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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 7386. (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 3817 | . . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑙 → (𝑏 <Q [〈𝑗, 1𝑜〉] ~Q ↔ 𝑙 <Q [〈𝑗, 1𝑜〉] ~Q )) | |
5 | 4 | cbvabv 2208 | . . . . . . . . . . . 12 ⊢ {𝑏 ∣ 𝑏 <Q [〈𝑗, 1𝑜〉] ~Q } = {𝑙 ∣ 𝑙 <Q [〈𝑗, 1𝑜〉] ~Q } |
6 | breq2 3818 | . . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑢 → ([〈𝑗, 1𝑜〉] ~Q <Q 𝑐 ↔ [〈𝑗, 1𝑜〉] ~Q <Q 𝑢)) | |
7 | 6 | cbvabv 2208 | . . . . . . . . . . . 12 ⊢ {𝑐 ∣ [〈𝑗, 1𝑜〉] ~Q <Q 𝑐} = {𝑢 ∣ [〈𝑗, 1𝑜〉] ~Q <Q 𝑢} |
8 | 5, 7 | opeq12i 3604 | . . . . . . . . . . 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 5604 | . . . . . . . . . 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 3602 | . . . . . . . . 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 6260 | . . . . . . . . 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 3602 | . . . . . . 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 5259 | . . . . . 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 3599 | . . . . 5 ⊢ (𝑎 = 𝑧 → 〈𝑎, 0R〉 = 〈𝑧, 0R〉) | |
17 | 15, 16 | eqeq12d 2099 | . . . 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 5561 | . . 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 3894 | . 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 7355 | 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 102 = wceq 1287 ∈ wcel 1436 {cab 2071 ∀wral 2355 ∃wrex 2356 〈cop 3428 ∩ cint 3665 class class class wbr 3814 ↦ cmpt 3868 ⟶wf 4968 ‘cfv 4972 ℩crio 5549 (class class class)co 5594 1𝑜c1o 6109 [cec 6223 Ncnpi 6752 ~Q ceq 6759 <Q cltq 6765 1Pc1p 6772 +P cpp 6773 ~R cer 6776 Rcnr 6777 0Rc0r 6778 ℝcr 7270 0cc0 7271 1c1 7272 + caddc 7274 <ℝ cltrr 7275 · cmul 7276 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-13 1447 ax-14 1448 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-coll 3922 ax-sep 3925 ax-nul 3933 ax-pow 3977 ax-pr 4003 ax-un 4227 ax-setind 4319 ax-iinf 4369 |
This theorem depends on definitions: df-bi 115 df-dc 779 df-3or 923 df-3an 924 df-tru 1290 df-fal 1293 df-nf 1393 df-sb 1690 df-eu 1948 df-mo 1949 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ne 2252 df-ral 2360 df-rex 2361 df-reu 2362 df-rmo 2363 df-rab 2364 df-v 2616 df-sbc 2829 df-csb 2922 df-dif 2988 df-un 2990 df-in 2992 df-ss 2999 df-nul 3273 df-pw 3411 df-sn 3431 df-pr 3432 df-op 3434 df-uni 3631 df-int 3666 df-iun 3709 df-br 3815 df-opab 3869 df-mpt 3870 df-tr 3905 df-eprel 4083 df-id 4087 df-po 4090 df-iso 4091 df-iord 4160 df-on 4162 df-suc 4165 df-iom 4372 df-xp 4410 df-rel 4411 df-cnv 4412 df-co 4413 df-dm 4414 df-rn 4415 df-res 4416 df-ima 4417 df-iota 4937 df-fun 4974 df-fn 4975 df-f 4976 df-f1 4977 df-fo 4978 df-f1o 4979 df-fv 4980 df-riota 5550 df-ov 5597 df-oprab 5598 df-mpt2 5599 df-1st 5849 df-2nd 5850 df-recs 6005 df-irdg 6070 df-1o 6116 df-2o 6117 df-oadd 6120 df-omul 6121 df-er 6225 df-ec 6227 df-qs 6231 df-ni 6784 df-pli 6785 df-mi 6786 df-lti 6787 df-plpq 6824 df-mpq 6825 df-enq 6827 df-nqqs 6828 df-plqqs 6829 df-mqqs 6830 df-1nqqs 6831 df-rq 6832 df-ltnqqs 6833 df-enq0 6904 df-nq0 6905 df-0nq0 6906 df-plq0 6907 df-mq0 6908 df-inp 6946 df-i1p 6947 df-iplp 6948 df-imp 6949 df-iltp 6950 df-enr 7193 df-nr 7194 df-plr 7195 df-mr 7196 df-ltr 7197 df-0r 7198 df-1r 7199 df-m1r 7200 df-c 7277 df-0 7278 df-1 7279 df-r 7281 df-add 7282 df-mul 7283 df-lt 7284 |
This theorem is referenced by: (None) |
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