ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  caucvgprprlemdisj GIF version

Theorem caucvgprprlemdisj 7251
Description: Lemma for caucvgprpr 7261. The putative limit is disjoint. (Contributed by Jim Kingdon, 21-Dec-2020.)
Hypotheses
Ref Expression
caucvgprpr.f (𝜑𝐹:NP)
caucvgprpr.cau (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))
caucvgprpr.bnd (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))
caucvgprpr.lim 𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩
Assertion
Ref Expression
caucvgprprlemdisj (𝜑 → ∀𝑠Q ¬ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿)))
Distinct variable groups:   𝐴,𝑚   𝑚,𝐹   𝑘,𝐹,𝑛,𝑙   𝐹,𝑟,𝑙   𝑢,𝐹,𝑟   𝑘,𝐿   𝑘,𝑝,𝑟,𝑠   𝜑,𝑟,𝑠   𝑘,𝑞,𝑟,𝑠   𝑝,𝑙,𝑠,𝑞   𝑢,𝑝,𝑠,𝑞   𝑢,𝑛   𝑛,𝑙,𝑘   𝑢,𝑘
Allowed substitution hints:   𝜑(𝑢,𝑘,𝑚,𝑛,𝑞,𝑝,𝑙)   𝐴(𝑢,𝑘,𝑛,𝑠,𝑟,𝑞,𝑝,𝑙)   𝐹(𝑠,𝑞,𝑝)   𝐿(𝑢,𝑚,𝑛,𝑠,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem caucvgprprlemdisj
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 caucvgprpr.lim . . . . . . . . 9 𝐿 = ⟨{𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)}, {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩}⟩
21caucvgprprlemell 7234 . . . . . . . 8 (𝑠 ∈ (1st𝐿) ↔ (𝑠Q ∧ ∃𝑎N ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑎)))
32simprbi 269 . . . . . . 7 (𝑠 ∈ (1st𝐿) → ∃𝑎N ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑎))
41caucvgprprlemelu 7235 . . . . . . . 8 (𝑠 ∈ (2nd𝐿) ↔ (𝑠Q ∧ ∃𝑏N ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩))
54simprbi 269 . . . . . . 7 (𝑠 ∈ (2nd𝐿) → ∃𝑏N ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩)
63, 5anim12i 331 . . . . . 6 ((𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿)) → (∃𝑎N ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑎) ∧ ∃𝑏N ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩))
7 reeanv 2536 . . . . . 6 (∃𝑎N𝑏N (⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑎) ∧ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩) ↔ (∃𝑎N ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑎) ∧ ∃𝑏N ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩))
86, 7sylibr 132 . . . . 5 ((𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿)) → ∃𝑎N𝑏N (⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑎) ∧ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩))
98adantl 271 . . . 4 ((𝜑 ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿))) → ∃𝑎N𝑏N (⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑎) ∧ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩))
10 caucvgprpr.f . . . . . . . 8 (𝜑𝐹:NP)
1110ad2antrr 472 . . . . . . 7 (((𝜑 ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿))) ∧ (𝑎N𝑏N)) → 𝐹:NP)
12 caucvgprpr.cau . . . . . . . 8 (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))
1312ad2antrr 472 . . . . . . 7 (((𝜑 ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿))) ∧ (𝑎N𝑏N)) → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))
14 simprl 498 . . . . . . 7 (((𝜑 ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿))) ∧ (𝑎N𝑏N)) → 𝑎N)
15 simprr 499 . . . . . . 7 (((𝜑 ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿))) ∧ (𝑎N𝑏N)) → 𝑏N)
161caucvgprprlemell 7234 . . . . . . . . . 10 (𝑠 ∈ (1st𝐿) ↔ (𝑠Q ∧ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)))
1716simplbi 268 . . . . . . . . 9 (𝑠 ∈ (1st𝐿) → 𝑠Q)
1817ad2antrl 474 . . . . . . . 8 ((𝜑 ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿))) → 𝑠Q)
1918adantr 270 . . . . . . 7 (((𝜑 ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿))) ∧ (𝑎N𝑏N)) → 𝑠Q)
2011, 13, 14, 15, 19caucvgprprlemnkj 7241 . . . . . 6 (((𝜑 ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿))) ∧ (𝑎N𝑏N)) → ¬ (⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑎) ∧ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩))
2120pm2.21d 584 . . . . 5 (((𝜑 ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿))) ∧ (𝑎N𝑏N)) → ((⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑎) ∧ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩) → ⊥))
2221rexlimdvva 2496 . . . 4 ((𝜑 ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿))) → (∃𝑎N𝑏N (⟨{𝑝𝑝 <Q (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑠 +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑎) ∧ ((𝐹𝑏) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑏, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑏, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑠}, {𝑞𝑠 <Q 𝑞}⟩) → ⊥))
239, 22mpd 13 . . 3 ((𝜑 ∧ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿))) → ⊥)
2423inegd 1308 . 2 (𝜑 → ¬ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿)))
2524ralrimivw 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  {cab 2074  wral 2359  wrex 2360  {crab 2363  cop 3447   class class class wbr 3843  wf 5006  cfv 5010  (class class class)co 5644  1st c1st 5901  2nd c2nd 5902  1𝑜c1o 6166  [cec 6280  Ncnpi 6821   <N clti 6824   ~Q ceq 6828  Qcnq 6829   +Q cplq 6831  *Qcrq 6833   <Q cltq 6834  Pcnp 6840   +P cpp 6842  <P cltp 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 3952  ax-sep 3955  ax-nul 3963  ax-pow 4007  ax-pr 4034  ax-un 4258  ax-setind 4351  ax-iinf 4401
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 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-int 3687  df-iun 3730  df-br 3844  df-opab 3898  df-mpt 3899  df-tr 3935  df-eprel 4114  df-id 4118  df-po 4121  df-iso 4122  df-iord 4191  df-on 4193  df-suc 4196  df-iom 4404  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-ima 4449  df-iota 4975  df-fun 5012  df-fn 5013  df-f 5014  df-f1 5015  df-fo 5016  df-f1o 5017  df-fv 5018  df-ov 5647  df-oprab 5648  df-mpt2 5649  df-1st 5903  df-2nd 5904  df-recs 6062  df-irdg 6127  df-1o 6173  df-2o 6174  df-oadd 6177  df-omul 6178  df-er 6282  df-ec 6284  df-qs 6288  df-ni 6853  df-pli 6854  df-mi 6855  df-lti 6856  df-plpq 6893  df-mpq 6894  df-enq 6896  df-nqqs 6897  df-plqqs 6898  df-mqqs 6899  df-1nqqs 6900  df-rq 6901  df-ltnqqs 6902  df-enq0 6973  df-nq0 6974  df-0nq0 6975  df-plq0 6976  df-mq0 6977  df-inp 7015  df-iplp 7017  df-iltp 7019
This theorem is referenced by:  caucvgprprlemcl  7253
  Copyright terms: Public domain W3C validator