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Theorem axcaucvg 6931
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 6961. (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,𝑗,𝑘,𝑦   𝑗,𝑁,𝑘,𝑛   𝑦,𝑁,𝑥   𝜑,𝑗,𝑘,𝑛   𝑘,𝑟,𝑛   𝜑,𝑥   𝑥,𝑦
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 3764 . . . . . . . . . . . . 13 (𝑏 = 𝑙 → (𝑏 <Q [⟨𝑗, 1𝑜⟩] ~Q𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q ))
54cbvabv 2161 . . . . . . . . . . . 12 {𝑏𝑏 <Q [⟨𝑗, 1𝑜⟩] ~Q } = {𝑙𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q }
6 breq2 3765 . . . . . . . . . . . . 13 (𝑐 = 𝑢 → ([⟨𝑗, 1𝑜⟩] ~Q <Q 𝑐 ↔ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢))
76cbvabv 2161 . . . . . . . . . . . 12 {𝑐 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑐} = {𝑢 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢}
85, 7opeq12i 3551 . . . . . . . . . . 11 ⟨{𝑏𝑏 <Q [⟨𝑗, 1𝑜⟩] ~Q }, {𝑐 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑐}⟩ = ⟨{𝑙𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢}⟩
98oveq1i 5485 . . . . . . . . . 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 3549 . . . . . . . . 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 6104 . . . . . . . . 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 3549 . . . . . . 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 5144 . . . . . 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 3546 . . . . 5 (𝑎 = 𝑧 → ⟨𝑎, 0R⟩ = ⟨𝑧, 0R⟩)
1715, 16eqeq12d 2054 . . . 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 5442 . . 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 3841 . 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 6930 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 97   = wceq 1243  wcel 1393  {cab 2026  wral 2303  wrex 2304  cop 3375   cint 3612   class class class wbr 3761  cmpt 3815  wf 4861  cfv 4865  crio 5430  (class class class)co 5475  1𝑜c1o 5957  [cec 6067  Ncnpi 6327   ~Q ceq 6334   <Q cltq 6340  1Pc1p 6347   +P cpp 6348   ~R cer 6351  Rcnr 6352  0Rc0r 6353  cr 6845  0cc0 6846  1c1 6847   + caddc 6849   < cltrr 6850   · cmul 6851
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 3869  ax-sep 3872  ax-nul 3880  ax-pow 3924  ax-pr 3941  ax-un 4142  ax-setind 4232  ax-iinf 4274
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 2308  df-rex 2309  df-reu 2310  df-rmo 2311  df-rab 2312  df-v 2556  df-sbc 2762  df-csb 2850  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-int 3613  df-iun 3656  df-br 3762  df-opab 3816  df-mpt 3817  df-tr 3852  df-eprel 4023  df-id 4027  df-po 4030  df-iso 4031  df-iord 4075  df-on 4077  df-suc 4080  df-iom 4277  df-xp 4314  df-rel 4315  df-cnv 4316  df-co 4317  df-dm 4318  df-rn 4319  df-res 4320  df-ima 4321  df-iota 4830  df-fun 4867  df-fn 4868  df-f 4869  df-f1 4870  df-fo 4871  df-f1o 4872  df-fv 4873  df-riota 5431  df-ov 5478  df-oprab 5479  df-mpt2 5480  df-1st 5730  df-2nd 5731  df-recs 5883  df-irdg 5920  df-1o 5964  df-2o 5965  df-oadd 5968  df-omul 5969  df-er 6069  df-ec 6071  df-qs 6075  df-ni 6359  df-pli 6360  df-mi 6361  df-lti 6362  df-plpq 6399  df-mpq 6400  df-enq 6402  df-nqqs 6403  df-plqqs 6404  df-mqqs 6405  df-1nqqs 6406  df-rq 6407  df-ltnqqs 6408  df-enq0 6479  df-nq0 6480  df-0nq0 6481  df-plq0 6482  df-mq0 6483  df-inp 6521  df-i1p 6522  df-iplp 6523  df-imp 6524  df-iltp 6525  df-enr 6768  df-nr 6769  df-plr 6770  df-mr 6771  df-ltr 6772  df-0r 6773  df-1r 6774  df-m1r 6775  df-c 6852  df-0 6853  df-1 6854  df-r 6856  df-add 6857  df-mul 6858  df-lt 6859
This theorem is referenced by: (None)
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