0.999... - Metamath Proof Explorer

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Theorem 0.999... 14854

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 11419	. . . . 5 ⊢ 9 ∈ ℂ
2		10re 11798	. . . . . . 7 ⊢ ;10 ∈ ℝ
3	2	recni 10349	. . . . . 6 ⊢ ;10 ∈ ℂ
4		nnnn0 11586	. . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
5		expcl 13121	. . . . . 6 ⊢ ((;10 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (;10↑𝑘) ∈ ℂ)
6	3, 4, 5	sylancr 577	. . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) ∈ ℂ)
7	3	a1i 11	. . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 ∈ ℂ)
8		10pos 11796	. . . . . . . 8 ⊢ 0 < ;10
9	2, 8	gt0ne0ii 10859	. . . . . . 7 ⊢ ;10 ≠ 0
10	9	a1i 11	. . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 ≠ 0)
11		nnz 11685	. . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
12	7, 10, 11	expne0d 13257	. . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) ≠ 0)
13		divrec 10996	. . . . 5 ⊢ ((9 ∈ ℂ ∧ (;10↑𝑘) ∈ ℂ ∧ (;10↑𝑘) ≠ 0) → (9 / (;10↑𝑘)) = (9 · (1 / (;10↑𝑘))))
14	1, 6, 12, 13	mp3an2i 1583	. . . 4 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · (1 / (;10↑𝑘))))
15	7, 10, 11	exprecd 13259	. . . . 5 ⊢ (𝑘 ∈ ℕ → ((1 / ;10)↑𝑘) = (1 / (;10↑𝑘)))
16	15	oveq2d 6900	. . . 4 ⊢ (𝑘 ∈ ℕ → (9 · ((1 / ;10)↑𝑘)) = (9 · (1 / (;10↑𝑘))))
17	14, 16	eqtr4d 2854	. . 3 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · ((1 / ;10)↑𝑘)))
18	17	sumeq2i 14672	. 2 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘))
19	2, 9	rereccli 11085	. . . . 5 ⊢ (1 / ;10) ∈ ℝ
20	19	recni 10349	. . . 4 ⊢ (1 / ;10) ∈ ℂ
21		0re 10337	. . . . . . 7 ⊢ 0 ∈ ℝ
22	2, 8	recgt0ii 11224	. . . . . . 7 ⊢ 0 < (1 / ;10)
23	21, 19, 22	ltleii 10455	. . . . . 6 ⊢ 0 ≤ (1 / ;10)
24	19	absidi 14360	. . . . . 6 ⊢ (0 ≤ (1 / ;10) → (abs‘(1 / ;10)) = (1 / ;10))
25	23, 24	ax-mp 5	. . . . 5 ⊢ (abs‘(1 / ;10)) = (1 / ;10)
26		1lt10 11918	. . . . . 6 ⊢ 1 < ;10
27		recgt1 11214	. . . . . . 7 ⊢ ((;10 ∈ ℝ ∧ 0 < ;10) → (1 < ;10 ↔ (1 / ;10) < 1))
28	2, 8, 27	mp2an 675	. . . . . 6 ⊢ (1 < ;10 ↔ (1 / ;10) < 1)
29	26, 28	mpbi 221	. . . . 5 ⊢ (1 / ;10) < 1
30	25, 29	eqbrtri 4876	. . . 4 ⊢ (abs‘(1 / ;10)) < 1
31		geoisum1c 14853	. . . 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 1578	. . 3 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = ((9 · (1 / ;10)) / (1 − (1 / ;10)))
33	1, 3, 9	divreci 11065	. . . 4 ⊢ (9 / ;10) = (9 · (1 / ;10))
34	1, 3, 9	divcan2i 11063	. . . . . 6 ⊢ (;10 · (9 / ;10)) = 9
35		ax-1cn 10289	. . . . . . . 8 ⊢ 1 ∈ ℂ
36	3, 35, 20	subdii 10774	. . . . . . 7 ⊢ (;10 · (1 − (1 / ;10))) = ((;10 · 1) − (;10 · (1 / ;10)))
37	3	mulid1i 10339	. . . . . . . 8 ⊢ (;10 · 1) = ;10
38	3, 9	recidi 11051	. . . . . . . 8 ⊢ (;10 · (1 / ;10)) = 1
39	37, 38	oveq12i 6896	. . . . . . 7 ⊢ ((;10 · 1) − (;10 · (1 / ;10))) = (;10 − 1)
40		10m1e9 11875	. . . . . . 7 ⊢ (;10 − 1) = 9
41	36, 39, 40	3eqtrri 2844	. . . . . 6 ⊢ 9 = (;10 · (1 − (1 / ;10)))
42	34, 41	eqtri 2839	. . . . 5 ⊢ (;10 · (9 / ;10)) = (;10 · (1 − (1 / ;10)))
43		9re 11418	. . . . . . . 8 ⊢ 9 ∈ ℝ
44	43, 2, 9	redivcli 11087	. . . . . . 7 ⊢ (9 / ;10) ∈ ℝ
45	44	recni 10349	. . . . . 6 ⊢ (9 / ;10) ∈ ℂ
46	35, 20	subcli 10652	. . . . . 6 ⊢ (1 − (1 / ;10)) ∈ ℂ
47	45, 46, 3, 9	mulcani 10961	. . . . 5 ⊢ ((;10 · (9 / ;10)) = (;10 · (1 − (1 / ;10))) ↔ (9 / ;10) = (1 − (1 / ;10)))
48	42, 47	mpbi 221	. . . 4 ⊢ (9 / ;10) = (1 − (1 / ;10))
49	33, 48	oveq12i 6896	. . 3 ⊢ ((9 / ;10) / (9 / ;10)) = ((9 · (1 / ;10)) / (1 − (1 / ;10)))
50		9pos 11433	. . . . . 6 ⊢ 0 < 9
51	43, 2, 50, 8	divgt0ii 11236	. . . . 5 ⊢ 0 < (9 / ;10)
52	44, 51	gt0ne0ii 10859	. . . 4 ⊢ (9 / ;10) ≠ 0
53	45, 52	dividi 11053	. . 3 ⊢ ((9 / ;10) / (9 / ;10)) = 1
54	32, 49, 53	3eqtr2i 2845	. 2 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = 1
55	18, 54	eqtri 2839	1 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = 1

Colors of variables: wff setvar class

Syntax hints: ↔ wb 197 = wceq 1637 ∈ wcel 2157 ≠ wne 2989 class class class wbr 4855 ‘cfv 6111 (class class class)co 6884 ℂcc 10229 ℝcr 10230 0cc0 10231 1c1 10232 · cmul 10236 < clt 10369 ≤ cle 10370 − cmin 10561 / cdiv 10979 ℕcn 11315 9c9 11375 ℕ₀cn0 11579 cdc 11779 ↑cexp 13103 abscabs 14217 Σcsu 14659

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1877 ax-4 1894 ax-5 2001 ax-6 2069 ax-7 2105 ax-8 2159 ax-9 2166 ax-10 2186 ax-11 2202 ax-12 2215 ax-13 2422 ax-ext 2795 ax-rep 4977 ax-sep 4988 ax-nul 4996 ax-pow 5048 ax-pr 5109 ax-un 7189 ax-inf2 8795 ax-cnex 10287 ax-resscn 10288 ax-1cn 10289 ax-icn 10290 ax-addcl 10291 ax-addrcl 10292 ax-mulcl 10293 ax-mulrcl 10294 ax-mulcom 10295 ax-addass 10296 ax-mulass 10297 ax-distr 10298 ax-i2m1 10299 ax-1ne0 10300 ax-1rid 10301 ax-rnegex 10302 ax-rrecex 10303 ax-cnre 10304 ax-pre-lttri 10305 ax-pre-lttrn 10306 ax-pre-ltadd 10307 ax-pre-mulgt0 10308 ax-pre-sup 10309

This theorem depends on definitions: df-bi 198 df-an 385 df-or 866 df-3or 1101 df-3an 1102 df-tru 1641 df-fal 1651 df-ex 1860 df-nf 1864 df-sb 2062 df-mo 2635 df-eu 2642 df-clab 2804 df-cleq 2810 df-clel 2813 df-nfc 2948 df-ne 2990 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3404 df-sbc 3645 df-csb 3740 df-dif 3783 df-un 3785 df-in 3787 df-ss 3794 df-pss 3796 df-nul 4128 df-if 4291 df-pw 4364 df-sn 4382 df-pr 4384 df-tp 4386 df-op 4388 df-uni 4642 df-int 4681 df-iun 4725 df-br 4856 df-opab 4918 df-mpt 4935 df-tr 4958 df-id 5232 df-eprel 5237 df-po 5245 df-so 5246 df-fr 5283 df-se 5284 df-we 5285 df-xp 5330 df-rel 5331 df-cnv 5332 df-co 5333 df-dm 5334 df-rn 5335 df-res 5336 df-ima 5337 df-pred 5907 df-ord 5953 df-on 5954 df-lim 5955 df-suc 5956 df-iota 6074 df-fun 6113 df-fn 6114 df-f 6115 df-f1 6116 df-fo 6117 df-f1o 6118 df-fv 6119 df-isom 6120 df-riota 6845 df-ov 6887 df-oprab 6888 df-mpt2 6889 df-om 7306 df-1st 7408 df-2nd 7409 df-wrecs 7652 df-recs 7714 df-rdg 7752 df-1o 7806 df-oadd 7810 df-er 7989 df-pm 8105 df-en 8203 df-dom 8204 df-sdom 8205 df-fin 8206 df-sup 8597 df-inf 8598 df-oi 8664 df-card 9058 df-pnf 10371 df-mnf 10372 df-xr 10373 df-ltxr 10374 df-le 10375 df-sub 10563 df-neg 10564 df-div 10980 df-nn 11316 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-7 11381 df-8 11382 df-9 11383 df-n0 11580 df-z 11664 df-dec 11780 df-uz 11925 df-rp 12067 df-fz 12570 df-fzo 12710 df-fl 12837 df-seq 13045 df-exp 13104 df-hash 13358 df-cj 14082 df-re 14083 df-im 14084 df-sqrt 14218 df-abs 14219 df-clim 14462 df-rlim 14463 df-sum 14660

This theorem is referenced by: (None)

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