0.999... - Intuitionistic Logic Explorer

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Theorem 0.999... 11531

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 9009	. . . . . 6 ⊢ 9 ∈ ℂ
2	1	a1i 9	. . . . 5 ⊢ (𝑘 ∈ ℕ → 9 ∈ ℂ)
3		10re 9404	. . . . . . . 8 ⊢ ;10 ∈ ℝ
4	3	recni 7971	. . . . . . 7 ⊢ ;10 ∈ ℂ
5	4	a1i 9	. . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 ∈ ℂ)
6		nnnn0 9185	. . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
7	5, 6	expcld 10656	. . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) ∈ ℂ)
8		10pos 9402	. . . . . . . 8 ⊢ 0 < ;10
9	3, 8	gt0ap0ii 8587	. . . . . . 7 ⊢ ;10 # 0
10	9	a1i 9	. . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 # 0)
11		nnz 9274	. . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
12	5, 10, 11	expap0d 10662	. . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) # 0)
13	2, 7, 12	divrecapd 8752	. . . 4 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · (1 / (;10↑𝑘))))
14	5, 10, 11	exprecapd 10664	. . . . 5 ⊢ (𝑘 ∈ ℕ → ((1 / ;10)↑𝑘) = (1 / (;10↑𝑘)))
15	14	oveq2d 5893	. . . 4 ⊢ (𝑘 ∈ ℕ → (9 · ((1 / ;10)↑𝑘)) = (9 · (1 / (;10↑𝑘))))
16	13, 15	eqtr4d 2213	. . 3 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · ((1 / ;10)↑𝑘)))
17	16	sumeq2i 11374	. 2 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘))
18	3, 9	rerecclapi 8736	. . . . 5 ⊢ (1 / ;10) ∈ ℝ
19	18	recni 7971	. . . 4 ⊢ (1 / ;10) ∈ ℂ
20		0re 7959	. . . . . . 7 ⊢ 0 ∈ ℝ
21	3, 8	recgt0ii 8866	. . . . . . 7 ⊢ 0 < (1 / ;10)
22	20, 18, 21	ltleii 8062	. . . . . 6 ⊢ 0 ≤ (1 / ;10)
23	18	absidi 11137	. . . . . 6 ⊢ (0 ≤ (1 / ;10) → (abs‘(1 / ;10)) = (1 / ;10))
24	22, 23	ax-mp 5	. . . . 5 ⊢ (abs‘(1 / ;10)) = (1 / ;10)
25		1lt10 9524	. . . . . 6 ⊢ 1 < ;10
26		recgt1 8856	. . . . . . 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 4026	. . . 4 ⊢ (abs‘(1 / ;10)) < 1
30		geoisum1c 11530	. . . 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 1337	. . 3 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = ((9 · (1 / ;10)) / (1 − (1 / ;10)))
32	1, 4, 9	divrecapi 8716	. . . 4 ⊢ (9 / ;10) = (9 · (1 / ;10))
33	1, 4, 9	divcanap2i 8714	. . . . . 6 ⊢ (;10 · (9 / ;10)) = 9
34		ax-1cn 7906	. . . . . . . 8 ⊢ 1 ∈ ℂ
35	4, 34, 19	subdii 8366	. . . . . . 7 ⊢ (;10 · (1 − (1 / ;10))) = ((;10 · 1) − (;10 · (1 / ;10)))
36	4	mulid1i 7961	. . . . . . . 8 ⊢ (;10 · 1) = ;10
37	4, 9	recidapi 8702	. . . . . . . 8 ⊢ (;10 · (1 / ;10)) = 1
38	36, 37	oveq12i 5889	. . . . . . 7 ⊢ ((;10 · 1) − (;10 · (1 / ;10))) = (;10 − 1)
39		10m1e9 9481	. . . . . . 7 ⊢ (;10 − 1) = 9
40	35, 38, 39	3eqtrri 2203	. . . . . 6 ⊢ 9 = (;10 · (1 − (1 / ;10)))
41	33, 40	eqtri 2198	. . . . 5 ⊢ (;10 · (9 / ;10)) = (;10 · (1 − (1 / ;10)))
42		9re 9008	. . . . . . . 8 ⊢ 9 ∈ ℝ
43	42, 3, 9	redivclapi 8738	. . . . . . 7 ⊢ (9 / ;10) ∈ ℝ
44	43	recni 7971	. . . . . 6 ⊢ (9 / ;10) ∈ ℂ
45	34, 19	subcli 8235	. . . . . 6 ⊢ (1 − (1 / ;10)) ∈ ℂ
46	44, 45, 4, 9	mulcanapi 8626	. . . . 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 5889	. . 3 ⊢ ((9 / ;10) / (9 / ;10)) = ((9 · (1 / ;10)) / (1 − (1 / ;10)))
49		9pos 9025	. . . . . 6 ⊢ 0 < 9
50	42, 3, 49, 8	divgt0ii 8878	. . . . 5 ⊢ 0 < (9 / ;10)
51	43, 50	gt0ap0ii 8587	. . . 4 ⊢ (9 / ;10) # 0
52	44, 51	dividapi 8704	. . 3 ⊢ ((9 / ;10) / (9 / ;10)) = 1
53	31, 48, 52	3eqtr2i 2204	. 2 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = 1
54	17, 53	eqtri 2198	1 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = 1

Colors of variables: wff set class

Syntax hints: ↔ wb 105 = wceq 1353 ∈ wcel 2148 class class class wbr 4005 ‘cfv 5218 (class class class)co 5877 ℂcc 7811 ℝcr 7812 0cc0 7813 1c1 7814 · cmul 7818 < clt 7994 ≤ cle 7995 − cmin 8130 # cap 8540 / cdiv 8631 ℕcn 8921 9c9 8979 cdc 9386 ↑cexp 10521 abscabs 11008 Σcsu 11363

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-precex 7923 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 ax-pre-mulgt0 7930 ax-pre-mulext 7931 ax-arch 7932 ax-caucvg 7933

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-isom 5227 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-recs 6308 df-irdg 6373 df-frec 6394 df-1o 6419 df-oadd 6423 df-er 6537 df-en 6743 df-dom 6744 df-fin 6745 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-reap 8534 df-ap 8541 df-div 8632 df-inn 8922 df-2 8980 df-3 8981 df-4 8982 df-5 8983 df-6 8984 df-7 8985 df-8 8986 df-9 8987 df-n0 9179 df-z 9256 df-dec 9387 df-uz 9531 df-q 9622 df-rp 9656 df-fz 10011 df-fzo 10145 df-seqfrec 10448 df-exp 10522 df-ihash 10758 df-cj 10853 df-re 10854 df-im 10855 df-rsqrt 11009 df-abs 11010 df-clim 11289 df-sumdc 11364

This theorem is referenced by: (None)

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