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Theorem caucvgprlemdisj 7636
Description: Lemma for caucvgpr 7644. 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 5860 . . . . . . . . . . . 12 (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
21breq1d 3999 . . . . . . . . . . 11 (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
32rexbidv 2471 . . . . . . . . . 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 5499 . . . . . . . . . . 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 7325 . . . . . . . . . . . . 13 Q ∈ V
76rabex 4133 . . . . . . . . . . . 12 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)} ∈ V
86rabex 4133 . . . . . . . . . . . 12 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ V
97, 8op1st 6125 . . . . . . . . . . 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 2191 . . . . . . . . . 10 (1st𝐿) = {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}
113, 10elrab2 2889 . . . . . . . . 9 (𝑠 ∈ (1st𝐿) ↔ (𝑠Q ∧ ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
1211simprbi 273 . . . . . . . 8 (𝑠 ∈ (1st𝐿) → ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗))
13 opeq1 3765 . . . . . . . . . . . . 13 (𝑗 = 𝑘 → ⟨𝑗, 1o⟩ = ⟨𝑘, 1o⟩)
1413eceq1d 6549 . . . . . . . . . . . 12 (𝑗 = 𝑘 → [⟨𝑗, 1o⟩] ~Q = [⟨𝑘, 1o⟩] ~Q )
1514fveq2d 5500 . . . . . . . . . . 11 (𝑗 = 𝑘 → (*Q‘[⟨𝑗, 1o⟩] ~Q ) = (*Q‘[⟨𝑘, 1o⟩] ~Q ))
1615oveq2d 5869 . . . . . . . . . 10 (𝑗 = 𝑘 → (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑘, 1o⟩] ~Q )))
17 fveq2 5496 . . . . . . . . . 10 (𝑗 = 𝑘 → (𝐹𝑗) = (𝐹𝑘))
1816, 17breq12d 4002 . . . . . . . . 9 (𝑗 = 𝑘 → ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑘, 1o⟩] ~Q )) <Q (𝐹𝑘)))
1918cbvrexv 2697 . . . . . . . 8 (∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) ↔ ∃𝑘N (𝑠 +Q (*Q‘[⟨𝑘, 1o⟩] ~Q )) <Q (𝐹𝑘))
2012, 19sylib 121 . . . . . . 7 (𝑠 ∈ (1st𝐿) → ∃𝑘N (𝑠 +Q (*Q‘[⟨𝑘, 1o⟩] ~Q )) <Q (𝐹𝑘))
21 breq2 3993 . . . . . . . . . 10 (𝑢 = 𝑠 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
2221rexbidv 2471 . . . . . . . . 9 (𝑢 = 𝑠 → (∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
234fveq2i 5499 . . . . . . . . . 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 6126 . . . . . . . . . 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 2191 . . . . . . . . 9 (2nd𝐿) = {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}
2622, 25elrab2 2889 . . . . . . . 8 (𝑠 ∈ (2nd𝐿) ↔ (𝑠Q ∧ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
2726simprbi 273 . . . . . . 7 (𝑠 ∈ (2nd𝐿) → ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠)
2820, 27anim12i 336 . . . . . 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 2639 . . . . . 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 133 . . . . 5 ((𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿)) → ∃𝑘N𝑗N ((𝑠 +Q (*Q‘[⟨𝑘, 1o⟩] ~Q )) <Q (𝐹𝑘) ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
3130adantl 275 . . . 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 485 . . . . . . 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 485 . . . . . . 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 526 . . . . . . 7 (((𝜑 ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿))) ∧ (𝑘N𝑗N)) → 𝑘N)
37 simprr 527 . . . . . . 7 (((𝜑 ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿))) ∧ (𝑘N𝑗N)) → 𝑗N)
3811simplbi 272 . . . . . . . . 9 (𝑠 ∈ (1st𝐿) → 𝑠Q)
3938ad2antrl 487 . . . . . . . 8 ((𝜑 ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿))) → 𝑠Q)
4039adantr 274 . . . . . . 7 (((𝜑 ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿))) ∧ (𝑘N𝑗N)) → 𝑠Q)
4133, 35, 36, 37, 40caucvgprlemnkj 7628 . . . . . 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 614 . . . . 5 (((𝜑 ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿))) ∧ (𝑘N𝑗N)) → (((𝑠 +Q (*Q‘[⟨𝑘, 1o⟩] ~Q )) <Q (𝐹𝑘) ∧ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠) → ⊥))
4342rexlimdvva 2595 . . . 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 1367 . 2 (𝜑 → ¬ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿)))
4645ralrimivw 2544 1 (𝜑 → ∀𝑠Q ¬ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103   = wceq 1348  wfal 1353  wcel 2141  wral 2448  wrex 2449  {crab 2452  cop 3586   class class class wbr 3989  wf 5194  cfv 5198  (class class class)co 5853  1st c1st 6117  2nd c2nd 6118  1oc1o 6388  [cec 6511  Ncnpi 7234   <N clti 7237   ~Q ceq 7241  Qcnq 7242   +Q cplq 7244  *Qcrq 7246   <Q cltq 7247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315
This theorem is referenced by:  caucvgprlemcl  7638
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