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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 10985 | . . . . 5 ⊢ 9 ∈ ℂ | |
2 | 10re 11393 | . . . . . . 7 ⊢ ;10 ∈ ℝ | |
3 | 2 | recni 9931 | . . . . . 6 ⊢ ;10 ∈ ℂ |
4 | nnnn0 11176 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0) | |
5 | expcl 12740 | . . . . . 6 ⊢ ((;10 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (;10↑𝑘) ∈ ℂ) | |
6 | 3, 4, 5 | sylancr 694 | . . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) ∈ ℂ) |
7 | 3 | a1i 11 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 ∈ ℂ) |
8 | 10pos 11391 | . . . . . . . 8 ⊢ 0 < ;10 | |
9 | 2, 8 | gt0ne0ii 10443 | . . . . . . 7 ⊢ ;10 ≠ 0 |
10 | 9 | a1i 11 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 ≠ 0) |
11 | nnz 11276 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ) | |
12 | 7, 10, 11 | expne0d 12876 | . . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) ≠ 0) |
13 | divrec 10580 | . . . . 5 ⊢ ((9 ∈ ℂ ∧ (;10↑𝑘) ∈ ℂ ∧ (;10↑𝑘) ≠ 0) → (9 / (;10↑𝑘)) = (9 · (1 / (;10↑𝑘)))) | |
14 | 1, 6, 12, 13 | mp3an2i 1421 | . . . 4 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · (1 / (;10↑𝑘)))) |
15 | 7, 10, 11 | exprecd 12878 | . . . . 5 ⊢ (𝑘 ∈ ℕ → ((1 / ;10)↑𝑘) = (1 / (;10↑𝑘))) |
16 | 15 | oveq2d 6565 | . . . 4 ⊢ (𝑘 ∈ ℕ → (9 · ((1 / ;10)↑𝑘)) = (9 · (1 / (;10↑𝑘)))) |
17 | 14, 16 | eqtr4d 2647 | . . 3 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · ((1 / ;10)↑𝑘))) |
18 | 17 | sumeq2i 14277 | . 2 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) |
19 | 2, 9 | rereccli 10669 | . . . . 5 ⊢ (1 / ;10) ∈ ℝ |
20 | 19 | recni 9931 | . . . 4 ⊢ (1 / ;10) ∈ ℂ |
21 | 0re 9919 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
22 | 2, 8 | recgt0ii 10808 | . . . . . . 7 ⊢ 0 < (1 / ;10) |
23 | 21, 19, 22 | ltleii 10039 | . . . . . 6 ⊢ 0 ≤ (1 / ;10) |
24 | 19 | absidi 13965 | . . . . . 6 ⊢ (0 ≤ (1 / ;10) → (abs‘(1 / ;10)) = (1 / ;10)) |
25 | 23, 24 | ax-mp 5 | . . . . 5 ⊢ (abs‘(1 / ;10)) = (1 / ;10) |
26 | 1lt10 11557 | . . . . . 6 ⊢ 1 < ;10 | |
27 | recgt1 10798 | . . . . . . 7 ⊢ ((;10 ∈ ℝ ∧ 0 < ;10) → (1 < ;10 ↔ (1 / ;10) < 1)) | |
28 | 2, 8, 27 | mp2an 704 | . . . . . 6 ⊢ (1 < ;10 ↔ (1 / ;10) < 1) |
29 | 26, 28 | mpbi 219 | . . . . 5 ⊢ (1 / ;10) < 1 |
30 | 25, 29 | eqbrtri 4604 | . . . 4 ⊢ (abs‘(1 / ;10)) < 1 |
31 | geoisum1c 14450 | . . . 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 1416 | . . 3 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = ((9 · (1 / ;10)) / (1 − (1 / ;10))) |
33 | 1, 3, 9 | divreci 10649 | . . . 4 ⊢ (9 / ;10) = (9 · (1 / ;10)) |
34 | 1, 3, 9 | divcan2i 10647 | . . . . . 6 ⊢ (;10 · (9 / ;10)) = 9 |
35 | ax-1cn 9873 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
36 | 3, 35, 20 | subdii 10358 | . . . . . . 7 ⊢ (;10 · (1 − (1 / ;10))) = ((;10 · 1) − (;10 · (1 / ;10))) |
37 | 3 | mulid1i 9921 | . . . . . . . 8 ⊢ (;10 · 1) = ;10 |
38 | 3, 9 | recidi 10635 | . . . . . . . 8 ⊢ (;10 · (1 / ;10)) = 1 |
39 | 37, 38 | oveq12i 6561 | . . . . . . 7 ⊢ ((;10 · 1) − (;10 · (1 / ;10))) = (;10 − 1) |
40 | 10m1e9 11506 | . . . . . . 7 ⊢ (;10 − 1) = 9 | |
41 | 36, 39, 40 | 3eqtrri 2637 | . . . . . 6 ⊢ 9 = (;10 · (1 − (1 / ;10))) |
42 | 34, 41 | eqtri 2632 | . . . . 5 ⊢ (;10 · (9 / ;10)) = (;10 · (1 − (1 / ;10))) |
43 | 9re 10984 | . . . . . . . 8 ⊢ 9 ∈ ℝ | |
44 | 43, 2, 9 | redivcli 10671 | . . . . . . 7 ⊢ (9 / ;10) ∈ ℝ |
45 | 44 | recni 9931 | . . . . . 6 ⊢ (9 / ;10) ∈ ℂ |
46 | 35, 20 | subcli 10236 | . . . . . 6 ⊢ (1 − (1 / ;10)) ∈ ℂ |
47 | 45, 46, 3, 9 | mulcani 10545 | . . . . 5 ⊢ ((;10 · (9 / ;10)) = (;10 · (1 − (1 / ;10))) ↔ (9 / ;10) = (1 − (1 / ;10))) |
48 | 42, 47 | mpbi 219 | . . . 4 ⊢ (9 / ;10) = (1 − (1 / ;10)) |
49 | 33, 48 | oveq12i 6561 | . . 3 ⊢ ((9 / ;10) / (9 / ;10)) = ((9 · (1 / ;10)) / (1 − (1 / ;10))) |
50 | 9pos 10999 | . . . . . 6 ⊢ 0 < 9 | |
51 | 43, 2, 50, 8 | divgt0ii 10820 | . . . . 5 ⊢ 0 < (9 / ;10) |
52 | 44, 51 | gt0ne0ii 10443 | . . . 4 ⊢ (9 / ;10) ≠ 0 |
53 | 45, 52 | dividi 10637 | . . 3 ⊢ ((9 / ;10) / (9 / ;10)) = 1 |
54 | 32, 49, 53 | 3eqtr2i 2638 | . 2 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = 1 |
55 | 18, 54 | eqtri 2632 | 1 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = 1 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 · cmul 9820 < clt 9953 ≤ cle 9954 − cmin 10145 / cdiv 10563 ℕcn 10897 9c9 10954 ℕ0cn0 11169 ;cdc 11369 ↑cexp 12722 abscabs 13822 Σcsu 14264 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-rlim 14068 df-sum 14265 |
This theorem is referenced by: (None) |
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