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Theorem caucvgprlemupu 7702
Description: Lemma for caucvgpr 7712. The upper cut of the putative limit is upper. (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
caucvgprlemupu ((𝜑𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)) → 𝑟 ∈ (2nd𝐿))
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 caucvgprlemupu
StepHypRef Expression
1 ltrelnq 7395 . . . . 5 <Q ⊆ (Q × Q)
21brel 4696 . . . 4 (𝑠 <Q 𝑟 → (𝑠Q𝑟Q))
32simprd 114 . . 3 (𝑠 <Q 𝑟𝑟Q)
433ad2ant2 1021 . 2 ((𝜑𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)) → 𝑟Q)
5 breq2 4022 . . . . . . 7 (𝑢 = 𝑠 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
65rexbidv 2491 . . . . . 6 (𝑢 = 𝑠 → (∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
7 caucvgpr.lim . . . . . . . 8 𝐿 = ⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩
87fveq2i 5537 . . . . . . 7 (2nd𝐿) = (2nd ‘⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩)
9 nqex 7393 . . . . . . . . 9 Q ∈ V
109rabex 4162 . . . . . . . 8 {𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)} ∈ V
119rabex 4162 . . . . . . . 8 {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ V
1210, 11op2nd 6173 . . . . . . 7 (2nd ‘⟨{𝑙Q ∣ ∃𝑗N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹𝑗)}, {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩) = {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}
138, 12eqtri 2210 . . . . . 6 (2nd𝐿) = {𝑢Q ∣ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}
146, 13elrab2 2911 . . . . 5 (𝑠 ∈ (2nd𝐿) ↔ (𝑠Q ∧ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
1514simprbi 275 . . . 4 (𝑠 ∈ (2nd𝐿) → ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠)
16153ad2ant3 1022 . . 3 ((𝜑𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)) → ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠)
17 ltsonq 7428 . . . . . . 7 <Q Or Q
1817, 1sotri 5042 . . . . . 6 ((((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠𝑠 <Q 𝑟) → ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟)
1918expcom 116 . . . . 5 (𝑠 <Q 𝑟 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠 → ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟))
20193ad2ant2 1021 . . . 4 ((𝜑𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)) → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠 → ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟))
2120reximdv 2591 . . 3 ((𝜑𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)) → (∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠 → ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟))
2216, 21mpd 13 . 2 ((𝜑𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)) → ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟)
23 breq2 4022 . . . 4 (𝑢 = 𝑟 → (((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟))
2423rexbidv 2491 . . 3 (𝑢 = 𝑟 → (∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟))
2524, 13elrab2 2911 . 2 (𝑟 ∈ (2nd𝐿) ↔ (𝑟Q ∧ ∃𝑗N ((𝐹𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑟))
264, 22, 25sylanbrc 417 1 ((𝜑𝑠 <Q 𝑟𝑠 ∈ (2nd𝐿)) → 𝑟 ∈ (2nd𝐿))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 980   = wceq 1364  wcel 2160  wral 2468  wrex 2469  {crab 2472  cop 3610   class class class wbr 4018  wf 5231  cfv 5235  (class class class)co 5897  2nd c2nd 6165  1oc1o 6435  [cec 6558  Ncnpi 7302   <N clti 7305   ~Q ceq 7309  Qcnq 7310   +Q cplq 7312  *Qcrq 7314   <Q cltq 7315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-irdg 6396  df-oadd 6446  df-omul 6447  df-er 6560  df-ec 6562  df-qs 6566  df-ni 7334  df-mi 7336  df-lti 7337  df-enq 7377  df-nqqs 7378  df-ltnqqs 7383
This theorem is referenced by:  caucvgprlemrnd  7703
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