0.999... - Metamath Proof Explorer

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Theorem 0.999... 15593

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.)

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

**Proof of Theorem 0.999...**
Step	Hyp	Ref	Expression
1		9cn 12073	. . . . 5 ⊢ 9 ∈ ℂ
2		10re 12456	. . . . . . 7 ⊢ ;10 ∈ ℝ
3	2	recni 10989	. . . . . 6 ⊢ ;10 ∈ ℂ
4		nnnn0 12240	. . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
5		expcl 13800	. . . . . 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 12454	. . . . . . . 8 ⊢ 0 < ;10
9	2, 8	gt0ne0ii 11511	. . . . . . 7 ⊢ ;10 ≠ 0
10	9	a1i 11	. . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 ≠ 0)
11		nnz 12342	. . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
12	7, 10, 11	expne0d 13870	. . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) ≠ 0)
13		divrec 11649	. . . . 5 ⊢ ((9 ∈ ℂ ∧ (;10↑𝑘) ∈ ℂ ∧ (;10↑𝑘) ≠ 0) → (9 / (;10↑𝑘)) = (9 · (1 / (;10↑𝑘))))
14	1, 6, 12, 13	mp3an2i 1465	. . . 4 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · (1 / (;10↑𝑘))))
15	7, 10, 11	exprecd 13872	. . . . 5 ⊢ (𝑘 ∈ ℕ → ((1 / ;10)↑𝑘) = (1 / (;10↑𝑘)))
16	15	oveq2d 7291	. . . 4 ⊢ (𝑘 ∈ ℕ → (9 · ((1 / ;10)↑𝑘)) = (9 · (1 / (;10↑𝑘))))
17	14, 16	eqtr4d 2781	. . 3 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · ((1 / ;10)↑𝑘)))
18	17	sumeq2i 15411	. 2 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘))
19	2, 9	rereccli 11740	. . . . 5 ⊢ (1 / ;10) ∈ ℝ
20	19	recni 10989	. . . 4 ⊢ (1 / ;10) ∈ ℂ
21		0re 10977	. . . . . . 7 ⊢ 0 ∈ ℝ
22	2, 8	recgt0ii 11881	. . . . . . 7 ⊢ 0 < (1 / ;10)
23	21, 19, 22	ltleii 11098	. . . . . 6 ⊢ 0 ≤ (1 / ;10)
24	19	absidi 15089	. . . . . 6 ⊢ (0 ≤ (1 / ;10) → (abs‘(1 / ;10)) = (1 / ;10))
25	23, 24	ax-mp 5	. . . . 5 ⊢ (abs‘(1 / ;10)) = (1 / ;10)
26		1lt10 12576	. . . . . 6 ⊢ 1 < ;10
27		recgt1 11871	. . . . . . 7 ⊢ ((;10 ∈ ℝ ∧ 0 < ;10) → (1 < ;10 ↔ (1 / ;10) < 1))
28	2, 8, 27	mp2an 689	. . . . . 6 ⊢ (1 < ;10 ↔ (1 / ;10) < 1)
29	26, 28	mpbi 229	. . . . 5 ⊢ (1 / ;10) < 1
30	25, 29	eqbrtri 5095	. . . 4 ⊢ (abs‘(1 / ;10)) < 1
31		geoisum1c 15592	. . . 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 1460	. . 3 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = ((9 · (1 / ;10)) / (1 − (1 / ;10)))
33	1, 3, 9	divreci 11720	. . . 4 ⊢ (9 / ;10) = (9 · (1 / ;10))
34	1, 3, 9	divcan2i 11718	. . . . . 6 ⊢ (;10 · (9 / ;10)) = 9
35		ax-1cn 10929	. . . . . . . 8 ⊢ 1 ∈ ℂ
36	3, 35, 20	subdii 11424	. . . . . . 7 ⊢ (;10 · (1 − (1 / ;10))) = ((;10 · 1) − (;10 · (1 / ;10)))
37	3	mulid1i 10979	. . . . . . . 8 ⊢ (;10 · 1) = ;10
38	3, 9	recidi 11706	. . . . . . . 8 ⊢ (;10 · (1 / ;10)) = 1
39	37, 38	oveq12i 7287	. . . . . . 7 ⊢ ((;10 · 1) − (;10 · (1 / ;10))) = (;10 − 1)
40		10m1e9 12533	. . . . . . 7 ⊢ (;10 − 1) = 9
41	36, 39, 40	3eqtrri 2771	. . . . . 6 ⊢ 9 = (;10 · (1 − (1 / ;10)))
42	34, 41	eqtri 2766	. . . . 5 ⊢ (;10 · (9 / ;10)) = (;10 · (1 − (1 / ;10)))
43		9re 12072	. . . . . . . 8 ⊢ 9 ∈ ℝ
44	43, 2, 9	redivcli 11742	. . . . . . 7 ⊢ (9 / ;10) ∈ ℝ
45	44	recni 10989	. . . . . 6 ⊢ (9 / ;10) ∈ ℂ
46	35, 20	subcli 11297	. . . . . 6 ⊢ (1 − (1 / ;10)) ∈ ℂ
47	45, 46, 3, 9	mulcani 11614	. . . . 5 ⊢ ((;10 · (9 / ;10)) = (;10 · (1 − (1 / ;10))) ↔ (9 / ;10) = (1 − (1 / ;10)))
48	42, 47	mpbi 229	. . . 4 ⊢ (9 / ;10) = (1 − (1 / ;10))
49	33, 48	oveq12i 7287	. . 3 ⊢ ((9 / ;10) / (9 / ;10)) = ((9 · (1 / ;10)) / (1 − (1 / ;10)))
50		9pos 12086	. . . . . 6 ⊢ 0 < 9
51	43, 2, 50, 8	divgt0ii 11892	. . . . 5 ⊢ 0 < (9 / ;10)
52	44, 51	gt0ne0ii 11511	. . . 4 ⊢ (9 / ;10) ≠ 0
53	45, 52	dividi 11708	. . 3 ⊢ ((9 / ;10) / (9 / ;10)) = 1
54	32, 49, 53	3eqtr2i 2772	. 2 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = 1
55	18, 54	eqtri 2766	1 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = 1

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 · cmul 10876 < clt 11009 ≤ cle 11010 − cmin 11205 / cdiv 11632 ℕcn 11973 9c9 12035 ℕ₀cn0 12233 cdc 12437 ↑cexp 13782 abscabs 14945 Σcsu 15397

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-pm 8618 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383 df-fl 13512 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-rlim 15198 df-sum 15398

This theorem is referenced by: (None)

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