0.999... - Metamath Proof Explorer

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Theorem 0.999... 15783

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 12220	. . . . 5 ⊢ 9 ∈ ℂ
2		10re 12602	. . . . . . 7 ⊢ ;10 ∈ ℝ
3	2	recni 11121	. . . . . 6 ⊢ ;10 ∈ ℂ
4		nnnn0 12383	. . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
5		expcl 13981	. . . . . 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 12600	. . . . . . . 8 ⊢ 0 < ;10
9	2, 8	gt0ne0ii 11648	. . . . . . 7 ⊢ ;10 ≠ 0
10	9	a1i 11	. . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 ≠ 0)
11		nnz 12484	. . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
12	7, 10, 11	expne0d 14054	. . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) ≠ 0)
13		divrec 11787	. . . . 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 14056	. . . . 5 ⊢ (𝑘 ∈ ℕ → ((1 / ;10)↑𝑘) = (1 / (;10↑𝑘)))
16	15	oveq2d 7357	. . . 4 ⊢ (𝑘 ∈ ℕ → (9 · ((1 / ;10)↑𝑘)) = (9 · (1 / (;10↑𝑘))))
17	14, 16	eqtr4d 2769	. . 3 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · ((1 / ;10)↑𝑘)))
18	17	sumeq2i 15600	. 2 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘))
19	2, 9	rereccli 11881	. . . . 5 ⊢ (1 / ;10) ∈ ℝ
20	19	recni 11121	. . . 4 ⊢ (1 / ;10) ∈ ℂ
21		0re 11109	. . . . . . 7 ⊢ 0 ∈ ℝ
22	2, 8	recgt0ii 12023	. . . . . . 7 ⊢ 0 < (1 / ;10)
23	21, 19, 22	ltleii 11231	. . . . . 6 ⊢ 0 ≤ (1 / ;10)
24	19	absidi 15280	. . . . . 6 ⊢ (0 ≤ (1 / ;10) → (abs‘(1 / ;10)) = (1 / ;10))
25	23, 24	ax-mp 5	. . . . 5 ⊢ (abs‘(1 / ;10)) = (1 / ;10)
26		1lt10 12722	. . . . . 6 ⊢ 1 < ;10
27		recgt1 12013	. . . . . . 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 5107	. . . 4 ⊢ (abs‘(1 / ;10)) < 1
31		geoisum1c 15782	. . . 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 11861	. . . 4 ⊢ (9 / ;10) = (9 · (1 / ;10))
34	1, 3, 9	divcan2i 11859	. . . . . 6 ⊢ (;10 · (9 / ;10)) = 9
35		ax-1cn 11059	. . . . . . . 8 ⊢ 1 ∈ ℂ
36	3, 35, 20	subdii 11561	. . . . . . 7 ⊢ (;10 · (1 − (1 / ;10))) = ((;10 · 1) − (;10 · (1 / ;10)))
37	3	mulridi 11111	. . . . . . . 8 ⊢ (;10 · 1) = ;10
38	3, 9	recidi 11847	. . . . . . . 8 ⊢ (;10 · (1 / ;10)) = 1
39	37, 38	oveq12i 7353	. . . . . . 7 ⊢ ((;10 · 1) − (;10 · (1 / ;10))) = (;10 − 1)
40		10m1e9 12679	. . . . . . 7 ⊢ (;10 − 1) = 9
41	36, 39, 40	3eqtrri 2759	. . . . . 6 ⊢ 9 = (;10 · (1 − (1 / ;10)))
42	34, 41	eqtri 2754	. . . . 5 ⊢ (;10 · (9 / ;10)) = (;10 · (1 − (1 / ;10)))
43		9re 12219	. . . . . . . 8 ⊢ 9 ∈ ℝ
44	43, 2, 9	redivcli 11883	. . . . . . 7 ⊢ (9 / ;10) ∈ ℝ
45	44	recni 11121	. . . . . 6 ⊢ (9 / ;10) ∈ ℂ
46	35, 20	subcli 11432	. . . . . 6 ⊢ (1 − (1 / ;10)) ∈ ℂ
47	45, 46, 3, 9	mulcani 11751	. . . . 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 7353	. . 3 ⊢ ((9 / ;10) / (9 / ;10)) = ((9 · (1 / ;10)) / (1 − (1 / ;10)))
50		9pos 12233	. . . . . 6 ⊢ 0 < 9
51	43, 2, 50, 8	divgt0ii 12034	. . . . 5 ⊢ 0 < (9 / ;10)
52	44, 51	gt0ne0ii 11648	. . . 4 ⊢ (9 / ;10) ≠ 0
53	45, 52	dividi 11849	. . 3 ⊢ ((9 / ;10) / (9 / ;10)) = 1
54	32, 49, 53	3eqtr2i 2760	. 2 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = 1
55	18, 54	eqtri 2754	1 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = 1

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 class class class wbr 5086 ‘cfv 6476 (class class class)co 7341 ℂcc 10999 ℝcr 11000 0cc0 11001 1c1 11002 · cmul 11006 < clt 11141 ≤ cle 11142 − cmin 11339 / cdiv 11769 ℕcn 12120 9c9 12182 ℕ₀cn0 12376 cdc 12583 ↑cexp 13963 abscabs 15136 Σcsu 15588

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5212 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-inf2 9526 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078 ax-pre-sup 11079

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-op 4578 df-uni 4855 df-int 4893 df-iun 4938 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-se 5565 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-isom 6485 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-1st 7916 df-2nd 7917 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-er 8617 df-pm 8748 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-sup 9321 df-inf 9322 df-oi 9391 df-card 9827 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-div 11770 df-nn 12121 df-2 12183 df-3 12184 df-4 12185 df-5 12186 df-6 12187 df-7 12188 df-8 12189 df-9 12190 df-n0 12377 df-z 12464 df-dec 12584 df-uz 12728 df-rp 12886 df-fz 13403 df-fzo 13550 df-fl 13691 df-seq 13904 df-exp 13964 df-hash 14233 df-cj 15001 df-re 15002 df-im 15003 df-sqrt 15137 df-abs 15138 df-clim 15390 df-rlim 15391 df-sum 15589

This theorem is referenced by: (None)

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