Theorem axcaucvg 8012

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 8044. (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 4046	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑙 → (𝑏 <Q [⟨𝑗, 1o⟩] ~Q ↔ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q ))
5	4	cbvabv 2329	. . . . . . . . . . . 12 ⊢ {𝑏 ∣ 𝑏 <Q [⟨𝑗, 1o⟩] ~Q } = {𝑙 ∣ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q }
6		breq2 4047	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑢 → ([⟨𝑗, 1o⟩] ~Q <Q 𝑐 ↔ [⟨𝑗, 1o⟩] ~Q <Q 𝑢))
7	6	cbvabv 2329	. . . . . . . . . . . 12 ⊢ {𝑐 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑐} = {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}
8	5, 7	opeq12i 3823	. . . . . . . . . . 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 5953	. . . . . . . . . 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 3821	. . . . . . . . 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 6654	. . . . . . . . 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 3821	. . . . . . 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 5578	. . . . . 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 3818	. . . . 5 ⊢ (𝑎 = 𝑧 → ⟨𝑎, 0R⟩ = ⟨𝑧, 0R⟩)
17	15, 16	eqeq12d 2219	. . . 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 5910	. . 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 4130	. 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 8011	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 1372   ∈ wcel 2175  {cab 2190  ∀wral 2483  ∃wrex 2484  ⟨cop 3635  ∩ cint 3884   class class class wbr 4043   ↦ cmpt 4104  ⟶wf 5266  ‘cfv 5270  ℩crio 5897  (class class class)co 5943  1_oc1o 6494  [cec 6617  Ncnpi 7384   ~_Q ceq 7391   <_Q cltq 7397  1_Pc1p 7404   +_P cpp 7405   ~_R cer 7408  Rcnr 7409  0_Rc0r 7410  ℝcr 7923  0cc0 7924  1c1 7925   + caddc 7927   <_ℝ cltrr 7928   · cmul 7929

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-eprel 4335  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-irdg 6455  df-1o 6501  df-2o 6502  df-oadd 6505  df-omul 6506  df-er 6619  df-ec 6621  df-qs 6625  df-ni 7416  df-pli 7417  df-mi 7418  df-lti 7419  df-plpq 7456  df-mpq 7457  df-enq 7459  df-nqqs 7460  df-plqqs 7461  df-mqqs 7462  df-1nqqs 7463  df-rq 7464  df-ltnqqs 7465  df-enq0 7536  df-nq0 7537  df-0nq0 7538  df-plq0 7539  df-mq0 7540  df-inp 7578  df-i1p 7579  df-iplp 7580  df-imp 7581  df-iltp 7582  df-enr 7838  df-nr 7839  df-plr 7840  df-mr 7841  df-ltr 7842  df-0r 7843  df-1r 7844  df-m1r 7845  df-c 7930  df-0 7931  df-1 7932  df-r 7934  df-add 7935  df-mul 7936  df-lt 7937

This theorem is referenced by: (None)