ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cauappcvgprlem2 GIF version

Theorem cauappcvgprlem2 6815
Description: Lemma for cauappcvgpr 6817. 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 6562 . . . . 5 ((𝑄Q𝑅Q) → 𝑄 <Q (𝑄 +Q 𝑅))
41, 2, 3syl2anc 397 . . . 4 (𝜑𝑄 <Q (𝑄 +Q 𝑅))
5 cauappcvgpr.f . . . . 5 (𝜑𝐹:QQ)
65, 1ffvelrnd 5330 . . . 4 (𝜑 → (𝐹𝑄) ∈ Q)
7 ltanqi 6557 . . . 4 ((𝑄 <Q (𝑄 +Q 𝑅) ∧ (𝐹𝑄) ∈ Q) → ((𝐹𝑄) +Q 𝑄) <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅)))
84, 6, 7syl2anc 397 . . 3 (𝜑 → ((𝐹𝑄) +Q 𝑄) <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅)))
9 ltbtwnnqq 6570 . . 3 (((𝐹𝑄) +Q 𝑄) <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) ↔ ∃𝑥Q (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))
108, 9sylib 131 . 2 (𝜑 → ∃𝑥Q (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))
11 simprl 491 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → 𝑥Q)
121adantr 265 . . . . . 6 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → 𝑄Q)
13 simprrl 499 . . . . . 6 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → ((𝐹𝑄) +Q 𝑄) <Q 𝑥)
14 fveq2 5205 . . . . . . . . 9 (𝑞 = 𝑄 → (𝐹𝑞) = (𝐹𝑄))
15 id 19 . . . . . . . . 9 (𝑞 = 𝑄𝑞 = 𝑄)
1614, 15oveq12d 5557 . . . . . . . 8 (𝑞 = 𝑄 → ((𝐹𝑞) +Q 𝑞) = ((𝐹𝑄) +Q 𝑄))
1716breq1d 3801 . . . . . . 7 (𝑞 = 𝑄 → (((𝐹𝑞) +Q 𝑞) <Q 𝑥 ↔ ((𝐹𝑄) +Q 𝑄) <Q 𝑥))
1817rspcev 2673 . . . . . 6 ((𝑄Q ∧ ((𝐹𝑄) +Q 𝑄) <Q 𝑥) → ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑥)
1912, 13, 18syl2anc 397 . . . . 5 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑥)
20 breq2 3795 . . . . . . 7 (𝑢 = 𝑥 → (((𝐹𝑞) +Q 𝑞) <Q 𝑢 ↔ ((𝐹𝑞) +Q 𝑞) <Q 𝑥))
2120rexbidv 2344 . . . . . 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 5208 . . . . . . 7 (2nd𝐿) = (2nd ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩)
24 nqex 6518 . . . . . . . . 9 Q ∈ V
2524rabex 3928 . . . . . . . 8 {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)} ∈ V
2624rabex 3928 . . . . . . . 8 {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢} ∈ V
2725, 26op2nd 5801 . . . . . . 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 2076 . . . . . 6 (2nd𝐿) = {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}
2921, 28elrab2 2722 . . . . 5 (𝑥 ∈ (2nd𝐿) ↔ (𝑥Q ∧ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑥))
3011, 19, 29sylanbrc 402 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → 𝑥 ∈ (2nd𝐿))
31 simprrr 500 . . . . . 6 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → 𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅)))
32 vex 2577 . . . . . . 7 𝑥 ∈ V
33 breq1 3794 . . . . . . 7 (𝑙 = 𝑥 → (𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) ↔ 𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))
3432, 33elab 2709 . . . . . 6 (𝑥 ∈ {𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))} ↔ 𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅)))
3531, 34sylibr 141 . . . . 5 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → 𝑥 ∈ {𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))})
36 ltnqex 6704 . . . . . 6 {𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))} ∈ V
37 gtnqex 6705 . . . . . 6 {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢} ∈ V
3836, 37op1st 5800 . . . . 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, 38syl6eleqr 2147 . . . 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 2387 . . . 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 1144 . . 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 6808 . . . . 5 (𝜑𝐿P)
4544adantr 265 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → 𝐿P)
46 addclnq 6530 . . . . . . . 8 ((𝑄Q𝑅Q) → (𝑄 +Q 𝑅) ∈ Q)
471, 2, 46syl2anc 397 . . . . . . 7 (𝜑 → (𝑄 +Q 𝑅) ∈ Q)
48 addclnq 6530 . . . . . . 7 (((𝐹𝑄) ∈ Q ∧ (𝑄 +Q 𝑅) ∈ Q) → ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) ∈ Q)
496, 47, 48syl2anc 397 . . . . . 6 (𝜑 → ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) ∈ Q)
50 nqprlu 6702 . . . . . 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 265 . . . 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 6661 . . . 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 397 . . 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 160 . 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 2454 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 101  wb 102   = wceq 1259  wcel 1409  {cab 2042  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  2nd c2nd 5793  Qcnq 6435   +Q cplq 6437   <Q cltq 6440  Pcnp 6446  <P cltp 6450
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  df-inp 6621  df-iltp 6625
This theorem is referenced by:  cauappcvgprlemlim  6816
  Copyright terms: Public domain W3C validator