0.999... - Intuitionistic Logic Explorer

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Theorem 0.999... 12018

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. (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 9186	. . . . . 6 ⊢ 9 ∈ ℂ
2	1	a1i 9	. . . . 5 ⊢ (𝑘 ∈ ℕ → 9 ∈ ℂ)
3		10re 9584	. . . . . . . 8 ⊢ ;10 ∈ ℝ
4	3	recni 8146	. . . . . . 7 ⊢ ;10 ∈ ℂ
5	4	a1i 9	. . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 ∈ ℂ)
6		nnnn0 9364	. . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
7	5, 6	expcld 10882	. . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) ∈ ℂ)
8		10pos 9582	. . . . . . . 8 ⊢ 0 < ;10
9	3, 8	gt0ap0ii 8763	. . . . . . 7 ⊢ ;10 # 0
10	9	a1i 9	. . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 # 0)
11		nnz 9453	. . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
12	5, 10, 11	expap0d 10888	. . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) # 0)
13	2, 7, 12	divrecapd 8928	. . . 4 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · (1 / (;10↑𝑘))))
14	5, 10, 11	exprecapd 10890	. . . . 5 ⊢ (𝑘 ∈ ℕ → ((1 / ;10)↑𝑘) = (1 / (;10↑𝑘)))
15	14	oveq2d 6010	. . . 4 ⊢ (𝑘 ∈ ℕ → (9 · ((1 / ;10)↑𝑘)) = (9 · (1 / (;10↑𝑘))))
16	13, 15	eqtr4d 2265	. . 3 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · ((1 / ;10)↑𝑘)))
17	16	sumeq2i 11861	. 2 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘))
18	3, 9	rerecclapi 8912	. . . . 5 ⊢ (1 / ;10) ∈ ℝ
19	18	recni 8146	. . . 4 ⊢ (1 / ;10) ∈ ℂ
20		0re 8134	. . . . . . 7 ⊢ 0 ∈ ℝ
21	3, 8	recgt0ii 9042	. . . . . . 7 ⊢ 0 < (1 / ;10)
22	20, 18, 21	ltleii 8237	. . . . . 6 ⊢ 0 ≤ (1 / ;10)
23	18	absidi 11623	. . . . . 6 ⊢ (0 ≤ (1 / ;10) → (abs‘(1 / ;10)) = (1 / ;10))
24	22, 23	ax-mp 5	. . . . 5 ⊢ (abs‘(1 / ;10)) = (1 / ;10)
25		1lt10 9704	. . . . . 6 ⊢ 1 < ;10
26		recgt1 9032	. . . . . . 7 ⊢ ((;10 ∈ ℝ ∧ 0 < ;10) → (1 < ;10 ↔ (1 / ;10) < 1))
27	3, 8, 26	mp2an 426	. . . . . 6 ⊢ (1 < ;10 ↔ (1 / ;10) < 1)
28	25, 27	mpbi 145	. . . . 5 ⊢ (1 / ;10) < 1
29	24, 28	eqbrtri 4103	. . . 4 ⊢ (abs‘(1 / ;10)) < 1
30		geoisum1c 12017	. . . 4 ⊢ ((9 ∈ ℂ ∧ (1 / ;10) ∈ ℂ ∧ (abs‘(1 / ;10)) < 1) → Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = ((9 · (1 / ;10)) / (1 − (1 / ;10))))
31	1, 19, 29, 30	mp3an 1371	. . 3 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = ((9 · (1 / ;10)) / (1 − (1 / ;10)))
32	1, 4, 9	divrecapi 8892	. . . 4 ⊢ (9 / ;10) = (9 · (1 / ;10))
33	1, 4, 9	divcanap2i 8890	. . . . . 6 ⊢ (;10 · (9 / ;10)) = 9
34		ax-1cn 8080	. . . . . . . 8 ⊢ 1 ∈ ℂ
35	4, 34, 19	subdii 8541	. . . . . . 7 ⊢ (;10 · (1 − (1 / ;10))) = ((;10 · 1) − (;10 · (1 / ;10)))
36	4	mulridi 8136	. . . . . . . 8 ⊢ (;10 · 1) = ;10
37	4, 9	recidapi 8878	. . . . . . . 8 ⊢ (;10 · (1 / ;10)) = 1
38	36, 37	oveq12i 6006	. . . . . . 7 ⊢ ((;10 · 1) − (;10 · (1 / ;10))) = (;10 − 1)
39		10m1e9 9661	. . . . . . 7 ⊢ (;10 − 1) = 9
40	35, 38, 39	3eqtrri 2255	. . . . . 6 ⊢ 9 = (;10 · (1 − (1 / ;10)))
41	33, 40	eqtri 2250	. . . . 5 ⊢ (;10 · (9 / ;10)) = (;10 · (1 − (1 / ;10)))
42		9re 9185	. . . . . . . 8 ⊢ 9 ∈ ℝ
43	42, 3, 9	redivclapi 8914	. . . . . . 7 ⊢ (9 / ;10) ∈ ℝ
44	43	recni 8146	. . . . . 6 ⊢ (9 / ;10) ∈ ℂ
45	34, 19	subcli 8410	. . . . . 6 ⊢ (1 − (1 / ;10)) ∈ ℂ
46	44, 45, 4, 9	mulcanapi 8802	. . . . 5 ⊢ ((;10 · (9 / ;10)) = (;10 · (1 − (1 / ;10))) ↔ (9 / ;10) = (1 − (1 / ;10)))
47	41, 46	mpbi 145	. . . 4 ⊢ (9 / ;10) = (1 − (1 / ;10))
48	32, 47	oveq12i 6006	. . 3 ⊢ ((9 / ;10) / (9 / ;10)) = ((9 · (1 / ;10)) / (1 − (1 / ;10)))
49		9pos 9202	. . . . . 6 ⊢ 0 < 9
50	42, 3, 49, 8	divgt0ii 9054	. . . . 5 ⊢ 0 < (9 / ;10)
51	43, 50	gt0ap0ii 8763	. . . 4 ⊢ (9 / ;10) # 0
52	44, 51	dividapi 8880	. . 3 ⊢ ((9 / ;10) / (9 / ;10)) = 1
53	31, 48, 52	3eqtr2i 2256	. 2 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = 1
54	17, 53	eqtri 2250	1 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = 1

Colors of variables: wff set class

Syntax hints: ↔ wb 105 = wceq 1395 ∈ wcel 2200 class class class wbr 4082 ‘cfv 5314 (class class class)co 5994 ℂcc 7985 ℝcr 7986 0cc0 7987 1c1 7988 · cmul 7992 < clt 8169 ≤ cle 8170 − cmin 8305 # cap 8716 / cdiv 8807 ℕcn 9098 9c9 9156 cdc 9566 ↑cexp 10747 abscabs 11494 Σcsu 11850

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4198 ax-sep 4201 ax-nul 4209 ax-pow 4257 ax-pr 4292 ax-un 4521 ax-setind 4626 ax-iinf 4677 ax-cnex 8078 ax-resscn 8079 ax-1cn 8080 ax-1re 8081 ax-icn 8082 ax-addcl 8083 ax-addrcl 8084 ax-mulcl 8085 ax-mulrcl 8086 ax-addcom 8087 ax-mulcom 8088 ax-addass 8089 ax-mulass 8090 ax-distr 8091 ax-i2m1 8092 ax-0lt1 8093 ax-1rid 8094 ax-0id 8095 ax-rnegex 8096 ax-precex 8097 ax-cnre 8098 ax-pre-ltirr 8099 ax-pre-ltwlin 8100 ax-pre-lttrn 8101 ax-pre-apti 8102 ax-pre-ltadd 8103 ax-pre-mulgt0 8104 ax-pre-mulext 8105 ax-arch 8106 ax-caucvg 8107

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3888 df-int 3923 df-iun 3966 df-br 4083 df-opab 4145 df-mpt 4146 df-tr 4182 df-id 4381 df-po 4384 df-iso 4385 df-iord 4454 df-on 4456 df-ilim 4457 df-suc 4459 df-iom 4680 df-xp 4722 df-rel 4723 df-cnv 4724 df-co 4725 df-dm 4726 df-rn 4727 df-res 4728 df-ima 4729 df-iota 5274 df-fun 5316 df-fn 5317 df-f 5318 df-f1 5319 df-fo 5320 df-f1o 5321 df-fv 5322 df-isom 5323 df-riota 5947 df-ov 5997 df-oprab 5998 df-mpo 5999 df-1st 6276 df-2nd 6277 df-recs 6441 df-irdg 6506 df-frec 6527 df-1o 6552 df-oadd 6556 df-er 6670 df-en 6878 df-dom 6879 df-fin 6880 df-pnf 8171 df-mnf 8172 df-xr 8173 df-ltxr 8174 df-le 8175 df-sub 8307 df-neg 8308 df-reap 8710 df-ap 8717 df-div 8808 df-inn 9099 df-2 9157 df-3 9158 df-4 9159 df-5 9160 df-6 9161 df-7 9162 df-8 9163 df-9 9164 df-n0 9358 df-z 9435 df-dec 9567 df-uz 9711 df-q 9803 df-rp 9838 df-fz 10193 df-fzo 10327 df-seqfrec 10657 df-exp 10748 df-ihash 10985 df-cj 11339 df-re 11340 df-im 11341 df-rsqrt 11495 df-abs 11496 df-clim 11776 df-sumdc 11851

This theorem is referenced by: (None)

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