0.999... - Intuitionistic Logic Explorer

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Theorem 0.999... 11882

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 9137	. . . . . 6 ⊢ 9 ∈ ℂ
2	1	a1i 9	. . . . 5 ⊢ (𝑘 ∈ ℕ → 9 ∈ ℂ)
3		10re 9535	. . . . . . . 8 ⊢ ;10 ∈ ℝ
4	3	recni 8097	. . . . . . 7 ⊢ ;10 ∈ ℂ
5	4	a1i 9	. . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 ∈ ℂ)
6		nnnn0 9315	. . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
7	5, 6	expcld 10831	. . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) ∈ ℂ)
8		10pos 9533	. . . . . . . 8 ⊢ 0 < ;10
9	3, 8	gt0ap0ii 8714	. . . . . . 7 ⊢ ;10 # 0
10	9	a1i 9	. . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 # 0)
11		nnz 9404	. . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
12	5, 10, 11	expap0d 10837	. . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) # 0)
13	2, 7, 12	divrecapd 8879	. . . 4 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · (1 / (;10↑𝑘))))
14	5, 10, 11	exprecapd 10839	. . . . 5 ⊢ (𝑘 ∈ ℕ → ((1 / ;10)↑𝑘) = (1 / (;10↑𝑘)))
15	14	oveq2d 5970	. . . 4 ⊢ (𝑘 ∈ ℕ → (9 · ((1 / ;10)↑𝑘)) = (9 · (1 / (;10↑𝑘))))
16	13, 15	eqtr4d 2242	. . 3 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · ((1 / ;10)↑𝑘)))
17	16	sumeq2i 11725	. 2 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘))
18	3, 9	rerecclapi 8863	. . . . 5 ⊢ (1 / ;10) ∈ ℝ
19	18	recni 8097	. . . 4 ⊢ (1 / ;10) ∈ ℂ
20		0re 8085	. . . . . . 7 ⊢ 0 ∈ ℝ
21	3, 8	recgt0ii 8993	. . . . . . 7 ⊢ 0 < (1 / ;10)
22	20, 18, 21	ltleii 8188	. . . . . 6 ⊢ 0 ≤ (1 / ;10)
23	18	absidi 11487	. . . . . 6 ⊢ (0 ≤ (1 / ;10) → (abs‘(1 / ;10)) = (1 / ;10))
24	22, 23	ax-mp 5	. . . . 5 ⊢ (abs‘(1 / ;10)) = (1 / ;10)
25		1lt10 9655	. . . . . 6 ⊢ 1 < ;10
26		recgt1 8983	. . . . . . 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 4069	. . . 4 ⊢ (abs‘(1 / ;10)) < 1
30		geoisum1c 11881	. . . 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 1350	. . 3 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = ((9 · (1 / ;10)) / (1 − (1 / ;10)))
32	1, 4, 9	divrecapi 8843	. . . 4 ⊢ (9 / ;10) = (9 · (1 / ;10))
33	1, 4, 9	divcanap2i 8841	. . . . . 6 ⊢ (;10 · (9 / ;10)) = 9
34		ax-1cn 8031	. . . . . . . 8 ⊢ 1 ∈ ℂ
35	4, 34, 19	subdii 8492	. . . . . . 7 ⊢ (;10 · (1 − (1 / ;10))) = ((;10 · 1) − (;10 · (1 / ;10)))
36	4	mulridi 8087	. . . . . . . 8 ⊢ (;10 · 1) = ;10
37	4, 9	recidapi 8829	. . . . . . . 8 ⊢ (;10 · (1 / ;10)) = 1
38	36, 37	oveq12i 5966	. . . . . . 7 ⊢ ((;10 · 1) − (;10 · (1 / ;10))) = (;10 − 1)
39		10m1e9 9612	. . . . . . 7 ⊢ (;10 − 1) = 9
40	35, 38, 39	3eqtrri 2232	. . . . . 6 ⊢ 9 = (;10 · (1 − (1 / ;10)))
41	33, 40	eqtri 2227	. . . . 5 ⊢ (;10 · (9 / ;10)) = (;10 · (1 − (1 / ;10)))
42		9re 9136	. . . . . . . 8 ⊢ 9 ∈ ℝ
43	42, 3, 9	redivclapi 8865	. . . . . . 7 ⊢ (9 / ;10) ∈ ℝ
44	43	recni 8097	. . . . . 6 ⊢ (9 / ;10) ∈ ℂ
45	34, 19	subcli 8361	. . . . . 6 ⊢ (1 − (1 / ;10)) ∈ ℂ
46	44, 45, 4, 9	mulcanapi 8753	. . . . 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 5966	. . 3 ⊢ ((9 / ;10) / (9 / ;10)) = ((9 · (1 / ;10)) / (1 − (1 / ;10)))
49		9pos 9153	. . . . . 6 ⊢ 0 < 9
50	42, 3, 49, 8	divgt0ii 9005	. . . . 5 ⊢ 0 < (9 / ;10)
51	43, 50	gt0ap0ii 8714	. . . 4 ⊢ (9 / ;10) # 0
52	44, 51	dividapi 8831	. . 3 ⊢ ((9 / ;10) / (9 / ;10)) = 1
53	31, 48, 52	3eqtr2i 2233	. 2 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = 1
54	17, 53	eqtri 2227	1 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = 1

Colors of variables: wff set class

Syntax hints: ↔ wb 105 = wceq 1373 ∈ wcel 2177 class class class wbr 4048 ‘cfv 5277 (class class class)co 5954 ℂcc 7936 ℝcr 7937 0cc0 7938 1c1 7939 · cmul 7943 < clt 8120 ≤ cle 8121 − cmin 8256 # cap 8667 / cdiv 8758 ℕcn 9049 9c9 9107 cdc 9517 ↑cexp 10696 abscabs 11358 Σcsu 11714

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 615 ax-in2 616 ax-io 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-coll 4164 ax-sep 4167 ax-nul 4175 ax-pow 4223 ax-pr 4258 ax-un 4485 ax-setind 4590 ax-iinf 4641 ax-cnex 8029 ax-resscn 8030 ax-1cn 8031 ax-1re 8032 ax-icn 8033 ax-addcl 8034 ax-addrcl 8035 ax-mulcl 8036 ax-mulrcl 8037 ax-addcom 8038 ax-mulcom 8039 ax-addass 8040 ax-mulass 8041 ax-distr 8042 ax-i2m1 8043 ax-0lt1 8044 ax-1rid 8045 ax-0id 8046 ax-rnegex 8047 ax-precex 8048 ax-cnre 8049 ax-pre-ltirr 8050 ax-pre-ltwlin 8051 ax-pre-lttrn 8052 ax-pre-apti 8053 ax-pre-ltadd 8054 ax-pre-mulgt0 8055 ax-pre-mulext 8056 ax-arch 8057 ax-caucvg 8058

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-nel 2473 df-ral 2490 df-rex 2491 df-reu 2492 df-rmo 2493 df-rab 2494 df-v 2775 df-sbc 3001 df-csb 3096 df-dif 3170 df-un 3172 df-in 3174 df-ss 3181 df-nul 3463 df-if 3574 df-pw 3620 df-sn 3641 df-pr 3642 df-op 3644 df-uni 3854 df-int 3889 df-iun 3932 df-br 4049 df-opab 4111 df-mpt 4112 df-tr 4148 df-id 4345 df-po 4348 df-iso 4349 df-iord 4418 df-on 4420 df-ilim 4421 df-suc 4423 df-iom 4644 df-xp 4686 df-rel 4687 df-cnv 4688 df-co 4689 df-dm 4690 df-rn 4691 df-res 4692 df-ima 4693 df-iota 5238 df-fun 5279 df-fn 5280 df-f 5281 df-f1 5282 df-fo 5283 df-f1o 5284 df-fv 5285 df-isom 5286 df-riota 5909 df-ov 5957 df-oprab 5958 df-mpo 5959 df-1st 6236 df-2nd 6237 df-recs 6401 df-irdg 6466 df-frec 6487 df-1o 6512 df-oadd 6516 df-er 6630 df-en 6838 df-dom 6839 df-fin 6840 df-pnf 8122 df-mnf 8123 df-xr 8124 df-ltxr 8125 df-le 8126 df-sub 8258 df-neg 8259 df-reap 8661 df-ap 8668 df-div 8759 df-inn 9050 df-2 9108 df-3 9109 df-4 9110 df-5 9111 df-6 9112 df-7 9113 df-8 9114 df-9 9115 df-n0 9309 df-z 9386 df-dec 9518 df-uz 9662 df-q 9754 df-rp 9789 df-fz 10144 df-fzo 10278 df-seqfrec 10606 df-exp 10697 df-ihash 10934 df-cj 11203 df-re 11204 df-im 11205 df-rsqrt 11359 df-abs 11360 df-clim 11640 df-sumdc 11715

This theorem is referenced by: (None)

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