0.999... - Intuitionistic Logic Explorer

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Theorem 0.999... 12075

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 9224	. . . . . 6 ⊢ 9 ∈ ℂ
2	1	a1i 9	. . . . 5 ⊢ (𝑘 ∈ ℕ → 9 ∈ ℂ)
3		10re 9622	. . . . . . . 8 ⊢ ;10 ∈ ℝ
4	3	recni 8184	. . . . . . 7 ⊢ ;10 ∈ ℂ
5	4	a1i 9	. . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 ∈ ℂ)
6		nnnn0 9402	. . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
7	5, 6	expcld 10928	. . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) ∈ ℂ)
8		10pos 9620	. . . . . . . 8 ⊢ 0 < ;10
9	3, 8	gt0ap0ii 8801	. . . . . . 7 ⊢ ;10 # 0
10	9	a1i 9	. . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 # 0)
11		nnz 9491	. . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
12	5, 10, 11	expap0d 10934	. . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) # 0)
13	2, 7, 12	divrecapd 8966	. . . 4 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · (1 / (;10↑𝑘))))
14	5, 10, 11	exprecapd 10936	. . . . 5 ⊢ (𝑘 ∈ ℕ → ((1 / ;10)↑𝑘) = (1 / (;10↑𝑘)))
15	14	oveq2d 6029	. . . 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 11918	. 2 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘))
18	3, 9	rerecclapi 8950	. . . . 5 ⊢ (1 / ;10) ∈ ℝ
19	18	recni 8184	. . . 4 ⊢ (1 / ;10) ∈ ℂ
20		0re 8172	. . . . . . 7 ⊢ 0 ∈ ℝ
21	3, 8	recgt0ii 9080	. . . . . . 7 ⊢ 0 < (1 / ;10)
22	20, 18, 21	ltleii 8275	. . . . . 6 ⊢ 0 ≤ (1 / ;10)
23	18	absidi 11680	. . . . . 6 ⊢ (0 ≤ (1 / ;10) → (abs‘(1 / ;10)) = (1 / ;10))
24	22, 23	ax-mp 5	. . . . 5 ⊢ (abs‘(1 / ;10)) = (1 / ;10)
25		1lt10 9742	. . . . . 6 ⊢ 1 < ;10
26		recgt1 9070	. . . . . . 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 4107	. . . 4 ⊢ (abs‘(1 / ;10)) < 1
30		geoisum1c 12074	. . . 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 8930	. . . 4 ⊢ (9 / ;10) = (9 · (1 / ;10))
33	1, 4, 9	divcanap2i 8928	. . . . . 6 ⊢ (;10 · (9 / ;10)) = 9
34		ax-1cn 8118	. . . . . . . 8 ⊢ 1 ∈ ℂ
35	4, 34, 19	subdii 8579	. . . . . . 7 ⊢ (;10 · (1 − (1 / ;10))) = ((;10 · 1) − (;10 · (1 / ;10)))
36	4	mulridi 8174	. . . . . . . 8 ⊢ (;10 · 1) = ;10
37	4, 9	recidapi 8916	. . . . . . . 8 ⊢ (;10 · (1 / ;10)) = 1
38	36, 37	oveq12i 6025	. . . . . . 7 ⊢ ((;10 · 1) − (;10 · (1 / ;10))) = (;10 − 1)
39		10m1e9 9699	. . . . . . 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 9223	. . . . . . . 8 ⊢ 9 ∈ ℝ
43	42, 3, 9	redivclapi 8952	. . . . . . 7 ⊢ (9 / ;10) ∈ ℝ
44	43	recni 8184	. . . . . 6 ⊢ (9 / ;10) ∈ ℂ
45	34, 19	subcli 8448	. . . . . 6 ⊢ (1 − (1 / ;10)) ∈ ℂ
46	44, 45, 4, 9	mulcanapi 8840	. . . . 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 6025	. . 3 ⊢ ((9 / ;10) / (9 / ;10)) = ((9 · (1 / ;10)) / (1 − (1 / ;10)))
49		9pos 9240	. . . . . 6 ⊢ 0 < 9
50	42, 3, 49, 8	divgt0ii 9092	. . . . 5 ⊢ 0 < (9 / ;10)
51	43, 50	gt0ap0ii 8801	. . . 4 ⊢ (9 / ;10) # 0
52	44, 51	dividapi 8918	. . 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 4086 ‘cfv 5324 (class class class)co 6013 ℂcc 8023 ℝcr 8024 0cc0 8025 1c1 8026 · cmul 8030 < clt 8207 ≤ cle 8208 − cmin 8343 # cap 8754 / cdiv 8845 ℕcn 9136 9c9 9194 cdc 9604 ↑cexp 10793 abscabs 11551 Σcsu 11907

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 4202 ax-sep 4205 ax-nul 4213 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-iinf 4684 ax-cnex 8116 ax-resscn 8117 ax-1cn 8118 ax-1re 8119 ax-icn 8120 ax-addcl 8121 ax-addrcl 8122 ax-mulcl 8123 ax-mulrcl 8124 ax-addcom 8125 ax-mulcom 8126 ax-addass 8127 ax-mulass 8128 ax-distr 8129 ax-i2m1 8130 ax-0lt1 8131 ax-1rid 8132 ax-0id 8133 ax-rnegex 8134 ax-precex 8135 ax-cnre 8136 ax-pre-ltirr 8137 ax-pre-ltwlin 8138 ax-pre-lttrn 8139 ax-pre-apti 8140 ax-pre-ltadd 8141 ax-pre-mulgt0 8142 ax-pre-mulext 8143 ax-arch 8144 ax-caucvg 8145

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 2802 df-sbc 3030 df-csb 3126 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-nul 3493 df-if 3604 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-int 3927 df-iun 3970 df-br 4087 df-opab 4149 df-mpt 4150 df-tr 4186 df-id 4388 df-po 4391 df-iso 4392 df-iord 4461 df-on 4463 df-ilim 4464 df-suc 4466 df-iom 4687 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-rn 4734 df-res 4735 df-ima 4736 df-iota 5284 df-fun 5326 df-fn 5327 df-f 5328 df-f1 5329 df-fo 5330 df-f1o 5331 df-fv 5332 df-isom 5333 df-riota 5966 df-ov 6016 df-oprab 6017 df-mpo 6018 df-1st 6298 df-2nd 6299 df-recs 6466 df-irdg 6531 df-frec 6552 df-1o 6577 df-oadd 6581 df-er 6697 df-en 6905 df-dom 6906 df-fin 6907 df-pnf 8209 df-mnf 8210 df-xr 8211 df-ltxr 8212 df-le 8213 df-sub 8345 df-neg 8346 df-reap 8748 df-ap 8755 df-div 8846 df-inn 9137 df-2 9195 df-3 9196 df-4 9197 df-5 9198 df-6 9199 df-7 9200 df-8 9201 df-9 9202 df-n0 9396 df-z 9473 df-dec 9605 df-uz 9749 df-q 9847 df-rp 9882 df-fz 10237 df-fzo 10371 df-seqfrec 10703 df-exp 10794 df-ihash 11031 df-cj 11396 df-re 11397 df-im 11398 df-rsqrt 11552 df-abs 11553 df-clim 11833 df-sumdc 11908

This theorem is referenced by: (None)

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