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Theorem caucvgprlemlol 6825
Description: Lemma for caucvgpr 6837. 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 6520 . . . . 5 <Q ⊆ (Q × Q)
21brel 4419 . . . 4 (𝑠 <Q 𝑟 → (𝑠Q𝑟Q))
32simpld 109 . . 3 (𝑠 <Q 𝑟𝑠Q)
433ad2ant2 937 . 2 ((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) → 𝑠Q)
5 oveq1 5546 . . . . . . . 8 (𝑙 = 𝑟 → (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )))
65breq1d 3801 . . . . . . 7 (𝑙 = 𝑟 → ((𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
76rexbidv 2344 . . . . . 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 5208 . . . . . . 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 6518 . . . . . . . . 9 Q ∈ V
1110rabex 3928 . . . . . . . 8 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)} ∈ V
1210rabex 3928 . . . . . . . 8 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q 𝑢} ∈ V
1311, 12op1st 5800 . . . . . . 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 2076 . . . . . 6 (1st𝐿) = {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)}
157, 14elrab2 2722 . . . . 5 (𝑟 ∈ (1st𝐿) ↔ (𝑟Q ∧ ∃𝑗N (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
1615simprbi 264 . . . 4 (𝑟 ∈ (1st𝐿) → ∃𝑗N (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))
17163ad2ant3 938 . . 3 ((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) → ∃𝑗N (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))
18 simpll2 955 . . . . . . 7 ((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑗N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)) → 𝑠 <Q 𝑟)
19 ltanqg 6555 . . . . . . . . 9 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
2019adantl 266 . . . . . . . 8 (((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑗N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)) ∧ (𝑓Q𝑔QQ)) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
214ad2antrr 465 . . . . . . . 8 ((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑗N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)) → 𝑠Q)
222simprd 111 . . . . . . . . . 10 (𝑠 <Q 𝑟𝑟Q)
23223ad2ant2 937 . . . . . . . . 9 ((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) → 𝑟Q)
2423ad2antrr 465 . . . . . . . 8 ((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑗N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)) → 𝑟Q)
25 simplr 490 . . . . . . . . 9 ((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑗N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)) → 𝑗N)
26 nnnq 6577 . . . . . . . . 9 (𝑗N → [⟨𝑗, 1𝑜⟩] ~QQ)
27 recclnq 6547 . . . . . . . . 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 6536 . . . . . . . . 9 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
3029adantl 266 . . . . . . . 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 5697 . . . . . . 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 139 . . . . . 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 6553 . . . . . . 7 <Q Or Q
3433, 1sotri 4747 . . . . . 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 405 . . . . 5 ((((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑗N) ∧ (𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)) → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗))
3635ex 112 . . . 4 (((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) ∧ 𝑗N) → ((𝑟 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) → (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
3736reximdva 2438 . . 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 5546 . . . . 5 (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )))
4039breq1d 3801 . . . 4 (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
4140rexbidv 2344 . . 3 (𝑙 = 𝑠 → (∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗) ↔ ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
4241, 14elrab2 2722 . 2 (𝑠 ∈ (1st𝐿) ↔ (𝑠Q ∧ ∃𝑗N (𝑠 +Q (*Q‘[⟨𝑗, 1𝑜⟩] ~Q )) <Q (𝐹𝑗)))
434, 38, 42sylanbrc 402 1 ((𝜑𝑠 <Q 𝑟𝑟 ∈ (1st𝐿)) → 𝑠 ∈ (1st𝐿))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  w3a 896   = wceq 1259  wcel 1409  wral 2323  wrex 2324  {crab 2327  cop 3405   class class class wbr 3791  wf 4925  cfv 4929  (class class class)co 5539  1st c1st 5792  1𝑜c1o 6024  [cec 6134  Ncnpi 6427   <N clti 6430   ~Q ceq 6434  Qcnq 6435   +Q cplq 6437  *Qcrq 6439   <Q cltq 6440
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-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
This theorem is referenced by:  caucvgprlemrnd  6828
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