Theorem axcaucvg 7832

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 7864. (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 3979	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑙 → (𝑏 <Q [⟨𝑗, 1o⟩] ~Q ↔ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q ))
5	4	cbvabv 2289	. . . . . . . . . . . 12 ⊢ {𝑏 ∣ 𝑏 <Q [⟨𝑗, 1o⟩] ~Q } = {𝑙 ∣ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q }
6		breq2 3980	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑢 → ([⟨𝑗, 1o⟩] ~Q <Q 𝑐 ↔ [⟨𝑗, 1o⟩] ~Q <Q 𝑢))
7	6	cbvabv 2289	. . . . . . . . . . . 12 ⊢ {𝑐 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑐} = {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}
8	5, 7	opeq12i 3757	. . . . . . . . . . 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 5846	. . . . . . . . . 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 3755	. . . . . . . . 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 6527	. . . . . . . . 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 3755	. . . . . . 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 5483	. . . . . 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 3752	. . . . 5 ⊢ (𝑎 = 𝑧 → ⟨𝑎, 0R⟩ = ⟨𝑧, 0R⟩)
17	15, 16	eqeq12d 2179	. . . 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 5803	. . 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 4063	. 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 7831	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 103   = wceq 1342   ∈ wcel 2135  {cab 2150  ∀wral 2442  ∃wrex 2443  ⟨cop 3573  ∩ cint 3818   class class class wbr 3976   ↦ cmpt 4037  ⟶wf 5178  ‘cfv 5182  ℩crio 5791  (class class class)co 5836  1_oc1o 6368  [cec 6490  Ncnpi 7204   ~_Q ceq 7211   <_Q cltq 7217  1_Pc1p 7224   +_P cpp 7225   ~_R cer 7228  Rcnr 7229  0_Rc0r 7230  ℝcr 7743  0cc0 7744  1c1 7745   + caddc 7747   <_ℝ cltrr 7748   · cmul 7749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4091  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-iinf 4559

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-reu 2449  df-rmo 2450  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-tr 4075  df-eprel 4261  df-id 4265  df-po 4268  df-iso 4269  df-iord 4338  df-on 4340  df-suc 4343  df-iom 4562  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-riota 5792  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101  df-recs 6264  df-irdg 6329  df-1o 6375  df-2o 6376  df-oadd 6379  df-omul 6380  df-er 6492  df-ec 6494  df-qs 6498  df-ni 7236  df-pli 7237  df-mi 7238  df-lti 7239  df-plpq 7276  df-mpq 7277  df-enq 7279  df-nqqs 7280  df-plqqs 7281  df-mqqs 7282  df-1nqqs 7283  df-rq 7284  df-ltnqqs 7285  df-enq0 7356  df-nq0 7357  df-0nq0 7358  df-plq0 7359  df-mq0 7360  df-inp 7398  df-i1p 7399  df-iplp 7400  df-imp 7401  df-iltp 7402  df-enr 7658  df-nr 7659  df-plr 7660  df-mr 7661  df-ltr 7662  df-0r 7663  df-1r 7664  df-m1r 7665  df-c 7750  df-0 7751  df-1 7752  df-r 7754  df-add 7755  df-mul 7756  df-lt 7757

This theorem is referenced by: (None)