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Theorem axcaucvg 7356
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 7386. (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 3817 . . . . . . . . . . . . 13 (𝑏 = 𝑙 → (𝑏 <Q [⟨𝑗, 1𝑜⟩] ~Q𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q ))
54cbvabv 2208 . . . . . . . . . . . 12 {𝑏𝑏 <Q [⟨𝑗, 1𝑜⟩] ~Q } = {𝑙𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q }
6 breq2 3818 . . . . . . . . . . . . 13 (𝑐 = 𝑢 → ([⟨𝑗, 1𝑜⟩] ~Q <Q 𝑐 ↔ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢))
76cbvabv 2208 . . . . . . . . . . . 12 {𝑐 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑐} = {𝑢 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢}
85, 7opeq12i 3604 . . . . . . . . . . 11 ⟨{𝑏𝑏 <Q [⟨𝑗, 1𝑜⟩] ~Q }, {𝑐 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑐}⟩ = ⟨{𝑙𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢}⟩
98oveq1i 5604 . . . . . . . . . 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 3602 . . . . . . . . 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 6260 . . . . . . . . 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 3602 . . . . . . 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 5259 . . . . . 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 3599 . . . . 5 (𝑎 = 𝑧 → ⟨𝑎, 0R⟩ = ⟨𝑧, 0R⟩)
1715, 16eqeq12d 2099 . . . 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 5561 . . 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 3894 . 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 7355 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 1287  wcel 1436  {cab 2071  wral 2355  wrex 2356  cop 3428   cint 3665   class class class wbr 3814  cmpt 3868  wf 4968  cfv 4972  crio 5549  (class class class)co 5594  1𝑜c1o 6109  [cec 6223  Ncnpi 6752   ~Q ceq 6759   <Q cltq 6765  1Pc1p 6772   +P cpp 6773   ~R cer 6776  Rcnr 6777  0Rc0r 6778  cr 7270  0cc0 7271  1c1 7272   + caddc 7274   < cltrr 7275   · cmul 7276
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3922  ax-sep 3925  ax-nul 3933  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319  ax-iinf 4369
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rmo 2363  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-nul 3273  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-int 3666  df-iun 3709  df-br 3815  df-opab 3869  df-mpt 3870  df-tr 3905  df-eprel 4083  df-id 4087  df-po 4090  df-iso 4091  df-iord 4160  df-on 4162  df-suc 4165  df-iom 4372  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-f1 4977  df-fo 4978  df-f1o 4979  df-fv 4980  df-riota 5550  df-ov 5597  df-oprab 5598  df-mpt2 5599  df-1st 5849  df-2nd 5850  df-recs 6005  df-irdg 6070  df-1o 6116  df-2o 6117  df-oadd 6120  df-omul 6121  df-er 6225  df-ec 6227  df-qs 6231  df-ni 6784  df-pli 6785  df-mi 6786  df-lti 6787  df-plpq 6824  df-mpq 6825  df-enq 6827  df-nqqs 6828  df-plqqs 6829  df-mqqs 6830  df-1nqqs 6831  df-rq 6832  df-ltnqqs 6833  df-enq0 6904  df-nq0 6905  df-0nq0 6906  df-plq0 6907  df-mq0 6908  df-inp 6946  df-i1p 6947  df-iplp 6948  df-imp 6949  df-iltp 6950  df-enr 7193  df-nr 7194  df-plr 7195  df-mr 7196  df-ltr 7197  df-0r 7198  df-1r 7199  df-m1r 7200  df-c 7277  df-0 7278  df-1 7279  df-r 7281  df-add 7282  df-mul 7283  df-lt 7284
This theorem is referenced by: (None)
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