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Theorem axcaucvg 7117
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 7147. (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 3790 . . . . . . . . . . . . 13 (𝑏 = 𝑙 → (𝑏 <Q [⟨𝑗, 1𝑜⟩] ~Q𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q ))
54cbvabv 2203 . . . . . . . . . . . 12 {𝑏𝑏 <Q [⟨𝑗, 1𝑜⟩] ~Q } = {𝑙𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q }
6 breq2 3791 . . . . . . . . . . . . 13 (𝑐 = 𝑢 → ([⟨𝑗, 1𝑜⟩] ~Q <Q 𝑐 ↔ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢))
76cbvabv 2203 . . . . . . . . . . . 12 {𝑐 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑐} = {𝑢 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢}
85, 7opeq12i 3577 . . . . . . . . . . 11 ⟨{𝑏𝑏 <Q [⟨𝑗, 1𝑜⟩] ~Q }, {𝑐 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑐}⟩ = ⟨{𝑙𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢}⟩
98oveq1i 5547 . . . . . . . . . 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 3575 . . . . . . . . 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 6200 . . . . . . . . 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 7 . . . . . . . 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 3575 . . . . . . 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 5206 . . . . . 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 3572 . . . . 5 (𝑎 = 𝑧 → ⟨𝑎, 0R⟩ = ⟨𝑧, 0R⟩)
1715, 16eqeq12d 2096 . . . 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 5504 . . 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 3867 . 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 7116 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 102   = wceq 1285  wcel 1434  {cab 2068  wral 2349  wrex 2350  cop 3403   cint 3638   class class class wbr 3787  cmpt 3841  wf 4922  cfv 4926  crio 5492  (class class class)co 5537  1𝑜c1o 6052  [cec 6163  Ncnpi 6513   ~Q ceq 6520   <Q cltq 6526  1Pc1p 6533   +P cpp 6534   ~R cer 6537  Rcnr 6538  0Rc0r 6539  cr 7031  0cc0 7032  1c1 7033   + caddc 7035   < cltrr 7036   · cmul 7037
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3895  ax-sep 3898  ax-nul 3906  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-iinf 4331
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3253  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-int 3639  df-iun 3682  df-br 3788  df-opab 3842  df-mpt 3843  df-tr 3878  df-eprel 4046  df-id 4050  df-po 4053  df-iso 4054  df-iord 4123  df-on 4125  df-suc 4128  df-iom 4334  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-f1 4931  df-fo 4932  df-f1o 4933  df-fv 4934  df-riota 5493  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-irdg 6013  df-1o 6059  df-2o 6060  df-oadd 6063  df-omul 6064  df-er 6165  df-ec 6167  df-qs 6171  df-ni 6545  df-pli 6546  df-mi 6547  df-lti 6548  df-plpq 6585  df-mpq 6586  df-enq 6588  df-nqqs 6589  df-plqqs 6590  df-mqqs 6591  df-1nqqs 6592  df-rq 6593  df-ltnqqs 6594  df-enq0 6665  df-nq0 6666  df-0nq0 6667  df-plq0 6668  df-mq0 6669  df-inp 6707  df-i1p 6708  df-iplp 6709  df-imp 6710  df-iltp 6711  df-enr 6954  df-nr 6955  df-plr 6956  df-mr 6957  df-ltr 6958  df-0r 6959  df-1r 6960  df-m1r 6961  df-c 7038  df-0 7039  df-1 7040  df-r 7042  df-add 7043  df-mul 7044  df-lt 7045
This theorem is referenced by: (None)
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