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Theorem caucvgprprlemcl 6859
Description: Lemma for caucvgprpr 6867. The putative limit is a positive real. (Contributed by Jim Kingdon, 21-Nov-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
caucvgprprlemcl (𝜑𝐿P)
Distinct variable groups:   𝐴,𝑚   𝑚,𝐹   𝐴,𝑟   𝐹,𝑙,𝑢,𝑟,𝑘   𝑛,𝐹,𝑘   𝑘,𝐿   𝜑,𝑟   𝑢,𝑙,𝑝,𝑞,𝑟   𝑚,𝑟   𝑘,𝑝,𝑞,𝑟   𝑢,𝑛,𝑙,𝑘
Allowed substitution hints:   𝜑(𝑢,𝑘,𝑚,𝑛,𝑞,𝑝,𝑙)   𝐴(𝑢,𝑘,𝑛,𝑞,𝑝,𝑙)   𝐹(𝑞,𝑝)   𝐿(𝑢,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem caucvgprprlemcl
Dummy variables 𝑠 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 caucvgprpr.f . . . 4 (𝜑𝐹:NP)
2 caucvgprpr.cau . . . 4 (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}⟩))))
3 caucvgprpr.bnd . . . 4 (𝜑 → ∀𝑚N 𝐴<P (𝐹𝑚))
4 caucvgprpr.lim . . . 4 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}⟩}⟩
51, 2, 3, 4caucvgprprlemm 6851 . . 3 (𝜑 → (∃𝑠Q 𝑠 ∈ (1st𝐿) ∧ ∃𝑡Q 𝑡 ∈ (2nd𝐿)))
6 ssrab2 3052 . . . . . 6 {𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)} ⊆ Q
7 nqex 6518 . . . . . . 7 Q ∈ V
87elpw2 3938 . . . . . 6 ({𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)} ∈ 𝒫 Q ↔ {𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)} ⊆ Q)
96, 8mpbir 138 . . . . 5 {𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)} ∈ 𝒫 Q
10 ssrab2 3052 . . . . . 6 {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩} ⊆ Q
117elpw2 3938 . . . . . 6 ({𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩} ∈ 𝒫 Q ↔ {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩} ⊆ Q)
1210, 11mpbir 138 . . . . 5 {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩} ∈ 𝒫 Q
13 opelxpi 4403 . . . . 5 (({𝑙Q ∣ ∃𝑟N ⟨{𝑝𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝑟)} ∈ 𝒫 Q ∧ {𝑢Q ∣ ∃𝑟N ((𝐹𝑟) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝑟, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑢}, {𝑞𝑢 <Q 𝑞}⟩} ∈ 𝒫 Q) → ⟨{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}⟩}⟩ ∈ (𝒫 Q × 𝒫 Q))
149, 12, 13mp2an 410 . . . 4 ⟨{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}⟩}⟩ ∈ (𝒫 Q × 𝒫 Q)
154, 14eqeltri 2126 . . 3 𝐿 ∈ (𝒫 Q × 𝒫 Q)
165, 15jctil 299 . 2 (𝜑 → (𝐿 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑠Q 𝑠 ∈ (1st𝐿) ∧ ∃𝑡Q 𝑡 ∈ (2nd𝐿))))
171, 2, 3, 4caucvgprprlemrnd 6856 . . 3 (𝜑 → (∀𝑠Q (𝑠 ∈ (1st𝐿) ↔ ∃𝑡Q (𝑠 <Q 𝑡𝑡 ∈ (1st𝐿))) ∧ ∀𝑡Q (𝑡 ∈ (2nd𝐿) ↔ ∃𝑠Q (𝑠 <Q 𝑡𝑠 ∈ (2nd𝐿)))))
181, 2, 3, 4caucvgprprlemdisj 6857 . . 3 (𝜑 → ∀𝑠Q ¬ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿)))
191, 2, 3, 4caucvgprprlemloc 6858 . . 3 (𝜑 → ∀𝑠Q𝑡Q (𝑠 <Q 𝑡 → (𝑠 ∈ (1st𝐿) ∨ 𝑡 ∈ (2nd𝐿))))
2017, 18, 193jca 1095 . 2 (𝜑 → ((∀𝑠Q (𝑠 ∈ (1st𝐿) ↔ ∃𝑡Q (𝑠 <Q 𝑡𝑡 ∈ (1st𝐿))) ∧ ∀𝑡Q (𝑡 ∈ (2nd𝐿) ↔ ∃𝑠Q (𝑠 <Q 𝑡𝑠 ∈ (2nd𝐿)))) ∧ ∀𝑠Q ¬ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿)) ∧ ∀𝑠Q𝑡Q (𝑠 <Q 𝑡 → (𝑠 ∈ (1st𝐿) ∨ 𝑡 ∈ (2nd𝐿)))))
21 elnp1st2nd 6631 . 2 (𝐿P ↔ ((𝐿 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑠Q 𝑠 ∈ (1st𝐿) ∧ ∃𝑡Q 𝑡 ∈ (2nd𝐿))) ∧ ((∀𝑠Q (𝑠 ∈ (1st𝐿) ↔ ∃𝑡Q (𝑠 <Q 𝑡𝑡 ∈ (1st𝐿))) ∧ ∀𝑡Q (𝑡 ∈ (2nd𝐿) ↔ ∃𝑠Q (𝑠 <Q 𝑡𝑠 ∈ (2nd𝐿)))) ∧ ∀𝑠Q ¬ (𝑠 ∈ (1st𝐿) ∧ 𝑠 ∈ (2nd𝐿)) ∧ ∀𝑠Q𝑡Q (𝑠 <Q 𝑡 → (𝑠 ∈ (1st𝐿) ∨ 𝑡 ∈ (2nd𝐿))))))
2216, 20, 21sylanbrc 402 1 (𝜑𝐿P)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102  wo 639  w3a 896   = wceq 1259  wcel 1409  {cab 2042  wral 2323  wrex 2324  {crab 2327  wss 2944  𝒫 cpw 3386  cop 3405   class class class wbr 3791   × cxp 4370  wf 4925  cfv 4929  (class class class)co 5539  1st c1st 5792  2nd c2nd 5793  1𝑜c1o 6024  [cec 6134  Ncnpi 6427   <N clti 6430   ~Q ceq 6434  Qcnq 6435   +Q cplq 6437  *Qcrq 6439   <Q cltq 6440  Pcnp 6446   +P cpp 6448  <P cltp 6450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-po 4060  df-iso 4061  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-2o 6032  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-pli 6460  df-mi 6461  df-lti 6462  df-plpq 6499  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-plqqs 6504  df-mqqs 6505  df-1nqqs 6506  df-rq 6507  df-ltnqqs 6508  df-enq0 6579  df-nq0 6580  df-0nq0 6581  df-plq0 6582  df-mq0 6583  df-inp 6621  df-iplp 6623  df-iltp 6625
This theorem is referenced by:  caucvgprprlemclphr  6860  caucvgprprlemaddq  6863  caucvgprprlem2  6865  caucvgprpr  6867
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