0.999... - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > 0.999...		GIF version

Theorem 0.999... 11484

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.)

**Assertion**
Ref	Expression
0.999...	⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = 1

**Proof of Theorem 0.999...**
Step	Hyp	Ref	Expression
1		9cn 8966	. . . . . 6 ⊢ 9 ∈ ℂ
2	1	a1i 9	. . . . 5 ⊢ (𝑘 ∈ ℕ → 9 ∈ ℂ)
3		10re 9361	. . . . . . . 8 ⊢ ;10 ∈ ℝ
4	3	recni 7932	. . . . . . 7 ⊢ ;10 ∈ ℂ
5	4	a1i 9	. . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 ∈ ℂ)
6		nnnn0 9142	. . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
7	5, 6	expcld 10609	. . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) ∈ ℂ)
8		10pos 9359	. . . . . . . 8 ⊢ 0 < ;10
9	3, 8	gt0ap0ii 8547	. . . . . . 7 ⊢ ;10 # 0
10	9	a1i 9	. . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 # 0)
11		nnz 9231	. . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
12	5, 10, 11	expap0d 10615	. . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) # 0)
13	2, 7, 12	divrecapd 8710	. . . 4 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · (1 / (;10↑𝑘))))
14	5, 10, 11	exprecapd 10617	. . . . 5 ⊢ (𝑘 ∈ ℕ → ((1 / ;10)↑𝑘) = (1 / (;10↑𝑘)))
15	14	oveq2d 5869	. . . 4 ⊢ (𝑘 ∈ ℕ → (9 · ((1 / ;10)↑𝑘)) = (9 · (1 / (;10↑𝑘))))
16	13, 15	eqtr4d 2206	. . 3 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · ((1 / ;10)↑𝑘)))
17	16	sumeq2i 11327	. 2 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘))
18	3, 9	rerecclapi 8694	. . . . 5 ⊢ (1 / ;10) ∈ ℝ
19	18	recni 7932	. . . 4 ⊢ (1 / ;10) ∈ ℂ
20		0re 7920	. . . . . . 7 ⊢ 0 ∈ ℝ
21	3, 8	recgt0ii 8823	. . . . . . 7 ⊢ 0 < (1 / ;10)
22	20, 18, 21	ltleii 8022	. . . . . 6 ⊢ 0 ≤ (1 / ;10)
23	18	absidi 11090	. . . . . 6 ⊢ (0 ≤ (1 / ;10) → (abs‘(1 / ;10)) = (1 / ;10))
24	22, 23	ax-mp 5	. . . . 5 ⊢ (abs‘(1 / ;10)) = (1 / ;10)
25		1lt10 9481	. . . . . 6 ⊢ 1 < ;10
26		recgt1 8813	. . . . . . 7 ⊢ ((;10 ∈ ℝ ∧ 0 < ;10) → (1 < ;10 ↔ (1 / ;10) < 1))
27	3, 8, 26	mp2an 424	. . . . . 6 ⊢ (1 < ;10 ↔ (1 / ;10) < 1)
28	25, 27	mpbi 144	. . . . 5 ⊢ (1 / ;10) < 1
29	24, 28	eqbrtri 4010	. . . 4 ⊢ (abs‘(1 / ;10)) < 1
30		geoisum1c 11483	. . . 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 1332	. . 3 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = ((9 · (1 / ;10)) / (1 − (1 / ;10)))
32	1, 4, 9	divrecapi 8674	. . . 4 ⊢ (9 / ;10) = (9 · (1 / ;10))
33	1, 4, 9	divcanap2i 8672	. . . . . 6 ⊢ (;10 · (9 / ;10)) = 9
34		ax-1cn 7867	. . . . . . . 8 ⊢ 1 ∈ ℂ
35	4, 34, 19	subdii 8326	. . . . . . 7 ⊢ (;10 · (1 − (1 / ;10))) = ((;10 · 1) − (;10 · (1 / ;10)))
36	4	mulid1i 7922	. . . . . . . 8 ⊢ (;10 · 1) = ;10
37	4, 9	recidapi 8660	. . . . . . . 8 ⊢ (;10 · (1 / ;10)) = 1
38	36, 37	oveq12i 5865	. . . . . . 7 ⊢ ((;10 · 1) − (;10 · (1 / ;10))) = (;10 − 1)
39		10m1e9 9438	. . . . . . 7 ⊢ (;10 − 1) = 9
40	35, 38, 39	3eqtrri 2196	. . . . . 6 ⊢ 9 = (;10 · (1 − (1 / ;10)))
41	33, 40	eqtri 2191	. . . . 5 ⊢ (;10 · (9 / ;10)) = (;10 · (1 − (1 / ;10)))
42		9re 8965	. . . . . . . 8 ⊢ 9 ∈ ℝ
43	42, 3, 9	redivclapi 8696	. . . . . . 7 ⊢ (9 / ;10) ∈ ℝ
44	43	recni 7932	. . . . . 6 ⊢ (9 / ;10) ∈ ℂ
45	34, 19	subcli 8195	. . . . . 6 ⊢ (1 − (1 / ;10)) ∈ ℂ
46	44, 45, 4, 9	mulcanapi 8585	. . . . 5 ⊢ ((;10 · (9 / ;10)) = (;10 · (1 − (1 / ;10))) ↔ (9 / ;10) = (1 − (1 / ;10)))
47	41, 46	mpbi 144	. . . 4 ⊢ (9 / ;10) = (1 − (1 / ;10))
48	32, 47	oveq12i 5865	. . 3 ⊢ ((9 / ;10) / (9 / ;10)) = ((9 · (1 / ;10)) / (1 − (1 / ;10)))
49		9pos 8982	. . . . . 6 ⊢ 0 < 9
50	42, 3, 49, 8	divgt0ii 8835	. . . . 5 ⊢ 0 < (9 / ;10)
51	43, 50	gt0ap0ii 8547	. . . 4 ⊢ (9 / ;10) # 0
52	44, 51	dividapi 8662	. . 3 ⊢ ((9 / ;10) / (9 / ;10)) = 1
53	31, 48, 52	3eqtr2i 2197	. 2 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = 1
54	17, 53	eqtri 2191	1 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = 1

Colors of variables: wff set class

Syntax hints: ↔ wb 104 = wceq 1348 ∈ wcel 2141 class class class wbr 3989 ‘cfv 5198 (class class class)co 5853 ℂcc 7772 ℝcr 7773 0cc0 7774 1c1 7775 · cmul 7779 < clt 7954 ≤ cle 7955 − cmin 8090 # cap 8500 / cdiv 8589 ℕcn 8878 9c9 8936 cdc 9343 ↑cexp 10475 abscabs 10961 Σcsu 11316

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894

This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-isom 5207 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-frec 6370 df-1o 6395 df-oadd 6399 df-er 6513 df-en 6719 df-dom 6720 df-fin 6721 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-5 8940 df-6 8941 df-7 8942 df-8 8943 df-9 8944 df-n0 9136 df-z 9213 df-dec 9344 df-uz 9488 df-q 9579 df-rp 9611 df-fz 9966 df-fzo 10099 df-seqfrec 10402 df-exp 10476 df-ihash 10710 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-clim 11242 df-sumdc 11317

This theorem is referenced by: (None)

W3C validator