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Mirrors > Home > ILE Home > Th. List > 0.999... | GIF version |
Description: The recurring decimal 0.999..., which is defined as the infinite sum 0.9 + 0.09 + 0.009 + ... i.e. 9 / 10↑1 + 9 / 10↑2 + 9 / 10↑3 + ..., is exactly equal to 1. (Contributed by NM, 2-Nov-2007.) (Revised by AV, 8-Sep-2021.) |
Ref | Expression |
---|---|
0.999... | ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 9cn 8776 | . . . . . 6 ⊢ 9 ∈ ℂ | |
2 | 1 | a1i 9 | . . . . 5 ⊢ (𝑘 ∈ ℕ → 9 ∈ ℂ) |
3 | 10re 9168 | . . . . . . . 8 ⊢ ;10 ∈ ℝ | |
4 | 3 | recni 7746 | . . . . . . 7 ⊢ ;10 ∈ ℂ |
5 | 4 | a1i 9 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 ∈ ℂ) |
6 | nnnn0 8952 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0) | |
7 | 5, 6 | expcld 10392 | . . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) ∈ ℂ) |
8 | 10pos 9166 | . . . . . . . 8 ⊢ 0 < ;10 | |
9 | 3, 8 | gt0ap0ii 8358 | . . . . . . 7 ⊢ ;10 # 0 |
10 | 9 | a1i 9 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 # 0) |
11 | nnz 9041 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ) | |
12 | 5, 10, 11 | expap0d 10398 | . . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) # 0) |
13 | 2, 7, 12 | divrecapd 8521 | . . . 4 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · (1 / (;10↑𝑘)))) |
14 | 5, 10, 11 | exprecapd 10400 | . . . . 5 ⊢ (𝑘 ∈ ℕ → ((1 / ;10)↑𝑘) = (1 / (;10↑𝑘))) |
15 | 14 | oveq2d 5758 | . . . 4 ⊢ (𝑘 ∈ ℕ → (9 · ((1 / ;10)↑𝑘)) = (9 · (1 / (;10↑𝑘)))) |
16 | 13, 15 | eqtr4d 2153 | . . 3 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · ((1 / ;10)↑𝑘))) |
17 | 16 | sumeq2i 11101 | . 2 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) |
18 | 3, 9 | rerecclapi 8505 | . . . . 5 ⊢ (1 / ;10) ∈ ℝ |
19 | 18 | recni 7746 | . . . 4 ⊢ (1 / ;10) ∈ ℂ |
20 | 0re 7734 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
21 | 3, 8 | recgt0ii 8633 | . . . . . . 7 ⊢ 0 < (1 / ;10) |
22 | 20, 18, 21 | ltleii 7834 | . . . . . 6 ⊢ 0 ≤ (1 / ;10) |
23 | 18 | absidi 10866 | . . . . . 6 ⊢ (0 ≤ (1 / ;10) → (abs‘(1 / ;10)) = (1 / ;10)) |
24 | 22, 23 | ax-mp 5 | . . . . 5 ⊢ (abs‘(1 / ;10)) = (1 / ;10) |
25 | 1lt10 9288 | . . . . . 6 ⊢ 1 < ;10 | |
26 | recgt1 8623 | . . . . . . 7 ⊢ ((;10 ∈ ℝ ∧ 0 < ;10) → (1 < ;10 ↔ (1 / ;10) < 1)) | |
27 | 3, 8, 26 | mp2an 422 | . . . . . 6 ⊢ (1 < ;10 ↔ (1 / ;10) < 1) |
28 | 25, 27 | mpbi 144 | . . . . 5 ⊢ (1 / ;10) < 1 |
29 | 24, 28 | eqbrtri 3919 | . . . 4 ⊢ (abs‘(1 / ;10)) < 1 |
30 | geoisum1c 11257 | . . . 4 ⊢ ((9 ∈ ℂ ∧ (1 / ;10) ∈ ℂ ∧ (abs‘(1 / ;10)) < 1) → Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = ((9 · (1 / ;10)) / (1 − (1 / ;10)))) | |
31 | 1, 19, 29, 30 | mp3an 1300 | . . 3 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = ((9 · (1 / ;10)) / (1 − (1 / ;10))) |
32 | 1, 4, 9 | divrecapi 8485 | . . . 4 ⊢ (9 / ;10) = (9 · (1 / ;10)) |
33 | 1, 4, 9 | divcanap2i 8483 | . . . . . 6 ⊢ (;10 · (9 / ;10)) = 9 |
34 | ax-1cn 7681 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
35 | 4, 34, 19 | subdii 8137 | . . . . . . 7 ⊢ (;10 · (1 − (1 / ;10))) = ((;10 · 1) − (;10 · (1 / ;10))) |
36 | 4 | mulid1i 7736 | . . . . . . . 8 ⊢ (;10 · 1) = ;10 |
37 | 4, 9 | recidapi 8471 | . . . . . . . 8 ⊢ (;10 · (1 / ;10)) = 1 |
38 | 36, 37 | oveq12i 5754 | . . . . . . 7 ⊢ ((;10 · 1) − (;10 · (1 / ;10))) = (;10 − 1) |
39 | 10m1e9 9245 | . . . . . . 7 ⊢ (;10 − 1) = 9 | |
40 | 35, 38, 39 | 3eqtrri 2143 | . . . . . 6 ⊢ 9 = (;10 · (1 − (1 / ;10))) |
41 | 33, 40 | eqtri 2138 | . . . . 5 ⊢ (;10 · (9 / ;10)) = (;10 · (1 − (1 / ;10))) |
42 | 9re 8775 | . . . . . . . 8 ⊢ 9 ∈ ℝ | |
43 | 42, 3, 9 | redivclapi 8507 | . . . . . . 7 ⊢ (9 / ;10) ∈ ℝ |
44 | 43 | recni 7746 | . . . . . 6 ⊢ (9 / ;10) ∈ ℂ |
45 | 34, 19 | subcli 8006 | . . . . . 6 ⊢ (1 − (1 / ;10)) ∈ ℂ |
46 | 44, 45, 4, 9 | mulcanapi 8396 | . . . . 5 ⊢ ((;10 · (9 / ;10)) = (;10 · (1 − (1 / ;10))) ↔ (9 / ;10) = (1 − (1 / ;10))) |
47 | 41, 46 | mpbi 144 | . . . 4 ⊢ (9 / ;10) = (1 − (1 / ;10)) |
48 | 32, 47 | oveq12i 5754 | . . 3 ⊢ ((9 / ;10) / (9 / ;10)) = ((9 · (1 / ;10)) / (1 − (1 / ;10))) |
49 | 9pos 8792 | . . . . . 6 ⊢ 0 < 9 | |
50 | 42, 3, 49, 8 | divgt0ii 8645 | . . . . 5 ⊢ 0 < (9 / ;10) |
51 | 43, 50 | gt0ap0ii 8358 | . . . 4 ⊢ (9 / ;10) # 0 |
52 | 44, 51 | dividapi 8473 | . . 3 ⊢ ((9 / ;10) / (9 / ;10)) = 1 |
53 | 31, 48, 52 | 3eqtr2i 2144 | . 2 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = 1 |
54 | 17, 53 | eqtri 2138 | 1 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = 1 |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1316 ∈ wcel 1465 class class class wbr 3899 ‘cfv 5093 (class class class)co 5742 ℂcc 7586 ℝcr 7587 0cc0 7588 1c1 7589 · cmul 7593 < clt 7768 ≤ cle 7769 − cmin 7901 # cap 8311 / cdiv 8400 ℕcn 8688 9c9 8746 ;cdc 9150 ↑cexp 10260 abscabs 10737 Σcsu 11090 |
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 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 ax-arch 7707 ax-caucvg 7708 |
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-nel 2381 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-if 3445 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-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-ilim 4261 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-isom 5102 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-frec 6256 df-1o 6281 df-oadd 6285 df-er 6397 df-en 6603 df-dom 6604 df-fin 6605 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8305 df-ap 8312 df-div 8401 df-inn 8689 df-2 8747 df-3 8748 df-4 8749 df-5 8750 df-6 8751 df-7 8752 df-8 8753 df-9 8754 df-n0 8946 df-z 9023 df-dec 9151 df-uz 9295 df-q 9380 df-rp 9410 df-fz 9759 df-fzo 9888 df-seqfrec 10187 df-exp 10261 df-ihash 10490 df-cj 10582 df-re 10583 df-im 10584 df-rsqrt 10738 df-abs 10739 df-clim 11016 df-sumdc 11091 |
This theorem is referenced by: (None) |
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