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Theorem cauappcvgpr 6758
Description: A Cauchy approximation has a limit. A Cauchy approximation, here 𝐹, is similar to a Cauchy sequence but is indexed by the desired tolerance (that is, how close together terms needs to be) rather than by natural numbers. This is basically Theorem 11.2.12 of [HoTT], p. (varies) with a few differences such as that we are proving the existence of a limit without anything about how fast it converges (that is, mere existence instead of existence, in HoTT terms), and that the codomain of 𝐹 is Q rather than P. We also specify that every term needs to be larger than a fraction 𝐴, to avoid the case where we have positive terms which "converge" to zero (which is not a positive real).

This proof (including its lemmas) is similar to the proofs of caucvgpr 6778 and caucvgprpr 6808 but is somewhat simpler, so reading this one first may help understanding the other two.

(Contributed by Jim Kingdon, 19-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))
Assertion
Ref Expression
cauappcvgpr (𝜑 → ∃𝑦P𝑞Q𝑟Q (⟨{𝑙𝑙 <Q (𝐹𝑞)}, {𝑢 ∣ (𝐹𝑞) <Q 𝑢}⟩<P (𝑦 +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩) ∧ 𝑦<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}⟩))
Distinct variable groups:   𝐴,𝑝   𝐹,𝑞,𝑦,𝑟,𝑢   𝐹,𝑝,𝑙,𝑞   𝑦,𝑙,𝑟   𝑢,𝑞,𝑦,𝑟   𝑢,𝑝,𝑟,𝑞,𝑙   𝜑,𝑞,𝑝
Allowed substitution hints:   𝜑(𝑦,𝑢,𝑟,𝑙)   𝐴(𝑦,𝑢,𝑟,𝑞,𝑙)

Proof of Theorem cauappcvgpr
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 cauappcvgpr.f . . 3 (𝜑𝐹:QQ)
2 cauappcvgpr.app . . 3 (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))
3 cauappcvgpr.bnd . . 3 (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))
4 oveq2 5520 . . . . . . . 8 (𝑧 = 𝑞 → (𝑙 +Q 𝑧) = (𝑙 +Q 𝑞))
5 fveq2 5178 . . . . . . . 8 (𝑧 = 𝑞 → (𝐹𝑧) = (𝐹𝑞))
64, 5breq12d 3777 . . . . . . 7 (𝑧 = 𝑞 → ((𝑙 +Q 𝑧) <Q (𝐹𝑧) ↔ (𝑙 +Q 𝑞) <Q (𝐹𝑞)))
76cbvrexv 2534 . . . . . 6 (∃𝑧Q (𝑙 +Q 𝑧) <Q (𝐹𝑧) ↔ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞))
87a1i 9 . . . . 5 (𝑙Q → (∃𝑧Q (𝑙 +Q 𝑧) <Q (𝐹𝑧) ↔ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)))
98rabbiia 2547 . . . 4 {𝑙Q ∣ ∃𝑧Q (𝑙 +Q 𝑧) <Q (𝐹𝑧)} = {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}
10 id 19 . . . . . . . . 9 (𝑧 = 𝑞𝑧 = 𝑞)
115, 10oveq12d 5530 . . . . . . . 8 (𝑧 = 𝑞 → ((𝐹𝑧) +Q 𝑧) = ((𝐹𝑞) +Q 𝑞))
1211breq1d 3774 . . . . . . 7 (𝑧 = 𝑞 → (((𝐹𝑧) +Q 𝑧) <Q 𝑢 ↔ ((𝐹𝑞) +Q 𝑞) <Q 𝑢))
1312cbvrexv 2534 . . . . . 6 (∃𝑧Q ((𝐹𝑧) +Q 𝑧) <Q 𝑢 ↔ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢)
1413a1i 9 . . . . 5 (𝑢Q → (∃𝑧Q ((𝐹𝑧) +Q 𝑧) <Q 𝑢 ↔ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢))
1514rabbiia 2547 . . . 4 {𝑢Q ∣ ∃𝑧Q ((𝐹𝑧) +Q 𝑧) <Q 𝑢} = {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}
169, 15opeq12i 3554 . . 3 ⟨{𝑙Q ∣ ∃𝑧Q (𝑙 +Q 𝑧) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧Q ((𝐹𝑧) +Q 𝑧) <Q 𝑢}⟩ = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩
171, 2, 3, 16cauappcvgprlemcl 6749 . 2 (𝜑 → ⟨{𝑙Q ∣ ∃𝑧Q (𝑙 +Q 𝑧) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧Q ((𝐹𝑧) +Q 𝑧) <Q 𝑢}⟩ ∈ P)
181, 2, 3, 16cauappcvgprlemlim 6757 . 2 (𝜑 → ∀𝑞Q𝑟Q (⟨{𝑙𝑙 <Q (𝐹𝑞)}, {𝑢 ∣ (𝐹𝑞) <Q 𝑢}⟩<P (⟨{𝑙Q ∣ ∃𝑧Q (𝑙 +Q 𝑧) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧Q ((𝐹𝑧) +Q 𝑧) <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩) ∧ ⟨{𝑙Q ∣ ∃𝑧Q (𝑙 +Q 𝑧) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧Q ((𝐹𝑧) +Q 𝑧) <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}⟩))
19 oveq1 5519 . . . . . 6 (𝑦 = ⟨{𝑙Q ∣ ∃𝑧Q (𝑙 +Q 𝑧) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧Q ((𝐹𝑧) +Q 𝑧) <Q 𝑢}⟩ → (𝑦 +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩) = (⟨{𝑙Q ∣ ∃𝑧Q (𝑙 +Q 𝑧) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧Q ((𝐹𝑧) +Q 𝑧) <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩))
2019breq2d 3776 . . . . 5 (𝑦 = ⟨{𝑙Q ∣ ∃𝑧Q (𝑙 +Q 𝑧) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧Q ((𝐹𝑧) +Q 𝑧) <Q 𝑢}⟩ → (⟨{𝑙𝑙 <Q (𝐹𝑞)}, {𝑢 ∣ (𝐹𝑞) <Q 𝑢}⟩<P (𝑦 +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩) ↔ ⟨{𝑙𝑙 <Q (𝐹𝑞)}, {𝑢 ∣ (𝐹𝑞) <Q 𝑢}⟩<P (⟨{𝑙Q ∣ ∃𝑧Q (𝑙 +Q 𝑧) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧Q ((𝐹𝑧) +Q 𝑧) <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩)))
21 breq1 3767 . . . . 5 (𝑦 = ⟨{𝑙Q ∣ ∃𝑧Q (𝑙 +Q 𝑧) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧Q ((𝐹𝑧) +Q 𝑧) <Q 𝑢}⟩ → (𝑦<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}⟩ ↔ ⟨{𝑙Q ∣ ∃𝑧Q (𝑙 +Q 𝑧) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧Q ((𝐹𝑧) +Q 𝑧) <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}⟩))
2220, 21anbi12d 442 . . . 4 (𝑦 = ⟨{𝑙Q ∣ ∃𝑧Q (𝑙 +Q 𝑧) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧Q ((𝐹𝑧) +Q 𝑧) <Q 𝑢}⟩ → ((⟨{𝑙𝑙 <Q (𝐹𝑞)}, {𝑢 ∣ (𝐹𝑞) <Q 𝑢}⟩<P (𝑦 +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩) ∧ 𝑦<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}⟩) ↔ (⟨{𝑙𝑙 <Q (𝐹𝑞)}, {𝑢 ∣ (𝐹𝑞) <Q 𝑢}⟩<P (⟨{𝑙Q ∣ ∃𝑧Q (𝑙 +Q 𝑧) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧Q ((𝐹𝑧) +Q 𝑧) <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩) ∧ ⟨{𝑙Q ∣ ∃𝑧Q (𝑙 +Q 𝑧) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧Q ((𝐹𝑧) +Q 𝑧) <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}⟩)))
23222ralbidv 2348 . . 3 (𝑦 = ⟨{𝑙Q ∣ ∃𝑧Q (𝑙 +Q 𝑧) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧Q ((𝐹𝑧) +Q 𝑧) <Q 𝑢}⟩ → (∀𝑞Q𝑟Q (⟨{𝑙𝑙 <Q (𝐹𝑞)}, {𝑢 ∣ (𝐹𝑞) <Q 𝑢}⟩<P (𝑦 +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩) ∧ 𝑦<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}⟩) ↔ ∀𝑞Q𝑟Q (⟨{𝑙𝑙 <Q (𝐹𝑞)}, {𝑢 ∣ (𝐹𝑞) <Q 𝑢}⟩<P (⟨{𝑙Q ∣ ∃𝑧Q (𝑙 +Q 𝑧) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧Q ((𝐹𝑧) +Q 𝑧) <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩) ∧ ⟨{𝑙Q ∣ ∃𝑧Q (𝑙 +Q 𝑧) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧Q ((𝐹𝑧) +Q 𝑧) <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}⟩)))
2423rspcev 2656 . 2 ((⟨{𝑙Q ∣ ∃𝑧Q (𝑙 +Q 𝑧) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧Q ((𝐹𝑧) +Q 𝑧) <Q 𝑢}⟩ ∈ P ∧ ∀𝑞Q𝑟Q (⟨{𝑙𝑙 <Q (𝐹𝑞)}, {𝑢 ∣ (𝐹𝑞) <Q 𝑢}⟩<P (⟨{𝑙Q ∣ ∃𝑧Q (𝑙 +Q 𝑧) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧Q ((𝐹𝑧) +Q 𝑧) <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩) ∧ ⟨{𝑙Q ∣ ∃𝑧Q (𝑙 +Q 𝑧) <Q (𝐹𝑧)}, {𝑢Q ∣ ∃𝑧Q ((𝐹𝑧) +Q 𝑧) <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}⟩)) → ∃𝑦P𝑞Q𝑟Q (⟨{𝑙𝑙 <Q (𝐹𝑞)}, {𝑢 ∣ (𝐹𝑞) <Q 𝑢}⟩<P (𝑦 +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩) ∧ 𝑦<P ⟨{𝑙𝑙 <Q ((𝐹𝑞) +Q (𝑞 +Q 𝑟))}, {𝑢 ∣ ((𝐹𝑞) +Q (𝑞 +Q 𝑟)) <Q 𝑢}⟩))
2517, 18, 24syl2anc 391 1 (𝜑 → ∃𝑦P𝑞Q𝑟Q (⟨{𝑙𝑙 <Q (𝐹𝑞)}, {𝑢 ∣ (𝐹𝑞) <Q 𝑢}⟩<P (𝑦 +P ⟨{𝑙𝑙 <Q (𝑞 +Q 𝑟)}, {𝑢 ∣ (𝑞 +Q 𝑟) <Q 𝑢}⟩) ∧ 𝑦<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 97  wb 98   = wceq 1243  wcel 1393  {cab 2026  wral 2306  wrex 2307  {crab 2310  cop 3378   class class class wbr 3764  wf 4898  cfv 4902  (class class class)co 5512  Qcnq 6376   +Q cplq 6378   <Q cltq 6381  Pcnp 6387   +P cpp 6389  <P cltp 6391
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6400  df-pli 6401  df-mi 6402  df-lti 6403  df-plpq 6440  df-mpq 6441  df-enq 6443  df-nqqs 6444  df-plqqs 6445  df-mqqs 6446  df-1nqqs 6447  df-rq 6448  df-ltnqqs 6449  df-enq0 6520  df-nq0 6521  df-0nq0 6522  df-plq0 6523  df-mq0 6524  df-inp 6562  df-iplp 6564  df-iltp 6566
This theorem is referenced by: (None)
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