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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 8811 | . . . . . 6 ⊢ 9 ∈ ℂ | |
2 | 1 | a1i 9 | . . . . 5 ⊢ (𝑘 ∈ ℕ → 9 ∈ ℂ) |
3 | 10re 9203 | . . . . . . . 8 ⊢ ;10 ∈ ℝ | |
4 | 3 | recni 7781 | . . . . . . 7 ⊢ ;10 ∈ ℂ |
5 | 4 | a1i 9 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 ∈ ℂ) |
6 | nnnn0 8987 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0) | |
7 | 5, 6 | expcld 10427 | . . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) ∈ ℂ) |
8 | 10pos 9201 | . . . . . . . 8 ⊢ 0 < ;10 | |
9 | 3, 8 | gt0ap0ii 8393 | . . . . . . 7 ⊢ ;10 # 0 |
10 | 9 | a1i 9 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 # 0) |
11 | nnz 9076 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ) | |
12 | 5, 10, 11 | expap0d 10433 | . . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) # 0) |
13 | 2, 7, 12 | divrecapd 8556 | . . . 4 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · (1 / (;10↑𝑘)))) |
14 | 5, 10, 11 | exprecapd 10435 | . . . . 5 ⊢ (𝑘 ∈ ℕ → ((1 / ;10)↑𝑘) = (1 / (;10↑𝑘))) |
15 | 14 | oveq2d 5790 | . . . 4 ⊢ (𝑘 ∈ ℕ → (9 · ((1 / ;10)↑𝑘)) = (9 · (1 / (;10↑𝑘)))) |
16 | 13, 15 | eqtr4d 2175 | . . 3 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · ((1 / ;10)↑𝑘))) |
17 | 16 | sumeq2i 11136 | . 2 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) |
18 | 3, 9 | rerecclapi 8540 | . . . . 5 ⊢ (1 / ;10) ∈ ℝ |
19 | 18 | recni 7781 | . . . 4 ⊢ (1 / ;10) ∈ ℂ |
20 | 0re 7769 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
21 | 3, 8 | recgt0ii 8668 | . . . . . . 7 ⊢ 0 < (1 / ;10) |
22 | 20, 18, 21 | ltleii 7869 | . . . . . 6 ⊢ 0 ≤ (1 / ;10) |
23 | 18 | absidi 10901 | . . . . . 6 ⊢ (0 ≤ (1 / ;10) → (abs‘(1 / ;10)) = (1 / ;10)) |
24 | 22, 23 | ax-mp 5 | . . . . 5 ⊢ (abs‘(1 / ;10)) = (1 / ;10) |
25 | 1lt10 9323 | . . . . . 6 ⊢ 1 < ;10 | |
26 | recgt1 8658 | . . . . . . 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 3949 | . . . 4 ⊢ (abs‘(1 / ;10)) < 1 |
30 | geoisum1c 11292 | . . . 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 1315 | . . 3 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = ((9 · (1 / ;10)) / (1 − (1 / ;10))) |
32 | 1, 4, 9 | divrecapi 8520 | . . . 4 ⊢ (9 / ;10) = (9 · (1 / ;10)) |
33 | 1, 4, 9 | divcanap2i 8518 | . . . . . 6 ⊢ (;10 · (9 / ;10)) = 9 |
34 | ax-1cn 7716 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
35 | 4, 34, 19 | subdii 8172 | . . . . . . 7 ⊢ (;10 · (1 − (1 / ;10))) = ((;10 · 1) − (;10 · (1 / ;10))) |
36 | 4 | mulid1i 7771 | . . . . . . . 8 ⊢ (;10 · 1) = ;10 |
37 | 4, 9 | recidapi 8506 | . . . . . . . 8 ⊢ (;10 · (1 / ;10)) = 1 |
38 | 36, 37 | oveq12i 5786 | . . . . . . 7 ⊢ ((;10 · 1) − (;10 · (1 / ;10))) = (;10 − 1) |
39 | 10m1e9 9280 | . . . . . . 7 ⊢ (;10 − 1) = 9 | |
40 | 35, 38, 39 | 3eqtrri 2165 | . . . . . 6 ⊢ 9 = (;10 · (1 − (1 / ;10))) |
41 | 33, 40 | eqtri 2160 | . . . . 5 ⊢ (;10 · (9 / ;10)) = (;10 · (1 − (1 / ;10))) |
42 | 9re 8810 | . . . . . . . 8 ⊢ 9 ∈ ℝ | |
43 | 42, 3, 9 | redivclapi 8542 | . . . . . . 7 ⊢ (9 / ;10) ∈ ℝ |
44 | 43 | recni 7781 | . . . . . 6 ⊢ (9 / ;10) ∈ ℂ |
45 | 34, 19 | subcli 8041 | . . . . . 6 ⊢ (1 − (1 / ;10)) ∈ ℂ |
46 | 44, 45, 4, 9 | mulcanapi 8431 | . . . . 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 5786 | . . 3 ⊢ ((9 / ;10) / (9 / ;10)) = ((9 · (1 / ;10)) / (1 − (1 / ;10))) |
49 | 9pos 8827 | . . . . . 6 ⊢ 0 < 9 | |
50 | 42, 3, 49, 8 | divgt0ii 8680 | . . . . 5 ⊢ 0 < (9 / ;10) |
51 | 43, 50 | gt0ap0ii 8393 | . . . 4 ⊢ (9 / ;10) # 0 |
52 | 44, 51 | dividapi 8508 | . . 3 ⊢ ((9 / ;10) / (9 / ;10)) = 1 |
53 | 31, 48, 52 | 3eqtr2i 2166 | . 2 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = 1 |
54 | 17, 53 | eqtri 2160 | 1 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = 1 |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1331 ∈ wcel 1480 class class class wbr 3929 ‘cfv 5123 (class class class)co 5774 ℂcc 7621 ℝcr 7622 0cc0 7623 1c1 7624 · cmul 7628 < clt 7803 ≤ cle 7804 − cmin 7936 # cap 8346 / cdiv 8435 ℕcn 8723 9c9 8781 ;cdc 9185 ↑cexp 10295 abscabs 10772 Σcsu 11125 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7714 ax-resscn 7715 ax-1cn 7716 ax-1re 7717 ax-icn 7718 ax-addcl 7719 ax-addrcl 7720 ax-mulcl 7721 ax-mulrcl 7722 ax-addcom 7723 ax-mulcom 7724 ax-addass 7725 ax-mulass 7726 ax-distr 7727 ax-i2m1 7728 ax-0lt1 7729 ax-1rid 7730 ax-0id 7731 ax-rnegex 7732 ax-precex 7733 ax-cnre 7734 ax-pre-ltirr 7735 ax-pre-ltwlin 7736 ax-pre-lttrn 7737 ax-pre-apti 7738 ax-pre-ltadd 7739 ax-pre-mulgt0 7740 ax-pre-mulext 7741 ax-arch 7742 ax-caucvg 7743 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-frec 6288 df-1o 6313 df-oadd 6317 df-er 6429 df-en 6635 df-dom 6636 df-fin 6637 df-pnf 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-le 7809 df-sub 7938 df-neg 7939 df-reap 8340 df-ap 8347 df-div 8436 df-inn 8724 df-2 8782 df-3 8783 df-4 8784 df-5 8785 df-6 8786 df-7 8787 df-8 8788 df-9 8789 df-n0 8981 df-z 9058 df-dec 9186 df-uz 9330 df-q 9415 df-rp 9445 df-fz 9794 df-fzo 9923 df-seqfrec 10222 df-exp 10296 df-ihash 10525 df-cj 10617 df-re 10618 df-im 10619 df-rsqrt 10773 df-abs 10774 df-clim 11051 df-sumdc 11126 |
This theorem is referenced by: (None) |
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