0.999... - Metamath Proof Explorer

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Theorem 0.999... 15824

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 12309	. . . . 5 ⊢ 9 ∈ ℂ
2		10re 12693	. . . . . . 7 ⊢ ;10 ∈ ℝ
3	2	recni 11225	. . . . . 6 ⊢ ;10 ∈ ℂ
4		nnnn0 12476	. . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
5		expcl 14042	. . . . . 6 ⊢ ((;10 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (;10↑𝑘) ∈ ℂ)
6	3, 4, 5	sylancr 588	. . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) ∈ ℂ)
7	3	a1i 11	. . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 ∈ ℂ)
8		10pos 12691	. . . . . . . 8 ⊢ 0 < ;10
9	2, 8	gt0ne0ii 11747	. . . . . . 7 ⊢ ;10 ≠ 0
10	9	a1i 11	. . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 ≠ 0)
11		nnz 12576	. . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
12	7, 10, 11	expne0d 14114	. . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) ≠ 0)
13		divrec 11885	. . . . 5 ⊢ ((9 ∈ ℂ ∧ (;10↑𝑘) ∈ ℂ ∧ (;10↑𝑘) ≠ 0) → (9 / (;10↑𝑘)) = (9 · (1 / (;10↑𝑘))))
14	1, 6, 12, 13	mp3an2i 1467	. . . 4 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · (1 / (;10↑𝑘))))
15	7, 10, 11	exprecd 14116	. . . . 5 ⊢ (𝑘 ∈ ℕ → ((1 / ;10)↑𝑘) = (1 / (;10↑𝑘)))
16	15	oveq2d 7422	. . . 4 ⊢ (𝑘 ∈ ℕ → (9 · ((1 / ;10)↑𝑘)) = (9 · (1 / (;10↑𝑘))))
17	14, 16	eqtr4d 2776	. . 3 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · ((1 / ;10)↑𝑘)))
18	17	sumeq2i 15642	. 2 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘))
19	2, 9	rereccli 11976	. . . . 5 ⊢ (1 / ;10) ∈ ℝ
20	19	recni 11225	. . . 4 ⊢ (1 / ;10) ∈ ℂ
21		0re 11213	. . . . . . 7 ⊢ 0 ∈ ℝ
22	2, 8	recgt0ii 12117	. . . . . . 7 ⊢ 0 < (1 / ;10)
23	21, 19, 22	ltleii 11334	. . . . . 6 ⊢ 0 ≤ (1 / ;10)
24	19	absidi 15321	. . . . . 6 ⊢ (0 ≤ (1 / ;10) → (abs‘(1 / ;10)) = (1 / ;10))
25	23, 24	ax-mp 5	. . . . 5 ⊢ (abs‘(1 / ;10)) = (1 / ;10)
26		1lt10 12813	. . . . . 6 ⊢ 1 < ;10
27		recgt1 12107	. . . . . . 7 ⊢ ((;10 ∈ ℝ ∧ 0 < ;10) → (1 < ;10 ↔ (1 / ;10) < 1))
28	2, 8, 27	mp2an 691	. . . . . 6 ⊢ (1 < ;10 ↔ (1 / ;10) < 1)
29	26, 28	mpbi 229	. . . . 5 ⊢ (1 / ;10) < 1
30	25, 29	eqbrtri 5169	. . . 4 ⊢ (abs‘(1 / ;10)) < 1
31		geoisum1c 15823	. . . 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 1462	. . 3 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = ((9 · (1 / ;10)) / (1 − (1 / ;10)))
33	1, 3, 9	divreci 11956	. . . 4 ⊢ (9 / ;10) = (9 · (1 / ;10))
34	1, 3, 9	divcan2i 11954	. . . . . 6 ⊢ (;10 · (9 / ;10)) = 9
35		ax-1cn 11165	. . . . . . . 8 ⊢ 1 ∈ ℂ
36	3, 35, 20	subdii 11660	. . . . . . 7 ⊢ (;10 · (1 − (1 / ;10))) = ((;10 · 1) − (;10 · (1 / ;10)))
37	3	mulridi 11215	. . . . . . . 8 ⊢ (;10 · 1) = ;10
38	3, 9	recidi 11942	. . . . . . . 8 ⊢ (;10 · (1 / ;10)) = 1
39	37, 38	oveq12i 7418	. . . . . . 7 ⊢ ((;10 · 1) − (;10 · (1 / ;10))) = (;10 − 1)
40		10m1e9 12770	. . . . . . 7 ⊢ (;10 − 1) = 9
41	36, 39, 40	3eqtrri 2766	. . . . . 6 ⊢ 9 = (;10 · (1 − (1 / ;10)))
42	34, 41	eqtri 2761	. . . . 5 ⊢ (;10 · (9 / ;10)) = (;10 · (1 − (1 / ;10)))
43		9re 12308	. . . . . . . 8 ⊢ 9 ∈ ℝ
44	43, 2, 9	redivcli 11978	. . . . . . 7 ⊢ (9 / ;10) ∈ ℝ
45	44	recni 11225	. . . . . 6 ⊢ (9 / ;10) ∈ ℂ
46	35, 20	subcli 11533	. . . . . 6 ⊢ (1 − (1 / ;10)) ∈ ℂ
47	45, 46, 3, 9	mulcani 11850	. . . . 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 7418	. . 3 ⊢ ((9 / ;10) / (9 / ;10)) = ((9 · (1 / ;10)) / (1 − (1 / ;10)))
50		9pos 12322	. . . . . 6 ⊢ 0 < 9
51	43, 2, 50, 8	divgt0ii 12128	. . . . 5 ⊢ 0 < (9 / ;10)
52	44, 51	gt0ne0ii 11747	. . . 4 ⊢ (9 / ;10) ≠ 0
53	45, 52	dividi 11944	. . 3 ⊢ ((9 / ;10) / (9 / ;10)) = 1
54	32, 49, 53	3eqtr2i 2767	. 2 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = 1
55	18, 54	eqtri 2761	1 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = 1

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 class class class wbr 5148 ‘cfv 6541 (class class class)co 7406 ℂcc 11105 ℝcr 11106 0cc0 11107 1c1 11108 · cmul 11112 < clt 11245 ≤ cle 11246 − cmin 11441 / cdiv 11868 ℕcn 12209 9c9 12271 ℕ₀cn0 12469 cdc 12674 ↑cexp 14024 abscabs 15178 Σcsu 15629

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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-inf2 9633 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-pm 8820 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-inf 9435 df-oi 9502 df-card 9931 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-rp 12972 df-fz 13482 df-fzo 13625 df-fl 13754 df-seq 13964 df-exp 14025 df-hash 14288 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-clim 15429 df-rlim 15430 df-sum 15630

This theorem is referenced by: (None)

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