Theorem axcaucvg 6974

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 7004. (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,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.

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 3767	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑙 → (𝑏 <Q [⟨𝑗, 1𝑜⟩] ~Q ↔ 𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q ))
5	4	cbvabv 2161	. . . . . . . . . . . 12 ⊢ {𝑏 ∣ 𝑏 <Q [⟨𝑗, 1𝑜⟩] ~Q } = {𝑙 ∣ 𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q }
6		breq2 3768	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑢 → ([⟨𝑗, 1𝑜⟩] ~Q <Q 𝑐 ↔ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢))
7	6	cbvabv 2161	. . . . . . . . . . . 12 ⊢ {𝑐 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑐} = {𝑢 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢}
8	5, 7	opeq12i 3554	. . . . . . . . . . 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 5522	. . . . . . . . . 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 3552	. . . . . . . . 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 6141	. . . . . . . . 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 7	. . . . . . . 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 3552	. . . . . . 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 5181	. . . . . 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 3549	. . . . 5 ⊢ (𝑎 = 𝑧 → ⟨𝑎, 0R⟩ = ⟨𝑧, 0R⟩)
17	15, 16	eqeq12d 2054	. . . 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 5479	. . 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 3844	. 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 6973	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 97   = wceq 1243   ∈ wcel 1393  {cab 2026  ∀wral 2306  ∃wrex 2307  ⟨cop 3378  ∩ cint 3615   class class class wbr 3764   ↦ cmpt 3818  ⟶wf 4898  ‘cfv 4902  ℩crio 5467  (class class class)co 5512  1_𝑜c1o 5994  [cec 6104  Ncnpi 6370   ~_Q ceq 6377   <_Q cltq 6383  1_Pc1p 6390   +_P cpp 6391   ~_R cer 6394  Rcnr 6395  0_Rc0r 6396  ℝcr 6888  0cc0 6889  1c1 6890   + caddc 6892   <_ℝ cltrr 6893   · cmul 6894

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 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

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 2311  df-rex 2312  df-reu 2313  df-rmo 2314  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-imp 6567  df-iltp 6568  df-enr 6811  df-nr 6812  df-plr 6813  df-mr 6814  df-ltr 6815  df-0r 6816  df-1r 6817  df-m1r 6818  df-c 6895  df-0 6896  df-1 6897  df-r 6899  df-add 6900  df-mul 6901  df-lt 6902

This theorem is referenced by: (None)