . . . . . 10 (2nd ‘⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}⟩) = {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}
2523, 24eqtri 2108 . . . . . . . . 9 (2nd𝐿) = {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢}
2622, 25elrab2 2774 . . . . . . . 8 (𝑠 ∈ (2nd𝐿) ↔ (𝑠Q ∧ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
2726simprbi 269 . . . . . . 7 (𝑠 ∈ (2nd𝐿) → ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠)
2820, 27anim12i 331 . . . . . 6 ((𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿)) → (∃𝑘N (𝑠 +Q (*Q‘[⟨𝑘, 1𝑜⟩] ~Q )) <Q (𝐹𝑘) ∧ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
29 reeanv 2536 . . . . . 6 (∃𝑘N𝑗N ((𝑠 +Q (*Q‘[⟨𝑘, 1𝑜⟩] ~Q )) <Q (𝐹𝑘) ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠) ↔ (∃𝑘N (𝑠 +Q (*Q‘[⟨𝑘, 1𝑜⟩] ~Q )) <Q (𝐹𝑘) ∧ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
3028, 29sylibr 132 . . . . 5 ((𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿)) → ∃𝑘N𝑗N ((𝑠 +Q (*Q‘[⟨𝑘, 1𝑜⟩] ~Q )) <Q (𝐹𝑘) ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
3130adantl 271 . . . 4 ((𝜑 ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿))) → ∃𝑘N𝑗N ((𝑠 +Q (*Q‘[⟨𝑘, 1𝑜⟩] ~Q )) <Q (𝐹𝑘) ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
32 caucvgpr.f . . . . . . . 8 (𝜑𝐹:NQ)
3332ad2antrr 472 . . . . . . 7 (((𝜑 ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿))) ∧ (𝑘N𝑗N)) → 𝐹:NQ)
34 caucvgpr.cau . . . . . . . 8 (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))
3534ad2antrr 472 . . . . . . 7 (((𝜑 ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿))) ∧ (𝑘N𝑗N)) → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))
36 simprl 498 . . . . . . 7 (((𝜑 ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿))) ∧ (𝑘N𝑗N)) → 𝑘N)
37 simprr 499 . . . . . . 7 (((𝜑 ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿))) ∧ (𝑘N𝑗N)) → 𝑗N)
3811simplbi 268 . . . . . . . . 9 (𝑠 ∈ (1st𝐿) → 𝑠Q)
3938ad2antrl 474 . . . . . . . 8 ((𝜑 ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿))) → 𝑠Q)
4039adantr 270 . . . . . . 7 (((𝜑 ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿))) ∧ (𝑘N𝑗N)) → 𝑠Q)
4133, 35, 36, 37, 40caucvgprlemnkj 7225 . . . . . 6 (((𝜑 ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿))) ∧ (𝑘N𝑗N)) → ¬ ((𝑠 +Q (*Q‘[⟨𝑘, 1𝑜⟩] ~Q )) <Q (𝐹𝑘) ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠))
4241pm2.21d 584 . . . . 5 (((𝜑 ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿))) ∧ (𝑘N𝑗N)) → (((𝑠 +Q (*Q‘[⟨𝑘, 1𝑜⟩] ~Q )) <Q (𝐹𝑘) ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑠) → ⊥))
4342rexlimdvva 2496 . . . 4 ((𝜑 ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿))) → (∃𝑘N𝑗N ((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 1308 . 2 (𝜑 → ¬ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿)))
4645ralrimivw 2447 1 (𝜑 → ∀𝑠Q ¬ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102   = wceq 1289  wfal 1294  wcel 1438  wral 2359  wrex 2360  {crab 2363  cop 3449   class class class wbr 3845  wf 5011  cfv 5015  (class class class)co 5652  1st c1st 5909  2nd c2nd 5910  1𝑜c1o 6174  [cec 6290  Ncnpi 6831   <N clti 6834   ~Q ceq 6838  Qcnq 6839   +Q cplq 6841  *Qcrq 6843   <Q cltq 6844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-eprel 4116  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-1o 6181  df-oadd 6185  df-omul 6186  df-er 6292  df-ec 6294  df-qs 6298  df-ni 6863  df-pli 6864  df-mi 6865  df-lti 6866  df-plpq 6903  df-mpq 6904  df-enq 6906  df-nqqs 6907  df-plqqs 6908  df-mqqs 6909  df-1nqqs 6910  df-rq 6911  df-ltnqqs 6912
This theorem is referenced by:  caucvgprlemcl  7235
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