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Theorem caucvgprlemlol 7326
Description: Lemma for caucvgpr 7338. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 20-Oct-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
caucvgprlemlol ((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) → 𝑠 ∈ (1st𝐿))
Distinct variable groups:   𝐴,𝑗   𝐹,𝑙,𝑟,𝑠   𝑢,𝐹   𝑗,𝐿,𝑟,𝑠   𝑗,𝑙,𝑠   𝜑,𝑗,𝑟,𝑠   𝑢,𝑗,𝑟,𝑠
Allowed substitution hints:   𝜑(𝑢,𝑘,𝑛,𝑙)   𝐴(𝑢,𝑘,𝑛,𝑠,𝑟,𝑙)   𝐹(𝑗,𝑘,𝑛)   𝐿(𝑢,𝑘,𝑛,𝑙)

Proof of Theorem caucvgprlemlol
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelnq 7021 . . . . 5 <Q ⊆ (Q × Q)
21brel 4519 . . . 4 (𝑠 <Q 𝑟 → (𝑠Q𝑟Q))
32simpld 111 . . 3 (𝑠 <Q 𝑟𝑠Q)
433ad2ant2 968 . 2 ((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) → 𝑠Q)
5 oveq1 5697 . . . . . . . 8 (𝑙 = 𝑟 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
65breq1d 3877 . . . . . . 7 (𝑙 = 𝑟 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
76rexbidv 2392 . . . . . 6 (𝑙 = 𝑟 → (∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) ↔ ∃𝑗N (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
8 caucvgpr.lim . . . . . . . 8 𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩
98fveq2i 5343 . . . . . . 7 (1st𝐿) = (1st ‘⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩)
10 nqex 7019 . . . . . . . . 9 Q ∈ V
1110rabex 4004 . . . . . . . 8 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)} ∈ V
1210rabex 4004 . . . . . . . 8 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ V
1311, 12op1st 5955 . . . . . . 7 (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)}
149, 13eqtri 2115 . . . . . 6 (1st𝐿) = {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}
157, 14elrab2 2788 . . . . 5 (𝑟 ∈ (1st𝐿) ↔ (𝑟Q ∧ ∃𝑗N (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
1615simprbi 270 . . . 4 (𝑟 ∈ (1st𝐿) → ∃𝑗N (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗))
17163ad2ant3 969 . . 3 ((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) → ∃𝑗N (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗))
18 simpll2 986 . . . . . . 7 ((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑗N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)) → 𝑠 <Q 𝑟)
19 ltanqg 7056 . . . . . . . . 9 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
2019adantl 272 . . . . . . . 8 (((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑗N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)) ∧ (𝑓Q𝑔QQ)) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
214ad2antrr 473 . . . . . . . 8 ((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑗N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)) → 𝑠Q)
222simprd 113 . . . . . . . . . 10 (𝑠 <Q 𝑟𝑟Q)
23223ad2ant2 968 . . . . . . . . 9 ((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) → 𝑟Q)
2423ad2antrr 473 . . . . . . . 8 ((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑗N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)) → 𝑟Q)
25 simplr 498 . . . . . . . . 9 ((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑗N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)) → 𝑗N)
26 nnnq 7078 . . . . . . . . 9 (𝑗N → [⟨𝑗, 1o⟩] ~QQ)
27 recclnq 7048 . . . . . . . . 9 ([⟨𝑗, 1o⟩] ~QQ → (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q)
2825, 26, 273syl 17 . . . . . . . 8 ((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑗N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)) → (*Q‘[⟨𝑗, 1o⟩] ~Q ) ∈ Q)
29 addcomnqg 7037 . . . . . . . . 9 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
3029adantl 272 . . . . . . . 8 (((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑗N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)) ∧ (𝑓Q𝑔Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
3120, 21, 24, 28, 30caovord2d 5852 . . . . . . 7 ((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑗N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)) → (𝑠 <Q 𝑟 ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q ))))
3218, 31mpbid 146 . . . . . 6 ((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑗N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)) → (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
33 ltsonq 7054 . . . . . . 7 <Q Or Q
3433, 1sotri 4860 . . . . . 6 (((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)) → (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗))
3532, 34sylancom 412 . . . . 5 ((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑗N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)) → (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗))
3635ex 114 . . . 4 (((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑗N) → ((𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) → (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
3736reximdva 2487 . . 3 ((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) → (∃𝑗N (𝑟 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) → ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
3817, 37mpd 13 . 2 ((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) → ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗))
39 oveq1 5697 . . . . 5 (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
4039breq1d 3877 . . . 4 (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
4140rexbidv 2392 . . 3 (𝑙 = 𝑠 → (∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗) ↔ ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
4241, 14elrab2 2788 . 2 (𝑠 ∈ (1st𝐿) ↔ (𝑠Q ∧ ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)))
434, 38, 42sylanbrc 409 1 ((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) → 𝑠 ∈ (1st𝐿))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 927   = wceq 1296  wcel 1445  wral 2370  wrex 2371  {crab 2374  cop 3469   class class class wbr 3867  wf 5045  cfv 5049  (class class class)co 5690  1st c1st 5947  1oc1o 6212  [cec 6330  Ncnpi 6928   <N clti 6931   ~Q ceq 6935  Qcnq 6936   +Q cplq 6938  *Qcrq 6940   <Q cltq 6941
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431
This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-eprel 4140  df-id 4144  df-po 4147  df-iso 4148  df-iord 4217  df-on 4219  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-irdg 6173  df-1o 6219  df-oadd 6223  df-omul 6224  df-er 6332  df-ec 6334  df-qs 6338  df-ni 6960  df-pli 6961  df-mi 6962  df-lti 6963  df-plpq 7000  df-mpq 7001  df-enq 7003  df-nqqs 7004  df-plqqs 7005  df-mqqs 7006  df-1nqqs 7007  df-rq 7008  df-ltnqqs 7009
This theorem is referenced by:  caucvgprlemrnd  7329
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