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Theorem caucvgprlemlol 6958
Description: Lemma for caucvgpr 6970. 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 6653 . . . . 5 <Q ⊆ (Q × Q)
21brel 4439 . . . 4 (𝑠 <Q 𝑟 → (𝑠Q𝑟Q))
32simpld 110 . . 3 (𝑠 <Q 𝑟𝑠Q)
433ad2ant2 961 . 2 ((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) → 𝑠Q)
5 oveq1 5571 . . . . . . . 8 (𝑙 = 𝑟 → (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )))
65breq1d 3816 . . . . . . 7 (𝑙 = 𝑟 → ((𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
76rexbidv 2374 . . . . . 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 5233 . . . . . . 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 6651 . . . . . . . . 9 Q ∈ V
1110rabex 3943 . . . . . . . 8 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)} ∈ V
1210rabex 3943 . . . . . . . 8 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢} ∈ V
1311, 12op1st 5825 . . . . . . 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 2103 . . . . . 6 (1st𝐿) = {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}
157, 14elrab2 2761 . . . . 5 (𝑟 ∈ (1st𝐿) ↔ (𝑟Q ∧ ∃𝑗N (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
1615simprbi 269 . . . 4 (𝑟 ∈ (1st𝐿) → ∃𝑗N (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))
17163ad2ant3 962 . . 3 ((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) → ∃𝑗N (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))
18 simpll2 979 . . . . . . 7 ((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑗N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)) → 𝑠 <Q 𝑟)
19 ltanqg 6688 . . . . . . . . 9 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
2019adantl 271 . . . . . . . 8 (((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑗N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)) ∧ (𝑓Q𝑔QQ)) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
214ad2antrr 472 . . . . . . . 8 ((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑗N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)) → 𝑠Q)
222simprd 112 . . . . . . . . . 10 (𝑠 <Q 𝑟𝑟Q)
23223ad2ant2 961 . . . . . . . . 9 ((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) → 𝑟Q)
2423ad2antrr 472 . . . . . . . 8 ((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑗N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)) → 𝑟Q)
25 simplr 497 . . . . . . . . 9 ((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑗N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)) → 𝑗N)
26 nnnq 6710 . . . . . . . . 9 (𝑗N → [⟨𝑗, 1𝑜⟩] ~QQ)
27 recclnq 6680 . . . . . . . . 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 6669 . . . . . . . . 9 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
3029adantl 271 . . . . . . . 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 5722 . . . . . . 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 145 . . . . . 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 6686 . . . . . . 7 <Q Or Q
3433, 1sotri 4771 . . . . . 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 411 . . . . 5 ((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑗N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)) → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))
3635ex 113 . . . 4 (((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑗N) → ((𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
3736reximdva 2468 . . 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 5571 . . . . 5 (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )))
4039breq1d 3816 . . . 4 (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
4140rexbidv 2374 . . 3 (𝑙 = 𝑠 → (∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
4241, 14elrab2 2761 . 2 (𝑠 ∈ (1st𝐿) ↔ (𝑠Q ∧ ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
434, 38, 42sylanbrc 408 1 ((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) → 𝑠 ∈ (1st𝐿))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 920   = wceq 1285  wcel 1434  wral 2353  wrex 2354  {crab 2357  cop 3420   class class class wbr 3806  wf 4949  cfv 4953  (class class class)co 5564  1st c1st 5817  1𝑜c1o 6079  [cec 6192  Ncnpi 6560   <N clti 6563   ~Q ceq 6567  Qcnq 6568   +Q cplq 6570  *Qcrq 6572   <Q cltq 6573
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3914  ax-sep 3917  ax-nul 3925  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-iinf 4358
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2826  df-csb 2919  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-nul 3269  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-int 3658  df-iun 3701  df-br 3807  df-opab 3861  df-mpt 3862  df-tr 3897  df-eprel 4073  df-id 4077  df-po 4080  df-iso 4081  df-iord 4150  df-on 4152  df-suc 4155  df-iom 4361  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fn 4956  df-f 4957  df-f1 4958  df-fo 4959  df-f1o 4960  df-fv 4961  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819  df-2nd 5820  df-recs 5975  df-irdg 6040  df-1o 6086  df-oadd 6090  df-omul 6091  df-er 6194  df-ec 6196  df-qs 6200  df-ni 6592  df-pli 6593  df-mi 6594  df-lti 6595  df-plpq 6632  df-mpq 6633  df-enq 6635  df-nqqs 6636  df-plqqs 6637  df-mqqs 6638  df-1nqqs 6639  df-rq 6640  df-ltnqqs 6641
This theorem is referenced by:  caucvgprlemrnd  6961
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