![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 0.999... | Structured version Visualization version 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, according to ZF set theory. Interestingly, about 40% of the people responding to a poll at http://forum.physorg.com/index.php?showtopic=13177 disagree. (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 12364 | . . . . 5 ⊢ 9 ∈ ℂ | |
2 | 10re 12748 | . . . . . . 7 ⊢ ;10 ∈ ℝ | |
3 | 2 | recni 11278 | . . . . . 6 ⊢ ;10 ∈ ℂ |
4 | nnnn0 12531 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0) | |
5 | expcl 14099 | . . . . . 6 ⊢ ((;10 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (;10↑𝑘) ∈ ℂ) | |
6 | 3, 4, 5 | sylancr 585 | . . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) ∈ ℂ) |
7 | 3 | a1i 11 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 ∈ ℂ) |
8 | 10pos 12746 | . . . . . . . 8 ⊢ 0 < ;10 | |
9 | 2, 8 | gt0ne0ii 11800 | . . . . . . 7 ⊢ ;10 ≠ 0 |
10 | 9 | a1i 11 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 ≠ 0) |
11 | nnz 12631 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ) | |
12 | 7, 10, 11 | expne0d 14171 | . . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) ≠ 0) |
13 | divrec 11939 | . . . . 5 ⊢ ((9 ∈ ℂ ∧ (;10↑𝑘) ∈ ℂ ∧ (;10↑𝑘) ≠ 0) → (9 / (;10↑𝑘)) = (9 · (1 / (;10↑𝑘)))) | |
14 | 1, 6, 12, 13 | mp3an2i 1463 | . . . 4 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · (1 / (;10↑𝑘)))) |
15 | 7, 10, 11 | exprecd 14173 | . . . . 5 ⊢ (𝑘 ∈ ℕ → ((1 / ;10)↑𝑘) = (1 / (;10↑𝑘))) |
16 | 15 | oveq2d 7440 | . . . 4 ⊢ (𝑘 ∈ ℕ → (9 · ((1 / ;10)↑𝑘)) = (9 · (1 / (;10↑𝑘)))) |
17 | 14, 16 | eqtr4d 2769 | . . 3 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · ((1 / ;10)↑𝑘))) |
18 | 17 | sumeq2i 15703 | . 2 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) |
19 | 2, 9 | rereccli 12030 | . . . . 5 ⊢ (1 / ;10) ∈ ℝ |
20 | 19 | recni 11278 | . . . 4 ⊢ (1 / ;10) ∈ ℂ |
21 | 0re 11266 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
22 | 2, 8 | recgt0ii 12172 | . . . . . . 7 ⊢ 0 < (1 / ;10) |
23 | 21, 19, 22 | ltleii 11387 | . . . . . 6 ⊢ 0 ≤ (1 / ;10) |
24 | 19 | absidi 15382 | . . . . . 6 ⊢ (0 ≤ (1 / ;10) → (abs‘(1 / ;10)) = (1 / ;10)) |
25 | 23, 24 | ax-mp 5 | . . . . 5 ⊢ (abs‘(1 / ;10)) = (1 / ;10) |
26 | 1lt10 12868 | . . . . . 6 ⊢ 1 < ;10 | |
27 | recgt1 12162 | . . . . . . 7 ⊢ ((;10 ∈ ℝ ∧ 0 < ;10) → (1 < ;10 ↔ (1 / ;10) < 1)) | |
28 | 2, 8, 27 | mp2an 690 | . . . . . 6 ⊢ (1 < ;10 ↔ (1 / ;10) < 1) |
29 | 26, 28 | mpbi 229 | . . . . 5 ⊢ (1 / ;10) < 1 |
30 | 25, 29 | eqbrtri 5174 | . . . 4 ⊢ (abs‘(1 / ;10)) < 1 |
31 | geoisum1c 15884 | . . . 4 ⊢ ((9 ∈ ℂ ∧ (1 / ;10) ∈ ℂ ∧ (abs‘(1 / ;10)) < 1) → Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = ((9 · (1 / ;10)) / (1 − (1 / ;10)))) | |
32 | 1, 20, 30, 31 | mp3an 1458 | . . 3 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = ((9 · (1 / ;10)) / (1 − (1 / ;10))) |
33 | 1, 3, 9 | divreci 12010 | . . . 4 ⊢ (9 / ;10) = (9 · (1 / ;10)) |
34 | 1, 3, 9 | divcan2i 12008 | . . . . . 6 ⊢ (;10 · (9 / ;10)) = 9 |
35 | ax-1cn 11216 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
36 | 3, 35, 20 | subdii 11713 | . . . . . . 7 ⊢ (;10 · (1 − (1 / ;10))) = ((;10 · 1) − (;10 · (1 / ;10))) |
37 | 3 | mulridi 11268 | . . . . . . . 8 ⊢ (;10 · 1) = ;10 |
38 | 3, 9 | recidi 11996 | . . . . . . . 8 ⊢ (;10 · (1 / ;10)) = 1 |
39 | 37, 38 | oveq12i 7436 | . . . . . . 7 ⊢ ((;10 · 1) − (;10 · (1 / ;10))) = (;10 − 1) |
40 | 10m1e9 12825 | . . . . . . 7 ⊢ (;10 − 1) = 9 | |
41 | 36, 39, 40 | 3eqtrri 2759 | . . . . . 6 ⊢ 9 = (;10 · (1 − (1 / ;10))) |
42 | 34, 41 | eqtri 2754 | . . . . 5 ⊢ (;10 · (9 / ;10)) = (;10 · (1 − (1 / ;10))) |
43 | 9re 12363 | . . . . . . . 8 ⊢ 9 ∈ ℝ | |
44 | 43, 2, 9 | redivcli 12032 | . . . . . . 7 ⊢ (9 / ;10) ∈ ℝ |
45 | 44 | recni 11278 | . . . . . 6 ⊢ (9 / ;10) ∈ ℂ |
46 | 35, 20 | subcli 11586 | . . . . . 6 ⊢ (1 − (1 / ;10)) ∈ ℂ |
47 | 45, 46, 3, 9 | mulcani 11903 | . . . . 5 ⊢ ((;10 · (9 / ;10)) = (;10 · (1 − (1 / ;10))) ↔ (9 / ;10) = (1 − (1 / ;10))) |
48 | 42, 47 | mpbi 229 | . . . 4 ⊢ (9 / ;10) = (1 − (1 / ;10)) |
49 | 33, 48 | oveq12i 7436 | . . 3 ⊢ ((9 / ;10) / (9 / ;10)) = ((9 · (1 / ;10)) / (1 − (1 / ;10))) |
50 | 9pos 12377 | . . . . . 6 ⊢ 0 < 9 | |
51 | 43, 2, 50, 8 | divgt0ii 12183 | . . . . 5 ⊢ 0 < (9 / ;10) |
52 | 44, 51 | gt0ne0ii 11800 | . . . 4 ⊢ (9 / ;10) ≠ 0 |
53 | 45, 52 | dividi 11998 | . . 3 ⊢ ((9 / ;10) / (9 / ;10)) = 1 |
54 | 32, 49, 53 | 3eqtr2i 2760 | . 2 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = 1 |
55 | 18, 54 | eqtri 2754 | 1 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = 1 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 class class class wbr 5153 ‘cfv 6554 (class class class)co 7424 ℂcc 11156 ℝcr 11157 0cc0 11158 1c1 11159 · cmul 11163 < clt 11298 ≤ cle 11299 − cmin 11494 / cdiv 11921 ℕcn 12264 9c9 12326 ℕ0cn0 12524 ;cdc 12729 ↑cexp 14081 abscabs 15239 Σcsu 15690 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-er 8734 df-pm 8858 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-sup 9485 df-inf 9486 df-oi 9553 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12611 df-dec 12730 df-uz 12875 df-rp 13029 df-fz 13539 df-fzo 13682 df-fl 13812 df-seq 14022 df-exp 14082 df-hash 14348 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-clim 15490 df-rlim 15491 df-sum 15691 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |