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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 8508 | . . . . . 6 ⊢ 9 ∈ ℂ | |
2 | 1 | a1i 9 | . . . . 5 ⊢ (𝑘 ∈ ℕ → 9 ∈ ℂ) |
3 | 10re 8893 | . . . . . . . 8 ⊢ ;10 ∈ ℝ | |
4 | 3 | recni 7498 | . . . . . . 7 ⊢ ;10 ∈ ℂ |
5 | 4 | a1i 9 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 ∈ ℂ) |
6 | nnnn0 8678 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0) | |
7 | 5, 6 | expcld 10082 | . . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) ∈ ℂ) |
8 | 10pos 8891 | . . . . . . . 8 ⊢ 0 < ;10 | |
9 | 3, 8 | gt0ap0ii 8102 | . . . . . . 7 ⊢ ;10 # 0 |
10 | 9 | a1i 9 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 # 0) |
11 | nnz 8767 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ) | |
12 | 5, 10, 11 | expap0d 10088 | . . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) # 0) |
13 | 2, 7, 12 | divrecapd 8258 | . . . 4 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · (1 / (;10↑𝑘)))) |
14 | 5, 10, 11 | exprecapd 10090 | . . . . 5 ⊢ (𝑘 ∈ ℕ → ((1 / ;10)↑𝑘) = (1 / (;10↑𝑘))) |
15 | 14 | oveq2d 5668 | . . . 4 ⊢ (𝑘 ∈ ℕ → (9 · ((1 / ;10)↑𝑘)) = (9 · (1 / (;10↑𝑘)))) |
16 | 13, 15 | eqtr4d 2123 | . . 3 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · ((1 / ;10)↑𝑘))) |
17 | 16 | sumeq2i 10749 | . 2 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) |
18 | 3, 9 | rerecclapi 8242 | . . . . 5 ⊢ (1 / ;10) ∈ ℝ |
19 | 18 | recni 7498 | . . . 4 ⊢ (1 / ;10) ∈ ℂ |
20 | 0re 7486 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
21 | 3, 8 | recgt0ii 8366 | . . . . . . 7 ⊢ 0 < (1 / ;10) |
22 | 20, 18, 21 | ltleii 7585 | . . . . . 6 ⊢ 0 ≤ (1 / ;10) |
23 | 18 | absidi 10555 | . . . . . 6 ⊢ (0 ≤ (1 / ;10) → (abs‘(1 / ;10)) = (1 / ;10)) |
24 | 22, 23 | ax-mp 7 | . . . . 5 ⊢ (abs‘(1 / ;10)) = (1 / ;10) |
25 | 1lt10 9013 | . . . . . 6 ⊢ 1 < ;10 | |
26 | recgt1 8356 | . . . . . . 7 ⊢ ((;10 ∈ ℝ ∧ 0 < ;10) → (1 < ;10 ↔ (1 / ;10) < 1)) | |
27 | 3, 8, 26 | mp2an 417 | . . . . . 6 ⊢ (1 < ;10 ↔ (1 / ;10) < 1) |
28 | 25, 27 | mpbi 143 | . . . . 5 ⊢ (1 / ;10) < 1 |
29 | 24, 28 | eqbrtri 3864 | . . . 4 ⊢ (abs‘(1 / ;10)) < 1 |
30 | geoisum1c 10910 | . . . 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 1273 | . . 3 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = ((9 · (1 / ;10)) / (1 − (1 / ;10))) |
32 | 1, 4, 9 | divrecapi 8222 | . . . 4 ⊢ (9 / ;10) = (9 · (1 / ;10)) |
33 | 1, 4, 9 | divcanap2i 8220 | . . . . . 6 ⊢ (;10 · (9 / ;10)) = 9 |
34 | ax-1cn 7436 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
35 | 4, 34, 19 | subdii 7883 | . . . . . . 7 ⊢ (;10 · (1 − (1 / ;10))) = ((;10 · 1) − (;10 · (1 / ;10))) |
36 | 4 | mulid1i 7488 | . . . . . . . 8 ⊢ (;10 · 1) = ;10 |
37 | 4, 9 | recidapi 8208 | . . . . . . . 8 ⊢ (;10 · (1 / ;10)) = 1 |
38 | 36, 37 | oveq12i 5664 | . . . . . . 7 ⊢ ((;10 · 1) − (;10 · (1 / ;10))) = (;10 − 1) |
39 | 10m1e9 8970 | . . . . . . 7 ⊢ (;10 − 1) = 9 | |
40 | 35, 38, 39 | 3eqtrri 2113 | . . . . . 6 ⊢ 9 = (;10 · (1 − (1 / ;10))) |
41 | 33, 40 | eqtri 2108 | . . . . 5 ⊢ (;10 · (9 / ;10)) = (;10 · (1 − (1 / ;10))) |
42 | 9re 8507 | . . . . . . . 8 ⊢ 9 ∈ ℝ | |
43 | 42, 3, 9 | redivclapi 8244 | . . . . . . 7 ⊢ (9 / ;10) ∈ ℝ |
44 | 43 | recni 7498 | . . . . . 6 ⊢ (9 / ;10) ∈ ℂ |
45 | 34, 19 | subcli 7756 | . . . . . 6 ⊢ (1 − (1 / ;10)) ∈ ℂ |
46 | 44, 45, 4, 9 | mulcanapi 8134 | . . . . 5 ⊢ ((;10 · (9 / ;10)) = (;10 · (1 − (1 / ;10))) ↔ (9 / ;10) = (1 − (1 / ;10))) |
47 | 41, 46 | mpbi 143 | . . . 4 ⊢ (9 / ;10) = (1 − (1 / ;10)) |
48 | 32, 47 | oveq12i 5664 | . . 3 ⊢ ((9 / ;10) / (9 / ;10)) = ((9 · (1 / ;10)) / (1 − (1 / ;10))) |
49 | 9pos 8524 | . . . . . 6 ⊢ 0 < 9 | |
50 | 42, 3, 49, 8 | divgt0ii 8378 | . . . . 5 ⊢ 0 < (9 / ;10) |
51 | 43, 50 | gt0ap0ii 8102 | . . . 4 ⊢ (9 / ;10) # 0 |
52 | 44, 51 | dividapi 8210 | . . 3 ⊢ ((9 / ;10) / (9 / ;10)) = 1 |
53 | 31, 48, 52 | 3eqtr2i 2114 | . 2 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = 1 |
54 | 17, 53 | eqtri 2108 | 1 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = 1 |
Colors of variables: wff set class |
Syntax hints: ↔ wb 103 = wceq 1289 ∈ wcel 1438 class class class wbr 3845 ‘cfv 5015 (class class class)co 5652 ℂcc 7346 ℝcr 7347 0cc0 7348 1c1 7349 · cmul 7353 < clt 7520 ≤ cle 7521 − cmin 7651 # cap 8056 / cdiv 8137 ℕcn 8420 9c9 8478 ;cdc 8875 ↑cexp 9950 abscabs 10426 Σcsu 10738 |
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 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-coll 3954 ax-sep 3957 ax-nul 3965 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-setind 4353 ax-iinf 4403 ax-cnex 7434 ax-resscn 7435 ax-1cn 7436 ax-1re 7437 ax-icn 7438 ax-addcl 7439 ax-addrcl 7440 ax-mulcl 7441 ax-mulrcl 7442 ax-addcom 7443 ax-mulcom 7444 ax-addass 7445 ax-mulass 7446 ax-distr 7447 ax-i2m1 7448 ax-0lt1 7449 ax-1rid 7450 ax-0id 7451 ax-rnegex 7452 ax-precex 7453 ax-cnre 7454 ax-pre-ltirr 7455 ax-pre-ltwlin 7456 ax-pre-lttrn 7457 ax-pre-apti 7458 ax-pre-ltadd 7459 ax-pre-mulgt0 7460 ax-pre-mulext 7461 ax-arch 7462 ax-caucvg 7463 |
This theorem depends on definitions: df-bi 115 df-dc 781 df-3or 925 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-nel 2351 df-ral 2364 df-rex 2365 df-reu 2366 df-rmo 2367 df-rab 2368 df-v 2621 df-sbc 2841 df-csb 2934 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-if 3394 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-int 3689 df-iun 3732 df-br 3846 df-opab 3900 df-mpt 3901 df-tr 3937 df-id 4120 df-po 4123 df-iso 4124 df-iord 4193 df-on 4195 df-ilim 4196 df-suc 4198 df-iom 4406 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-res 4450 df-ima 4451 df-iota 4980 df-fun 5017 df-fn 5018 df-f 5019 df-f1 5020 df-fo 5021 df-f1o 5022 df-fv 5023 df-isom 5024 df-riota 5608 df-ov 5655 df-oprab 5656 df-mpt2 5657 df-1st 5911 df-2nd 5912 df-recs 6070 df-irdg 6135 df-frec 6156 df-1o 6181 df-oadd 6185 df-er 6290 df-en 6456 df-dom 6457 df-fin 6458 df-pnf 7522 df-mnf 7523 df-xr 7524 df-ltxr 7525 df-le 7526 df-sub 7653 df-neg 7654 df-reap 8050 df-ap 8057 df-div 8138 df-inn 8421 df-2 8479 df-3 8480 df-4 8481 df-5 8482 df-6 8483 df-7 8484 df-8 8485 df-9 8486 df-n0 8672 df-z 8749 df-dec 8876 df-uz 9018 df-q 9103 df-rp 9133 df-fz 9423 df-fzo 9550 df-iseq 9849 df-seq3 9850 df-exp 9951 df-ihash 10180 df-cj 10272 df-re 10273 df-im 10274 df-rsqrt 10427 df-abs 10428 df-clim 10663 df-isum 10739 |
This theorem is referenced by: (None) |
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