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Mirrors > Home > ILE Home > Th. List > axcaucvg | GIF version |
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 7708. (Contributed by Jim Kingdon, 8-Jul-2021.) (New usage is discouraged.) |
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))))) |
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))))) |
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 3902 | . . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑙 → (𝑏 <Q [〈𝑗, 1o〉] ~Q ↔ 𝑙 <Q [〈𝑗, 1o〉] ~Q )) | |
5 | 4 | cbvabv 2241 | . . . . . . . . . . . 12 ⊢ {𝑏 ∣ 𝑏 <Q [〈𝑗, 1o〉] ~Q } = {𝑙 ∣ 𝑙 <Q [〈𝑗, 1o〉] ~Q } |
6 | breq2 3903 | . . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑢 → ([〈𝑗, 1o〉] ~Q <Q 𝑐 ↔ [〈𝑗, 1o〉] ~Q <Q 𝑢)) | |
7 | 6 | cbvabv 2241 | . . . . . . . . . . . 12 ⊢ {𝑐 ∣ [〈𝑗, 1o〉] ~Q <Q 𝑐} = {𝑢 ∣ [〈𝑗, 1o〉] ~Q <Q 𝑢} |
8 | 5, 7 | opeq12i 3680 | . . . . . . . . . . 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 5752 | . . . . . . . . . 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 3678 | . . . . . . . . 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 6432 | . . . . . . . . 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 3678 | . . . . . . 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 5392 | . . . . . 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 3675 | . . . . 5 ⊢ (𝑎 = 𝑧 → 〈𝑎, 0R〉 = 〈𝑧, 0R〉) | |
17 | 15, 16 | eqeq12d 2132 | . . . 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 5709 | . . 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 3985 | . 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 7675 | 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))))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1316 ∈ wcel 1465 {cab 2103 ∀wral 2393 ∃wrex 2394 〈cop 3500 ∩ cint 3741 class class class wbr 3899 ↦ cmpt 3959 ⟶wf 5089 ‘cfv 5093 ℩crio 5697 (class class class)co 5742 1oc1o 6274 [cec 6395 Ncnpi 7048 ~Q ceq 7055 <Q cltq 7061 1Pc1p 7068 +P cpp 7069 ~R cer 7072 Rcnr 7073 0Rc0r 7074 ℝcr 7587 0cc0 7588 1c1 7589 + caddc 7591 <ℝ cltrr 7592 · cmul 7593 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-eprel 4181 df-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-irdg 6235 df-1o 6281 df-2o 6282 df-oadd 6285 df-omul 6286 df-er 6397 df-ec 6399 df-qs 6403 df-ni 7080 df-pli 7081 df-mi 7082 df-lti 7083 df-plpq 7120 df-mpq 7121 df-enq 7123 df-nqqs 7124 df-plqqs 7125 df-mqqs 7126 df-1nqqs 7127 df-rq 7128 df-ltnqqs 7129 df-enq0 7200 df-nq0 7201 df-0nq0 7202 df-plq0 7203 df-mq0 7204 df-inp 7242 df-i1p 7243 df-iplp 7244 df-imp 7245 df-iltp 7246 df-enr 7502 df-nr 7503 df-plr 7504 df-mr 7505 df-ltr 7506 df-0r 7507 df-1r 7508 df-m1r 7509 df-c 7594 df-0 7595 df-1 7596 df-r 7598 df-add 7599 df-mul 7600 df-lt 7601 |
This theorem is referenced by: (None) |
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