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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 12366 | . . . . 5 ⊢ 9 ∈ ℂ | |
| 2 | 10re 12752 | . . . . . . 7 ⊢ ;10 ∈ ℝ | |
| 3 | 2 | recni 11275 | . . . . . 6 ⊢ ;10 ∈ ℂ |
| 4 | nnnn0 12533 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0) | |
| 5 | expcl 14120 | . . . . . 6 ⊢ ((;10 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (;10↑𝑘) ∈ ℂ) | |
| 6 | 3, 4, 5 | sylancr 587 | . . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) ∈ ℂ) |
| 7 | 3 | a1i 11 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 ∈ ℂ) |
| 8 | 10pos 12750 | . . . . . . . 8 ⊢ 0 < ;10 | |
| 9 | 2, 8 | gt0ne0ii 11799 | . . . . . . 7 ⊢ ;10 ≠ 0 |
| 10 | 9 | a1i 11 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 ≠ 0) |
| 11 | nnz 12634 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ) | |
| 12 | 7, 10, 11 | expne0d 14192 | . . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) ≠ 0) |
| 13 | divrec 11938 | . . . . 5 ⊢ ((9 ∈ ℂ ∧ (;10↑𝑘) ∈ ℂ ∧ (;10↑𝑘) ≠ 0) → (9 / (;10↑𝑘)) = (9 · (1 / (;10↑𝑘)))) | |
| 14 | 1, 6, 12, 13 | mp3an2i 1468 | . . . 4 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · (1 / (;10↑𝑘)))) |
| 15 | 7, 10, 11 | exprecd 14194 | . . . . 5 ⊢ (𝑘 ∈ ℕ → ((1 / ;10)↑𝑘) = (1 / (;10↑𝑘))) |
| 16 | 15 | oveq2d 7447 | . . . 4 ⊢ (𝑘 ∈ ℕ → (9 · ((1 / ;10)↑𝑘)) = (9 · (1 / (;10↑𝑘)))) |
| 17 | 14, 16 | eqtr4d 2780 | . . 3 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · ((1 / ;10)↑𝑘))) |
| 18 | 17 | sumeq2i 15734 | . 2 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) |
| 19 | 2, 9 | rereccli 12032 | . . . . 5 ⊢ (1 / ;10) ∈ ℝ |
| 20 | 19 | recni 11275 | . . . 4 ⊢ (1 / ;10) ∈ ℂ |
| 21 | 0re 11263 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
| 22 | 2, 8 | recgt0ii 12174 | . . . . . . 7 ⊢ 0 < (1 / ;10) |
| 23 | 21, 19, 22 | ltleii 11384 | . . . . . 6 ⊢ 0 ≤ (1 / ;10) |
| 24 | 19 | absidi 15416 | . . . . . 6 ⊢ (0 ≤ (1 / ;10) → (abs‘(1 / ;10)) = (1 / ;10)) |
| 25 | 23, 24 | ax-mp 5 | . . . . 5 ⊢ (abs‘(1 / ;10)) = (1 / ;10) |
| 26 | 1lt10 12872 | . . . . . 6 ⊢ 1 < ;10 | |
| 27 | recgt1 12164 | . . . . . . 7 ⊢ ((;10 ∈ ℝ ∧ 0 < ;10) → (1 < ;10 ↔ (1 / ;10) < 1)) | |
| 28 | 2, 8, 27 | mp2an 692 | . . . . . 6 ⊢ (1 < ;10 ↔ (1 / ;10) < 1) |
| 29 | 26, 28 | mpbi 230 | . . . . 5 ⊢ (1 / ;10) < 1 |
| 30 | 25, 29 | eqbrtri 5164 | . . . 4 ⊢ (abs‘(1 / ;10)) < 1 |
| 31 | geoisum1c 15916 | . . . 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 1463 | . . 3 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = ((9 · (1 / ;10)) / (1 − (1 / ;10))) |
| 33 | 1, 3, 9 | divreci 12012 | . . . 4 ⊢ (9 / ;10) = (9 · (1 / ;10)) |
| 34 | 1, 3, 9 | divcan2i 12010 | . . . . . 6 ⊢ (;10 · (9 / ;10)) = 9 |
| 35 | ax-1cn 11213 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
| 36 | 3, 35, 20 | subdii 11712 | . . . . . . 7 ⊢ (;10 · (1 − (1 / ;10))) = ((;10 · 1) − (;10 · (1 / ;10))) |
| 37 | 3 | mulridi 11265 | . . . . . . . 8 ⊢ (;10 · 1) = ;10 |
| 38 | 3, 9 | recidi 11998 | . . . . . . . 8 ⊢ (;10 · (1 / ;10)) = 1 |
| 39 | 37, 38 | oveq12i 7443 | . . . . . . 7 ⊢ ((;10 · 1) − (;10 · (1 / ;10))) = (;10 − 1) |
| 40 | 10m1e9 12829 | . . . . . . 7 ⊢ (;10 − 1) = 9 | |
| 41 | 36, 39, 40 | 3eqtrri 2770 | . . . . . 6 ⊢ 9 = (;10 · (1 − (1 / ;10))) |
| 42 | 34, 41 | eqtri 2765 | . . . . 5 ⊢ (;10 · (9 / ;10)) = (;10 · (1 − (1 / ;10))) |
| 43 | 9re 12365 | . . . . . . . 8 ⊢ 9 ∈ ℝ | |
| 44 | 43, 2, 9 | redivcli 12034 | . . . . . . 7 ⊢ (9 / ;10) ∈ ℝ |
| 45 | 44 | recni 11275 | . . . . . 6 ⊢ (9 / ;10) ∈ ℂ |
| 46 | 35, 20 | subcli 11585 | . . . . . 6 ⊢ (1 − (1 / ;10)) ∈ ℂ |
| 47 | 45, 46, 3, 9 | mulcani 11902 | . . . . 5 ⊢ ((;10 · (9 / ;10)) = (;10 · (1 − (1 / ;10))) ↔ (9 / ;10) = (1 − (1 / ;10))) |
| 48 | 42, 47 | mpbi 230 | . . . 4 ⊢ (9 / ;10) = (1 − (1 / ;10)) |
| 49 | 33, 48 | oveq12i 7443 | . . 3 ⊢ ((9 / ;10) / (9 / ;10)) = ((9 · (1 / ;10)) / (1 − (1 / ;10))) |
| 50 | 9pos 12379 | . . . . . 6 ⊢ 0 < 9 | |
| 51 | 43, 2, 50, 8 | divgt0ii 12185 | . . . . 5 ⊢ 0 < (9 / ;10) |
| 52 | 44, 51 | gt0ne0ii 11799 | . . . 4 ⊢ (9 / ;10) ≠ 0 |
| 53 | 45, 52 | dividi 12000 | . . 3 ⊢ ((9 / ;10) / (9 / ;10)) = 1 |
| 54 | 32, 49, 53 | 3eqtr2i 2771 | . 2 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = 1 |
| 55 | 18, 54 | eqtri 2765 | 1 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = 1 |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 · cmul 11160 < clt 11295 ≤ cle 11296 − cmin 11492 / cdiv 11920 ℕcn 12266 9c9 12328 ℕ0cn0 12526 ;cdc 12733 ↑cexp 14102 abscabs 15273 Σcsu 15722 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-rlim 15525 df-sum 15723 |
| This theorem is referenced by: (None) |
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