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Theorem cauappcvgprlem2 7690
Description: Lemma for cauappcvgpr 7692. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 23-Jun-2020.)
Hypotheses
Ref Expression
cauappcvgpr.f (𝜑𝐹:QQ)
cauappcvgpr.app (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))
cauappcvgpr.bnd (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))
cauappcvgpr.lim 𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩
cauappcvgprlem.q (𝜑𝑄Q)
cauappcvgprlem.r (𝜑𝑅Q)
Assertion
Ref Expression
cauappcvgprlem2 (𝜑𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩)
Distinct variable groups:   𝐴,𝑝   𝐿,𝑝,𝑞   𝜑,𝑝,𝑞   𝐹,𝑝,𝑞,𝑙,𝑢   𝑄,𝑝,𝑞,𝑙,𝑢   𝑅,𝑝,𝑞,𝑙,𝑢
Allowed substitution hints:   𝜑(𝑢,𝑙)   𝐴(𝑢,𝑞,𝑙)   𝐿(𝑢,𝑙)

Proof of Theorem cauappcvgprlem2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 cauappcvgprlem.q . . . . 5 (𝜑𝑄Q)
2 cauappcvgprlem.r . . . . 5 (𝜑𝑅Q)
3 ltaddnq 7437 . . . . 5 ((𝑄Q𝑅Q) → 𝑄 <Q (𝑄 +Q 𝑅))
41, 2, 3syl2anc 411 . . . 4 (𝜑𝑄 <Q (𝑄 +Q 𝑅))
5 cauappcvgpr.f . . . . 5 (𝜑𝐹:QQ)
65, 1ffvelcdmd 5673 . . . 4 (𝜑 → (𝐹𝑄) ∈ Q)
7 ltanqi 7432 . . . 4 ((𝑄 <Q (𝑄 +Q 𝑅) ∧ (𝐹𝑄) ∈ Q) → ((𝐹𝑄) +Q 𝑄) <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅)))
84, 6, 7syl2anc 411 . . 3 (𝜑 → ((𝐹𝑄) +Q 𝑄) <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅)))
9 ltbtwnnqq 7445 . . 3 (((𝐹𝑄) +Q 𝑄) <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) ↔ ∃𝑥Q (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))
108, 9sylib 122 . 2 (𝜑 → ∃𝑥Q (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))
11 simprl 529 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → 𝑥Q)
121adantr 276 . . . . . 6 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → 𝑄Q)
13 simprrl 539 . . . . . 6 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → ((𝐹𝑄) +Q 𝑄) <Q 𝑥)
14 fveq2 5534 . . . . . . . . 9 (𝑞 = 𝑄 → (𝐹𝑞) = (𝐹𝑄))
15 id 19 . . . . . . . . 9 (𝑞 = 𝑄𝑞 = 𝑄)
1614, 15oveq12d 5915 . . . . . . . 8 (𝑞 = 𝑄 → ((𝐹𝑞) +Q 𝑞) = ((𝐹𝑄) +Q 𝑄))
1716breq1d 4028 . . . . . . 7 (𝑞 = 𝑄 → (((𝐹𝑞) +Q 𝑞) <Q 𝑥 ↔ ((𝐹𝑄) +Q 𝑄) <Q 𝑥))
1817rspcev 2856 . . . . . 6 ((𝑄Q ∧ ((𝐹𝑄) +Q 𝑄) <Q 𝑥) → ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑥)
1912, 13, 18syl2anc 411 . . . . 5 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑥)
20 breq2 4022 . . . . . . 7 (𝑢 = 𝑥 → (((𝐹𝑞) +Q 𝑞) <Q 𝑢 ↔ ((𝐹𝑞) +Q 𝑞) <Q 𝑥))
2120rexbidv 2491 . . . . . 6 (𝑢 = 𝑥 → (∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢 ↔ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑥))
22 cauappcvgpr.lim . . . . . . . 8 𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩
2322fveq2i 5537 . . . . . . 7 (2nd𝐿) = (2nd ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩)
24 nqex 7393 . . . . . . . . 9 Q ∈ V
2524rabex 4162 . . . . . . . 8 {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)} ∈ V
2624rabex 4162 . . . . . . . 8 {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢} ∈ V
2725, 26op2nd 6173 . . . . . . 7 (2nd ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩) = {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}
2823, 27eqtri 2210 . . . . . 6 (2nd𝐿) = {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}
2921, 28elrab2 2911 . . . . 5 (𝑥 ∈ (2nd𝐿) ↔ (𝑥Q ∧ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑥))
3011, 19, 29sylanbrc 417 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → 𝑥 ∈ (2nd𝐿))
31 simprrr 540 . . . . . 6 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → 𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅)))
32 vex 2755 . . . . . . 7 𝑥 ∈ V
33 breq1 4021 . . . . . . 7 (𝑙 = 𝑥 → (𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) ↔ 𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))
3432, 33elab 2896 . . . . . 6 (𝑥 ∈ {𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))} ↔ 𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅)))
3531, 34sylibr 134 . . . . 5 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → 𝑥 ∈ {𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))})
36 ltnqex 7579 . . . . . 6 {𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))} ∈ V
37 gtnqex 7580 . . . . . 6 {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢} ∈ V
3836, 37op1st 6172 . . . . 5 (1st ‘⟨{𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩) = {𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))}
3935, 38eleqtrrdi 2283 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩))
40 rspe 2539 . . . 4 ((𝑥Q ∧ (𝑥 ∈ (2nd𝐿) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩))) → ∃𝑥Q (𝑥 ∈ (2nd𝐿) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩)))
4111, 30, 39, 40syl12anc 1247 . . 3 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → ∃𝑥Q (𝑥 ∈ (2nd𝐿) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩)))
42 cauappcvgpr.app . . . . . 6 (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))
43 cauappcvgpr.bnd . . . . . 6 (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))
445, 42, 43, 22cauappcvgprlemcl 7683 . . . . 5 (𝜑𝐿P)
4544adantr 276 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → 𝐿P)
46 addclnq 7405 . . . . . . . 8 ((𝑄Q𝑅Q) → (𝑄 +Q 𝑅) ∈ Q)
471, 2, 46syl2anc 411 . . . . . . 7 (𝜑 → (𝑄 +Q 𝑅) ∈ Q)
48 addclnq 7405 . . . . . . 7 (((𝐹𝑄) ∈ Q ∧ (𝑄 +Q 𝑅) ∈ Q) → ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) ∈ Q)
496, 47, 48syl2anc 411 . . . . . 6 (𝜑 → ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) ∈ Q)
50 nqprlu 7577 . . . . . 6 (((𝐹𝑄) +Q (𝑄 +Q 𝑅)) ∈ Q → ⟨{𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩ ∈ P)
5149, 50syl 14 . . . . 5 (𝜑 → ⟨{𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩ ∈ P)
5251adantr 276 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → ⟨{𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩ ∈ P)
53 ltdfpr 7536 . . . 4 ((𝐿P ∧ ⟨{𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩ ∈ P) → (𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩ ↔ ∃𝑥Q (𝑥 ∈ (2nd𝐿) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩))))
5445, 52, 53syl2anc 411 . . 3 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → (𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩ ↔ ∃𝑥Q (𝑥 ∈ (2nd𝐿) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩))))
5541, 54mpbird 167 . 2 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩)
5610, 55rexlimddv 2612 1 (𝜑𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wcel 2160  {cab 2175  wral 2468  wrex 2469  {crab 2472  cop 3610   class class class wbr 4018  wf 5231  cfv 5235  (class class class)co 5897  1st c1st 6164  2nd c2nd 6165  Qcnq 7310   +Q cplq 7312   <Q cltq 7315  Pcnp 7321  <P cltp 7325
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-1o 6442  df-oadd 6446  df-omul 6447  df-er 6560  df-ec 6562  df-qs 6566  df-ni 7334  df-pli 7335  df-mi 7336  df-lti 7337  df-plpq 7374  df-mpq 7375  df-enq 7377  df-nqqs 7378  df-plqqs 7379  df-mqqs 7380  df-1nqqs 7381  df-rq 7382  df-ltnqqs 7383  df-inp 7496  df-iltp 7500
This theorem is referenced by:  cauappcvgprlemlim  7691
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