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Theorem axcaucvg 7676
Description: Real number completeness axiom. A Cauchy sequence with a modulus of convergence converges. This is basically Corollary 11.2.13 of [HoTT], p. (varies). The HoTT book theorem has a modulus of convergence (that is, a rate of convergence) specified by (11.2.9) in HoTT whereas this theorem fixes the rate of convergence to say that all terms after the nth term must be within 1 / 𝑛 of the nth term (it should later be able to prove versions of this theorem with a different fixed rate or a modulus of convergence supplied as a hypothesis).

Because we are stating this axiom before we have introduced notations for or division, we use 𝑁 for the natural numbers and express a reciprocal in terms of .

This construction-dependent theorem should not be referenced directly; instead, use ax-caucvg 7708. (Contributed by Jim Kingdon, 8-Jul-2021.) (New usage is discouraged.)

Hypotheses
Ref Expression
axcaucvg.n 𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
axcaucvg.f (𝜑𝐹:𝑁⟶ℝ)
axcaucvg.cau (𝜑 → ∀𝑛𝑁𝑘𝑁 (𝑛 < 𝑘 → ((𝐹𝑛) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹𝑘) < ((𝐹𝑛) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))
Assertion
Ref Expression
axcaucvg (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ℝ (0 < 𝑥 → ∃𝑗𝑁𝑘𝑁 (𝑗 < 𝑘 → ((𝐹𝑘) < (𝑦 + 𝑥) ∧ 𝑦 < ((𝐹𝑘) + 𝑥)))))
Distinct variable groups:   𝑗,𝐹,𝑘,𝑛   𝑥,𝐹,𝑦,𝑗,𝑘   𝑗,𝑁,𝑘,𝑛   𝑥,𝑁,𝑦   𝜑,𝑗,𝑘,𝑛   𝑘,𝑟,𝑛   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑦,𝑟)   𝐹(𝑟)   𝑁(𝑟)

Proof of Theorem axcaucvg
Dummy variables 𝑎 𝑙 𝑢 𝑧 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 axcaucvg.n . 2 𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
2 axcaucvg.f . 2 (𝜑𝐹:𝑁⟶ℝ)
3 axcaucvg.cau . 2 (𝜑 → ∀𝑛𝑁𝑘𝑁 (𝑛 < 𝑘 → ((𝐹𝑛) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹𝑘) < ((𝐹𝑛) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))
4 breq1 3902 . . . . . . . . . . . . 13 (𝑏 = 𝑙 → (𝑏 <Q [⟨𝑗, 1o⟩] ~Q𝑙 <Q [⟨𝑗, 1o⟩] ~Q ))
54cbvabv 2241 . . . . . . . . . . . 12 {𝑏𝑏 <Q [⟨𝑗, 1o⟩] ~Q } = {𝑙𝑙 <Q [⟨𝑗, 1o⟩] ~Q }
6 breq2 3903 . . . . . . . . . . . . 13 (𝑐 = 𝑢 → ([⟨𝑗, 1o⟩] ~Q <Q 𝑐 ↔ [⟨𝑗, 1o⟩] ~Q <Q 𝑢))
76cbvabv 2241 . . . . . . . . . . . 12 {𝑐 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑐} = {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}
85, 7opeq12i 3680 . . . . . . . . . . 11 ⟨{𝑏𝑏 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑐 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑐}⟩ = ⟨{𝑙𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩
98oveq1i 5752 . . . . . . . . . 10 (⟨{𝑏𝑏 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑐 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑐}⟩ +P 1P) = (⟨{𝑙𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P)
109opeq1i 3678 . . . . . . . . 9 ⟨(⟨{𝑏𝑏 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑐 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑐}⟩ +P 1P), 1P⟩ = ⟨(⟨{𝑙𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P
11 eceq1 6432 . . . . . . . . 9 (⟨(⟨{𝑏𝑏 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑐 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑐}⟩ +P 1P), 1P⟩ = ⟨(⟨{𝑙𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ → [⟨(⟨{𝑏𝑏 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑐 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑐}⟩ +P 1P), 1P⟩] ~R = [⟨(⟨{𝑙𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
1210, 11ax-mp 5 . . . . . . . 8 [⟨(⟨{𝑏𝑏 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑐 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑐}⟩ +P 1P), 1P⟩] ~R = [⟨(⟨{𝑙𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R
1312opeq1i 3678 . . . . . . 7 ⟨[⟨(⟨{𝑏𝑏 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑐 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑐}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R
1413fveq2i 5392 . . . . . 6 (𝐹‘⟨[⟨(⟨{𝑏𝑏 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑐 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑐}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
1514a1i 9 . . . . 5 (𝑎 = 𝑧 → (𝐹‘⟨[⟨(⟨{𝑏𝑏 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑐 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑐}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩))
16 opeq1 3675 . . . . 5 (𝑎 = 𝑧 → ⟨𝑎, 0R⟩ = ⟨𝑧, 0R⟩)
1715, 16eqeq12d 2132 . . . 4 (𝑎 = 𝑧 → ((𝐹‘⟨[⟨(⟨{𝑏𝑏 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑐 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑐}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑎, 0R⟩ ↔ (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩))
1817cbvriotav 5709 . . 3 (𝑎R (𝐹‘⟨[⟨(⟨{𝑏𝑏 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑐 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑐}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑎, 0R⟩) = (𝑧R (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩)
1918mpteq2i 3985 . 2 (𝑗N ↦ (𝑎R (𝐹‘⟨[⟨(⟨{𝑏𝑏 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑐 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑐}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑎, 0R⟩)) = (𝑗N ↦ (𝑧R (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩))
201, 2, 3, 19axcaucvglemres 7675 1 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ℝ (0 < 𝑥 → ∃𝑗𝑁𝑘𝑁 (𝑗 < 𝑘 → ((𝐹𝑘) < (𝑦 + 𝑥) ∧ 𝑦 < ((𝐹𝑘) + 𝑥)))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  wcel 1465  {cab 2103  wral 2393  wrex 2394  cop 3500   cint 3741   class class class wbr 3899  cmpt 3959  wf 5089  cfv 5093  crio 5697  (class class class)co 5742  1oc1o 6274  [cec 6395  Ncnpi 7048   ~Q ceq 7055   <Q cltq 7061  1Pc1p 7068   +P cpp 7069   ~R cer 7072  Rcnr 7073  0Rc0r 7074  cr 7587  0cc0 7588  1c1 7589   + caddc 7591   < cltrr 7592   · cmul 7593
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-2o 6282  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-plpq 7120  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-mqqs 7126  df-1nqqs 7127  df-rq 7128  df-ltnqqs 7129  df-enq0 7200  df-nq0 7201  df-0nq0 7202  df-plq0 7203  df-mq0 7204  df-inp 7242  df-i1p 7243  df-iplp 7244  df-imp 7245  df-iltp 7246  df-enr 7502  df-nr 7503  df-plr 7504  df-mr 7505  df-ltr 7506  df-0r 7507  df-1r 7508  df-m1r 7509  df-c 7594  df-0 7595  df-1 7596  df-r 7598  df-add 7599  df-mul 7600  df-lt 7601
This theorem is referenced by: (None)
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