![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 9042 | . . . . . 6 ⊢ 9 ∈ ℂ | |
2 | 1 | a1i 9 | . . . . 5 ⊢ (𝑘 ∈ ℕ → 9 ∈ ℂ) |
3 | 10re 9437 | . . . . . . . 8 ⊢ ;10 ∈ ℝ | |
4 | 3 | recni 8004 | . . . . . . 7 ⊢ ;10 ∈ ℂ |
5 | 4 | a1i 9 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 ∈ ℂ) |
6 | nnnn0 9218 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0) | |
7 | 5, 6 | expcld 10694 | . . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) ∈ ℂ) |
8 | 10pos 9435 | . . . . . . . 8 ⊢ 0 < ;10 | |
9 | 3, 8 | gt0ap0ii 8620 | . . . . . . 7 ⊢ ;10 # 0 |
10 | 9 | a1i 9 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 # 0) |
11 | nnz 9307 | . . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ) | |
12 | 5, 10, 11 | expap0d 10700 | . . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) # 0) |
13 | 2, 7, 12 | divrecapd 8785 | . . . 4 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · (1 / (;10↑𝑘)))) |
14 | 5, 10, 11 | exprecapd 10702 | . . . . 5 ⊢ (𝑘 ∈ ℕ → ((1 / ;10)↑𝑘) = (1 / (;10↑𝑘))) |
15 | 14 | oveq2d 5916 | . . . 4 ⊢ (𝑘 ∈ ℕ → (9 · ((1 / ;10)↑𝑘)) = (9 · (1 / (;10↑𝑘)))) |
16 | 13, 15 | eqtr4d 2225 | . . 3 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · ((1 / ;10)↑𝑘))) |
17 | 16 | sumeq2i 11413 | . 2 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) |
18 | 3, 9 | rerecclapi 8769 | . . . . 5 ⊢ (1 / ;10) ∈ ℝ |
19 | 18 | recni 8004 | . . . 4 ⊢ (1 / ;10) ∈ ℂ |
20 | 0re 7992 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
21 | 3, 8 | recgt0ii 8899 | . . . . . . 7 ⊢ 0 < (1 / ;10) |
22 | 20, 18, 21 | ltleii 8095 | . . . . . 6 ⊢ 0 ≤ (1 / ;10) |
23 | 18 | absidi 11176 | . . . . . 6 ⊢ (0 ≤ (1 / ;10) → (abs‘(1 / ;10)) = (1 / ;10)) |
24 | 22, 23 | ax-mp 5 | . . . . 5 ⊢ (abs‘(1 / ;10)) = (1 / ;10) |
25 | 1lt10 9557 | . . . . . 6 ⊢ 1 < ;10 | |
26 | recgt1 8889 | . . . . . . 7 ⊢ ((;10 ∈ ℝ ∧ 0 < ;10) → (1 < ;10 ↔ (1 / ;10) < 1)) | |
27 | 3, 8, 26 | mp2an 426 | . . . . . 6 ⊢ (1 < ;10 ↔ (1 / ;10) < 1) |
28 | 25, 27 | mpbi 145 | . . . . 5 ⊢ (1 / ;10) < 1 |
29 | 24, 28 | eqbrtri 4042 | . . . 4 ⊢ (abs‘(1 / ;10)) < 1 |
30 | geoisum1c 11569 | . . . 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 1348 | . . 3 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = ((9 · (1 / ;10)) / (1 − (1 / ;10))) |
32 | 1, 4, 9 | divrecapi 8749 | . . . 4 ⊢ (9 / ;10) = (9 · (1 / ;10)) |
33 | 1, 4, 9 | divcanap2i 8747 | . . . . . 6 ⊢ (;10 · (9 / ;10)) = 9 |
34 | ax-1cn 7939 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
35 | 4, 34, 19 | subdii 8399 | . . . . . . 7 ⊢ (;10 · (1 − (1 / ;10))) = ((;10 · 1) − (;10 · (1 / ;10))) |
36 | 4 | mulid1i 7994 | . . . . . . . 8 ⊢ (;10 · 1) = ;10 |
37 | 4, 9 | recidapi 8735 | . . . . . . . 8 ⊢ (;10 · (1 / ;10)) = 1 |
38 | 36, 37 | oveq12i 5912 | . . . . . . 7 ⊢ ((;10 · 1) − (;10 · (1 / ;10))) = (;10 − 1) |
39 | 10m1e9 9514 | . . . . . . 7 ⊢ (;10 − 1) = 9 | |
40 | 35, 38, 39 | 3eqtrri 2215 | . . . . . 6 ⊢ 9 = (;10 · (1 − (1 / ;10))) |
41 | 33, 40 | eqtri 2210 | . . . . 5 ⊢ (;10 · (9 / ;10)) = (;10 · (1 − (1 / ;10))) |
42 | 9re 9041 | . . . . . . . 8 ⊢ 9 ∈ ℝ | |
43 | 42, 3, 9 | redivclapi 8771 | . . . . . . 7 ⊢ (9 / ;10) ∈ ℝ |
44 | 43 | recni 8004 | . . . . . 6 ⊢ (9 / ;10) ∈ ℂ |
45 | 34, 19 | subcli 8268 | . . . . . 6 ⊢ (1 − (1 / ;10)) ∈ ℂ |
46 | 44, 45, 4, 9 | mulcanapi 8659 | . . . . 5 ⊢ ((;10 · (9 / ;10)) = (;10 · (1 − (1 / ;10))) ↔ (9 / ;10) = (1 − (1 / ;10))) |
47 | 41, 46 | mpbi 145 | . . . 4 ⊢ (9 / ;10) = (1 − (1 / ;10)) |
48 | 32, 47 | oveq12i 5912 | . . 3 ⊢ ((9 / ;10) / (9 / ;10)) = ((9 · (1 / ;10)) / (1 − (1 / ;10))) |
49 | 9pos 9058 | . . . . . 6 ⊢ 0 < 9 | |
50 | 42, 3, 49, 8 | divgt0ii 8911 | . . . . 5 ⊢ 0 < (9 / ;10) |
51 | 43, 50 | gt0ap0ii 8620 | . . . 4 ⊢ (9 / ;10) # 0 |
52 | 44, 51 | dividapi 8737 | . . 3 ⊢ ((9 / ;10) / (9 / ;10)) = 1 |
53 | 31, 48, 52 | 3eqtr2i 2216 | . 2 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = 1 |
54 | 17, 53 | eqtri 2210 | 1 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = 1 |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 = wceq 1364 ∈ wcel 2160 class class class wbr 4021 ‘cfv 5238 (class class class)co 5900 ℂcc 7844 ℝcr 7845 0cc0 7846 1c1 7847 · cmul 7851 < clt 8027 ≤ cle 8028 − cmin 8163 # cap 8573 / cdiv 8664 ℕcn 8954 9c9 9012 ;cdc 9419 ↑cexp 10559 abscabs 11047 Σcsu 11402 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4136 ax-sep 4139 ax-nul 4147 ax-pow 4195 ax-pr 4230 ax-un 4454 ax-setind 4557 ax-iinf 4608 ax-cnex 7937 ax-resscn 7938 ax-1cn 7939 ax-1re 7940 ax-icn 7941 ax-addcl 7942 ax-addrcl 7943 ax-mulcl 7944 ax-mulrcl 7945 ax-addcom 7946 ax-mulcom 7947 ax-addass 7948 ax-mulass 7949 ax-distr 7950 ax-i2m1 7951 ax-0lt1 7952 ax-1rid 7953 ax-0id 7954 ax-rnegex 7955 ax-precex 7956 ax-cnre 7957 ax-pre-ltirr 7958 ax-pre-ltwlin 7959 ax-pre-lttrn 7960 ax-pre-apti 7961 ax-pre-ltadd 7962 ax-pre-mulgt0 7963 ax-pre-mulext 7964 ax-arch 7965 ax-caucvg 7966 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-int 3863 df-iun 3906 df-br 4022 df-opab 4083 df-mpt 4084 df-tr 4120 df-id 4314 df-po 4317 df-iso 4318 df-iord 4387 df-on 4389 df-ilim 4390 df-suc 4392 df-iom 4611 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-rn 4658 df-res 4659 df-ima 4660 df-iota 5199 df-fun 5240 df-fn 5241 df-f 5242 df-f1 5243 df-fo 5244 df-f1o 5245 df-fv 5246 df-isom 5247 df-riota 5855 df-ov 5903 df-oprab 5904 df-mpo 5905 df-1st 6169 df-2nd 6170 df-recs 6334 df-irdg 6399 df-frec 6420 df-1o 6445 df-oadd 6449 df-er 6563 df-en 6771 df-dom 6772 df-fin 6773 df-pnf 8029 df-mnf 8030 df-xr 8031 df-ltxr 8032 df-le 8033 df-sub 8165 df-neg 8166 df-reap 8567 df-ap 8574 df-div 8665 df-inn 8955 df-2 9013 df-3 9014 df-4 9015 df-5 9016 df-6 9017 df-7 9018 df-8 9019 df-9 9020 df-n0 9212 df-z 9289 df-dec 9420 df-uz 9564 df-q 9656 df-rp 9690 df-fz 10045 df-fzo 10179 df-seqfrec 10485 df-exp 10560 df-ihash 10797 df-cj 10892 df-re 10893 df-im 10894 df-rsqrt 11048 df-abs 11049 df-clim 11328 df-sumdc 11403 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |