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Theorem cauappcvgprlem2 7490
Description: Lemma for cauappcvgpr 7492. 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 7237 . . . . 5 ((𝑄Q𝑅Q) → 𝑄 <Q (𝑄 +Q 𝑅))
41, 2, 3syl2anc 409 . . . 4 (𝜑𝑄 <Q (𝑄 +Q 𝑅))
5 cauappcvgpr.f . . . . 5 (𝜑𝐹:QQ)
65, 1ffvelrnd 5562 . . . 4 (𝜑 → (𝐹𝑄) ∈ Q)
7 ltanqi 7232 . . . 4 ((𝑄 <Q (𝑄 +Q 𝑅) ∧ (𝐹𝑄) ∈ Q) → ((𝐹𝑄) +Q 𝑄) <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅)))
84, 6, 7syl2anc 409 . . 3 (𝜑 → ((𝐹𝑄) +Q 𝑄) <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅)))
9 ltbtwnnqq 7245 . . 3 (((𝐹𝑄) +Q 𝑄) <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) ↔ ∃𝑥Q (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))
108, 9sylib 121 . 2 (𝜑 → ∃𝑥Q (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))
11 simprl 521 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → 𝑥Q)
121adantr 274 . . . . . 6 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → 𝑄Q)
13 simprrl 529 . . . . . 6 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → ((𝐹𝑄) +Q 𝑄) <Q 𝑥)
14 fveq2 5427 . . . . . . . . 9 (𝑞 = 𝑄 → (𝐹𝑞) = (𝐹𝑄))
15 id 19 . . . . . . . . 9 (𝑞 = 𝑄𝑞 = 𝑄)
1614, 15oveq12d 5798 . . . . . . . 8 (𝑞 = 𝑄 → ((𝐹𝑞) +Q 𝑞) = ((𝐹𝑄) +Q 𝑄))
1716breq1d 3945 . . . . . . 7 (𝑞 = 𝑄 → (((𝐹𝑞) +Q 𝑞) <Q 𝑥 ↔ ((𝐹𝑄) +Q 𝑄) <Q 𝑥))
1817rspcev 2792 . . . . . 6 ((𝑄Q ∧ ((𝐹𝑄) +Q 𝑄) <Q 𝑥) → ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑥)
1912, 13, 18syl2anc 409 . . . . 5 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑥)
20 breq2 3939 . . . . . . 7 (𝑢 = 𝑥 → (((𝐹𝑞) +Q 𝑞) <Q 𝑢 ↔ ((𝐹𝑞) +Q 𝑞) <Q 𝑥))
2120rexbidv 2439 . . . . . 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 5430 . . . . . . 7 (2nd𝐿) = (2nd ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩)
24 nqex 7193 . . . . . . . . 9 Q ∈ V
2524rabex 4078 . . . . . . . 8 {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)} ∈ V
2624rabex 4078 . . . . . . . 8 {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢} ∈ V
2725, 26op2nd 6051 . . . . . . 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 2161 . . . . . 6 (2nd𝐿) = {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}
2921, 28elrab2 2846 . . . . 5 (𝑥 ∈ (2nd𝐿) ↔ (𝑥Q ∧ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑥))
3011, 19, 29sylanbrc 414 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → 𝑥 ∈ (2nd𝐿))
31 simprrr 530 . . . . . 6 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → 𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅)))
32 vex 2692 . . . . . . 7 𝑥 ∈ V
33 breq1 3938 . . . . . . 7 (𝑙 = 𝑥 → (𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) ↔ 𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))
3432, 33elab 2831 . . . . . 6 (𝑥 ∈ {𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))} ↔ 𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅)))
3531, 34sylibr 133 . . . . 5 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → 𝑥 ∈ {𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))})
36 ltnqex 7379 . . . . . 6 {𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))} ∈ V
37 gtnqex 7380 . . . . . 6 {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢} ∈ V
3836, 37op1st 6050 . . . . 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 2234 . . . 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 2484 . . . 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 1215 . . 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 7483 . . . . 5 (𝜑𝐿P)
4544adantr 274 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → 𝐿P)
46 addclnq 7205 . . . . . . . 8 ((𝑄Q𝑅Q) → (𝑄 +Q 𝑅) ∈ Q)
471, 2, 46syl2anc 409 . . . . . . 7 (𝜑 → (𝑄 +Q 𝑅) ∈ Q)
48 addclnq 7205 . . . . . . 7 (((𝐹𝑄) ∈ Q ∧ (𝑄 +Q 𝑅) ∈ Q) → ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) ∈ Q)
496, 47, 48syl2anc 409 . . . . . 6 (𝜑 → ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) ∈ Q)
50 nqprlu 7377 . . . . . 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 274 . . . 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 7336 . . . 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 409 . . 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 166 . 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 2557 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 103  wb 104   = wceq 1332  wcel 1481  {cab 2126  wral 2417  wrex 2418  {crab 2421  cop 3533   class class class wbr 3935  wf 5125  cfv 5129  (class class class)co 5780  1st c1st 6042  2nd c2nd 6043  Qcnq 7110   +Q cplq 7112   <Q cltq 7115  Pcnp 7121  <P cltp 7125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4049  ax-sep 4052  ax-nul 4060  ax-pow 4104  ax-pr 4137  ax-un 4361  ax-setind 4458  ax-iinf 4508
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3076  df-un 3078  df-in 3080  df-ss 3087  df-nul 3367  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-uni 3743  df-int 3778  df-iun 3821  df-br 3936  df-opab 3996  df-mpt 3997  df-tr 4033  df-eprel 4217  df-id 4221  df-po 4224  df-iso 4225  df-iord 4294  df-on 4296  df-suc 4299  df-iom 4511  df-xp 4551  df-rel 4552  df-cnv 4553  df-co 4554  df-dm 4555  df-rn 4556  df-res 4557  df-ima 4558  df-iota 5094  df-fun 5131  df-fn 5132  df-f 5133  df-f1 5134  df-fo 5135  df-f1o 5136  df-fv 5137  df-ov 5783  df-oprab 5784  df-mpo 5785  df-1st 6044  df-2nd 6045  df-recs 6208  df-irdg 6273  df-1o 6319  df-oadd 6323  df-omul 6324  df-er 6435  df-ec 6437  df-qs 6441  df-ni 7134  df-pli 7135  df-mi 7136  df-lti 7137  df-plpq 7174  df-mpq 7175  df-enq 7177  df-nqqs 7178  df-plqqs 7179  df-mqqs 7180  df-1nqqs 7181  df-rq 7182  df-ltnqqs 7183  df-inp 7296  df-iltp 7300
This theorem is referenced by:  cauappcvgprlemlim  7491
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