0.999... - Metamath Proof Explorer

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Theorem 0.999... 14986

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 11457	. . . . 5 ⊢ 9 ∈ ℂ
2		10re 11840	. . . . . . 7 ⊢ ;10 ∈ ℝ
3	2	recni 10371	. . . . . 6 ⊢ ;10 ∈ ℂ
4		nnnn0 11626	. . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
5		expcl 13172	. . . . . 6 ⊢ ((;10 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (;10↑𝑘) ∈ ℂ)
6	3, 4, 5	sylancr 583	. . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) ∈ ℂ)
7	3	a1i 11	. . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 ∈ ℂ)
8		10pos 11838	. . . . . . . 8 ⊢ 0 < ;10
9	2, 8	gt0ne0ii 10888	. . . . . . 7 ⊢ ;10 ≠ 0
10	9	a1i 11	. . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 ≠ 0)
11		nnz 11727	. . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
12	7, 10, 11	expne0d 13308	. . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) ≠ 0)
13		divrec 11026	. . . . 5 ⊢ ((9 ∈ ℂ ∧ (;10↑𝑘) ∈ ℂ ∧ (;10↑𝑘) ≠ 0) → (9 / (;10↑𝑘)) = (9 · (1 / (;10↑𝑘))))
14	1, 6, 12, 13	mp3an2i 1596	. . . 4 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · (1 / (;10↑𝑘))))
15	7, 10, 11	exprecd 13310	. . . . 5 ⊢ (𝑘 ∈ ℕ → ((1 / ;10)↑𝑘) = (1 / (;10↑𝑘)))
16	15	oveq2d 6921	. . . 4 ⊢ (𝑘 ∈ ℕ → (9 · ((1 / ;10)↑𝑘)) = (9 · (1 / (;10↑𝑘))))
17	14, 16	eqtr4d 2864	. . 3 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · ((1 / ;10)↑𝑘)))
18	17	sumeq2i 14806	. 2 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘))
19	2, 9	rereccli 11116	. . . . 5 ⊢ (1 / ;10) ∈ ℝ
20	19	recni 10371	. . . 4 ⊢ (1 / ;10) ∈ ℂ
21		0re 10358	. . . . . . 7 ⊢ 0 ∈ ℝ
22	2, 8	recgt0ii 11259	. . . . . . 7 ⊢ 0 < (1 / ;10)
23	21, 19, 22	ltleii 10479	. . . . . 6 ⊢ 0 ≤ (1 / ;10)
24	19	absidi 14494	. . . . . 6 ⊢ (0 ≤ (1 / ;10) → (abs‘(1 / ;10)) = (1 / ;10))
25	23, 24	ax-mp 5	. . . . 5 ⊢ (abs‘(1 / ;10)) = (1 / ;10)
26		1lt10 11962	. . . . . 6 ⊢ 1 < ;10
27		recgt1 11249	. . . . . . 7 ⊢ ((;10 ∈ ℝ ∧ 0 < ;10) → (1 < ;10 ↔ (1 / ;10) < 1))
28	2, 8, 27	mp2an 685	. . . . . 6 ⊢ (1 < ;10 ↔ (1 / ;10) < 1)
29	26, 28	mpbi 222	. . . . 5 ⊢ (1 / ;10) < 1
30	25, 29	eqbrtri 4894	. . . 4 ⊢ (abs‘(1 / ;10)) < 1
31		geoisum1c 14985	. . . 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 1591	. . 3 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = ((9 · (1 / ;10)) / (1 − (1 / ;10)))
33	1, 3, 9	divreci 11096	. . . 4 ⊢ (9 / ;10) = (9 · (1 / ;10))
34	1, 3, 9	divcan2i 11094	. . . . . 6 ⊢ (;10 · (9 / ;10)) = 9
35		ax-1cn 10310	. . . . . . . 8 ⊢ 1 ∈ ℂ
36	3, 35, 20	subdii 10803	. . . . . . 7 ⊢ (;10 · (1 − (1 / ;10))) = ((;10 · 1) − (;10 · (1 / ;10)))
37	3	mulid1i 10361	. . . . . . . 8 ⊢ (;10 · 1) = ;10
38	3, 9	recidi 11082	. . . . . . . 8 ⊢ (;10 · (1 / ;10)) = 1
39	37, 38	oveq12i 6917	. . . . . . 7 ⊢ ((;10 · 1) − (;10 · (1 / ;10))) = (;10 − 1)
40		10m1e9 11919	. . . . . . 7 ⊢ (;10 − 1) = 9
41	36, 39, 40	3eqtrri 2854	. . . . . 6 ⊢ 9 = (;10 · (1 − (1 / ;10)))
42	34, 41	eqtri 2849	. . . . 5 ⊢ (;10 · (9 / ;10)) = (;10 · (1 − (1 / ;10)))
43		9re 11456	. . . . . . . 8 ⊢ 9 ∈ ℝ
44	43, 2, 9	redivcli 11118	. . . . . . 7 ⊢ (9 / ;10) ∈ ℝ
45	44	recni 10371	. . . . . 6 ⊢ (9 / ;10) ∈ ℂ
46	35, 20	subcli 10678	. . . . . 6 ⊢ (1 − (1 / ;10)) ∈ ℂ
47	45, 46, 3, 9	mulcani 10991	. . . . 5 ⊢ ((;10 · (9 / ;10)) = (;10 · (1 − (1 / ;10))) ↔ (9 / ;10) = (1 − (1 / ;10)))
48	42, 47	mpbi 222	. . . 4 ⊢ (9 / ;10) = (1 − (1 / ;10))
49	33, 48	oveq12i 6917	. . 3 ⊢ ((9 / ;10) / (9 / ;10)) = ((9 · (1 / ;10)) / (1 − (1 / ;10)))
50		9pos 11471	. . . . . 6 ⊢ 0 < 9
51	43, 2, 50, 8	divgt0ii 11271	. . . . 5 ⊢ 0 < (9 / ;10)
52	44, 51	gt0ne0ii 10888	. . . 4 ⊢ (9 / ;10) ≠ 0
53	45, 52	dividi 11084	. . 3 ⊢ ((9 / ;10) / (9 / ;10)) = 1
54	32, 49, 53	3eqtr2i 2855	. 2 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = 1
55	18, 54	eqtri 2849	1 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = 1

Colors of variables: wff setvar class

Syntax hints: ↔ wb 198 = wceq 1658 ∈ wcel 2166 ≠ wne 2999 class class class wbr 4873 ‘cfv 6123 (class class class)co 6905 ℂcc 10250 ℝcr 10251 0cc0 10252 1c1 10253 · cmul 10257 < clt 10391 ≤ cle 10392 − cmin 10585 / cdiv 11009 ℕcn 11350 9c9 11413 ℕ₀cn0 11618 cdc 11821 ↑cexp 13154 abscabs 14351 Σcsu 14793

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-inf2 8815 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330

This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-se 5302 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-pm 8125 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-sup 8617 df-inf 8618 df-oi 8684 df-card 9078 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-n0 11619 df-z 11705 df-dec 11822 df-uz 11969 df-rp 12113 df-fz 12620 df-fzo 12761 df-fl 12888 df-seq 13096 df-exp 13155 df-hash 13411 df-cj 14216 df-re 14217 df-im 14218 df-sqrt 14352 df-abs 14353 df-clim 14596 df-rlim 14597 df-sum 14794

This theorem is referenced by: (None)

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