Theorem axcaucvg 8119

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 8151. (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 4091	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑙 → (𝑏 <Q [⟨𝑗, 1o⟩] ~Q ↔ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q ))
5	4	cbvabv 2356	. . . . . . . . . . . 12 ⊢ {𝑏 ∣ 𝑏 <Q [⟨𝑗, 1o⟩] ~Q } = {𝑙 ∣ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q }
6		breq2 4092	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑢 → ([⟨𝑗, 1o⟩] ~Q <Q 𝑐 ↔ [⟨𝑗, 1o⟩] ~Q <Q 𝑢))
7	6	cbvabv 2356	. . . . . . . . . . . 12 ⊢ {𝑐 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑐} = {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}
8	5, 7	opeq12i 3867	. . . . . . . . . . 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 6027	. . . . . . . . . 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 3865	. . . . . . . . 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 6736	. . . . . . . . 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 3865	. . . . . . 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 5642	. . . . . 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 3862	. . . . 5 ⊢ (𝑎 = 𝑧 → ⟨𝑎, 0R⟩ = ⟨𝑧, 0R⟩)
17	15, 16	eqeq12d 2246	. . . 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 5983	. . 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 4176	. 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 8118	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)))))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397   ∈ wcel 2202  {cab 2217  ∀wral 2510  ∃wrex 2511  ⟨cop 3672  ∩ cint 3928   class class class wbr 4088   ↦ cmpt 4150  ⟶wf 5322  ‘cfv 5326  ℩crio 5969  (class class class)co 6017  1_oc1o 6574  [cec 6699  Ncnpi 7491   ~_Q ceq 7498   <_Q cltq 7504  1_Pc1p 7511   +_P cpp 7512   ~_R cer 7515  Rcnr 7516  0_Rc0r 7517  ℝcr 8030  0cc0 8031  1c1 8032   + caddc 8034   <_ℝ cltrr 8035   · cmul 8036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-eprel 4386  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-1o 6581  df-2o 6582  df-oadd 6585  df-omul 6586  df-er 6701  df-ec 6703  df-qs 6707  df-ni 7523  df-pli 7524  df-mi 7525  df-lti 7526  df-plpq 7563  df-mpq 7564  df-enq 7566  df-nqqs 7567  df-plqqs 7568  df-mqqs 7569  df-1nqqs 7570  df-rq 7571  df-ltnqqs 7572  df-enq0 7643  df-nq0 7644  df-0nq0 7645  df-plq0 7646  df-mq0 7647  df-inp 7685  df-i1p 7686  df-iplp 7687  df-imp 7688  df-iltp 7689  df-enr 7945  df-nr 7946  df-plr 7947  df-mr 7948  df-ltr 7949  df-0r 7950  df-1r 7951  df-m1r 7952  df-c 8037  df-0 8038  df-1 8039  df-r 8041  df-add 8042  df-mul 8043  df-lt 8044

This theorem is referenced by: (None)