Theorem axcaucvg 8273

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 ℕ 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 8305. (Contributed by Jim Kingdon, 8-Jul-2021.) (New usage is discouraged.)

Hypotheses

Ref	Expression
axcaucvg.n	⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}
axcaucvg.f	⊢ (𝜑 → 𝐹:𝑁⟶ℝ)
axcaucvg.cau	⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∀𝑘 ∈ 𝑁 (𝑛 <ℝ 𝑘 → ((𝐹‘𝑛) <ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹‘𝑘) <ℝ ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))

Assertion

Ref	Expression
axcaucvg	⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ℝ (0 <ℝ 𝑥 → ∃𝑗 ∈ 𝑁 ∀𝑘 ∈ 𝑁 (𝑗 <ℝ 𝑘 → ((𝐹‘𝑘) <ℝ (𝑦 + 𝑥) ∧ 𝑦 <ℝ ((𝐹‘𝑘) + 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.

Step	Hyp	Ref	Expression
1		axcaucvg.n	. 2 ⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}
2		axcaucvg.f	. 2 ⊢ (𝜑 → 𝐹:𝑁⟶ℝ)
3		axcaucvg.cau	. 2 ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∀𝑘 ∈ 𝑁 (𝑛 <ℝ 𝑘 → ((𝐹‘𝑛) <ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹‘𝑘) <ℝ ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))
4		breq1 4134	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑙 → (𝑏 <Q [⟨𝑗, 1o⟩] ~Q ↔ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q ))
5	4	cbvabv 2365	. . . . . . . . . . . 12 ⊢ {𝑏 ∣ 𝑏 <Q [⟨𝑗, 1o⟩] ~Q } = {𝑙 ∣ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q }
6		breq2 4135	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑢 → ([⟨𝑗, 1o⟩] ~Q <Q 𝑐 ↔ [⟨𝑗, 1o⟩] ~Q <Q 𝑢))
7	6	cbvabv 2365	. . . . . . . . . . . 12 ⊢ {𝑐 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑐} = {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}
8	5, 7	opeq12i 3909	. . . . . . . . . . 11 ⊢ ⟨{𝑏 ∣ 𝑏 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑐 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑐}⟩ = ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩
9	8	oveq1i 6098	. . . . . . . . . 10 ⊢ (⟨{𝑏 ∣ 𝑏 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑐 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑐}⟩ +P 1P) = (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P)
10	9	opeq1i 3907	. . . . . . . . 9 ⊢ ⟨(⟨{𝑏 ∣ 𝑏 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑐 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑐}⟩ +P 1P), 1P⟩ = ⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩
11		eceq1 6846	. . . . . . . . 9 ⊢ (⟨(⟨{𝑏 ∣ 𝑏 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑐 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑐}⟩ +P 1P), 1P⟩ = ⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ → [⟨(⟨{𝑏 ∣ 𝑏 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑐 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑐}⟩ +P 1P), 1P⟩] ~R = [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
12	10, 11	ax-mp 5	. . . . . . . 8 ⊢ [⟨(⟨{𝑏 ∣ 𝑏 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑐 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑐}⟩ +P 1P), 1P⟩] ~R = [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R
13	12	opeq1i 3907	. . . . . . 7 ⊢ ⟨[⟨(⟨{𝑏 ∣ 𝑏 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑐 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑐}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩
14	13	fveq2i 5701	. . . . . 6 ⊢ (𝐹‘⟨[⟨(⟨{𝑏 ∣ 𝑏 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑐 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑐}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
15	14	a1i 9	. . . . 5 ⊢ (𝑎 = 𝑧 → (𝐹‘⟨[⟨(⟨{𝑏 ∣ 𝑏 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑐 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑐}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩))
16		opeq1 3904	. . . . 5 ⊢ (𝑎 = 𝑧 → ⟨𝑎, 0R⟩ = ⟨𝑧, 0R⟩)
17	15, 16	eqeq12d 2253	. . . 4 ⊢ (𝑎 = 𝑧 → ((𝐹‘⟨[⟨(⟨{𝑏 ∣ 𝑏 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑐 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑐}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑎, 0R⟩ ↔ (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩))
18	17	cbvriotav 6054	. . 3 ⊢ (℩𝑎 ∈ R (𝐹‘⟨[⟨(⟨{𝑏 ∣ 𝑏 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑐 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑐}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑎, 0R⟩) = (℩𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩)
19	18	mpteq2i 4220	. 2 ⊢ (𝑗 ∈ N ↦ (℩𝑎 ∈ R (𝐹‘⟨[⟨(⟨{𝑏 ∣ 𝑏 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑐 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑐}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑎, 0R⟩)) = (𝑗 ∈ N ↦ (℩𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩))
20	1, 2, 3, 19	axcaucvglemres 8272	1 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ ℝ (0 <ℝ 𝑥 → ∃𝑗 ∈ 𝑁 ∀𝑘 ∈ 𝑁 (𝑗 <ℝ 𝑘 → ((𝐹‘𝑘) <ℝ (𝑦 + 𝑥) ∧ 𝑦 <ℝ ((𝐹‘𝑘) + 𝑥)))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  {cab 2224  ∀wral 2528  ∃wrex 2529  ⟨cop 3712  ∩ cint 3970   class class class wbr 4131   ↦ cmpt 4193  ⟶wf 5376  ‘cfv 5380  ℩crio 6040  (class class class)co 6088  1_oc1o 6684  [cec 6809  Ncnpi 7645   ~_Q ceq 7652   <_Q cltq 7658  1_Pc1p 7665   +_P cpp 7666   ~_R cer 7669  Rcnr 7670  0_Rc0r 7671  ℝcr 8184  0cc0 8185  1c1 8186   + caddc 8188   <_ℝ cltrr 8189   · cmul 8190

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-eprel 4436  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-1o 6691  df-2o 6692  df-oadd 6695  df-omul 6696  df-er 6811  df-ec 6813  df-qs 6817  df-ni 7677  df-pli 7678  df-mi 7679  df-lti 7680  df-plpq 7717  df-mpq 7718  df-enq 7720  df-nqqs 7721  df-plqqs 7722  df-mqqs 7723  df-1nqqs 7724  df-rq 7725  df-ltnqqs 7726  df-enq0 7797  df-nq0 7798  df-0nq0 7799  df-plq0 7800  df-mq0 7801  df-inp 7839  df-i1p 7840  df-iplp 7841  df-imp 7842  df-iltp 7843  df-enr 8099  df-nr 8100  df-plr 8101  df-mr 8102  df-ltr 8103  df-0r 8104  df-1r 8105  df-m1r 8106  df-c 8191  df-0 8192  df-1 8193  df-r 8195  df-add 8196  df-mul 8197  df-lt 8198

This theorem is used by: (None)