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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 7665. (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 3898 | . . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑙 → (𝑏 <Q [〈𝑗, 1o〉] ~Q ↔ 𝑙 <Q [〈𝑗, 1o〉] ~Q )) | |
5 | 4 | cbvabv 2238 | . . . . . . . . . . . 12 ⊢ {𝑏 ∣ 𝑏 <Q [〈𝑗, 1o〉] ~Q } = {𝑙 ∣ 𝑙 <Q [〈𝑗, 1o〉] ~Q } |
6 | breq2 3899 | . . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑢 → ([〈𝑗, 1o〉] ~Q <Q 𝑐 ↔ [〈𝑗, 1o〉] ~Q <Q 𝑢)) | |
7 | 6 | cbvabv 2238 | . . . . . . . . . . . 12 ⊢ {𝑐 ∣ [〈𝑗, 1o〉] ~Q <Q 𝑐} = {𝑢 ∣ [〈𝑗, 1o〉] ~Q <Q 𝑢} |
8 | 5, 7 | opeq12i 3676 | . . . . . . . . . . 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 5738 | . . . . . . . . . 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 3674 | . . . . . . . . 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 6418 | . . . . . . . . 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 3674 | . . . . . . 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 5378 | . . . . . 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 3671 | . . . . 5 ⊢ (𝑎 = 𝑧 → 〈𝑎, 0R〉 = 〈𝑧, 0R〉) | |
17 | 15, 16 | eqeq12d 2129 | . . . 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 5695 | . . 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 3975 | . 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 7634 | 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 1314 ∈ wcel 1463 {cab 2101 ∀wral 2390 ∃wrex 2391 〈cop 3496 ∩ cint 3737 class class class wbr 3895 ↦ cmpt 3949 ⟶wf 5077 ‘cfv 5081 ℩crio 5683 (class class class)co 5728 1oc1o 6260 [cec 6381 Ncnpi 7028 ~Q ceq 7035 <Q cltq 7041 1Pc1p 7048 +P cpp 7049 ~R cer 7052 Rcnr 7053 0Rc0r 7054 ℝcr 7546 0cc0 7547 1c1 7548 + caddc 7550 <ℝ cltrr 7551 · cmul 7552 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-coll 4003 ax-sep 4006 ax-nul 4014 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-setind 4412 ax-iinf 4462 |
This theorem depends on definitions: df-bi 116 df-dc 803 df-3or 946 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-ral 2395 df-rex 2396 df-reu 2397 df-rmo 2398 df-rab 2399 df-v 2659 df-sbc 2879 df-csb 2972 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-nul 3330 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-int 3738 df-iun 3781 df-br 3896 df-opab 3950 df-mpt 3951 df-tr 3987 df-eprel 4171 df-id 4175 df-po 4178 df-iso 4179 df-iord 4248 df-on 4250 df-suc 4253 df-iom 4465 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-rn 4510 df-res 4511 df-ima 4512 df-iota 5046 df-fun 5083 df-fn 5084 df-f 5085 df-f1 5086 df-fo 5087 df-f1o 5088 df-fv 5089 df-riota 5684 df-ov 5731 df-oprab 5732 df-mpo 5733 df-1st 5992 df-2nd 5993 df-recs 6156 df-irdg 6221 df-1o 6267 df-2o 6268 df-oadd 6271 df-omul 6272 df-er 6383 df-ec 6385 df-qs 6389 df-ni 7060 df-pli 7061 df-mi 7062 df-lti 7063 df-plpq 7100 df-mpq 7101 df-enq 7103 df-nqqs 7104 df-plqqs 7105 df-mqqs 7106 df-1nqqs 7107 df-rq 7108 df-ltnqqs 7109 df-enq0 7180 df-nq0 7181 df-0nq0 7182 df-plq0 7183 df-mq0 7184 df-inp 7222 df-i1p 7223 df-iplp 7224 df-imp 7225 df-iltp 7226 df-enr 7469 df-nr 7470 df-plr 7471 df-mr 7472 df-ltr 7473 df-0r 7474 df-1r 7475 df-m1r 7476 df-c 7553 df-0 7554 df-1 7555 df-r 7557 df-add 7558 df-mul 7559 df-lt 7560 |
This theorem is referenced by: (None) |
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