0.999... - Intuitionistic Logic Explorer

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Theorem 0.999... 10911

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 8508	. . . . . 6 ⊢ 9 ∈ ℂ
2	1	a1i 9	. . . . 5 ⊢ (𝑘 ∈ ℕ → 9 ∈ ℂ)
3		10re 8893	. . . . . . . 8 ⊢ ;10 ∈ ℝ
4	3	recni 7498	. . . . . . 7 ⊢ ;10 ∈ ℂ
5	4	a1i 9	. . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 ∈ ℂ)
6		nnnn0 8678	. . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
7	5, 6	expcld 10082	. . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) ∈ ℂ)
8		10pos 8891	. . . . . . . 8 ⊢ 0 < ;10
9	3, 8	gt0ap0ii 8102	. . . . . . 7 ⊢ ;10 # 0
10	9	a1i 9	. . . . . 6 ⊢ (𝑘 ∈ ℕ → ;10 # 0)
11		nnz 8767	. . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
12	5, 10, 11	expap0d 10088	. . . . 5 ⊢ (𝑘 ∈ ℕ → (;10↑𝑘) # 0)
13	2, 7, 12	divrecapd 8258	. . . 4 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · (1 / (;10↑𝑘))))
14	5, 10, 11	exprecapd 10090	. . . . 5 ⊢ (𝑘 ∈ ℕ → ((1 / ;10)↑𝑘) = (1 / (;10↑𝑘)))
15	14	oveq2d 5668	. . . 4 ⊢ (𝑘 ∈ ℕ → (9 · ((1 / ;10)↑𝑘)) = (9 · (1 / (;10↑𝑘))))
16	13, 15	eqtr4d 2123	. . 3 ⊢ (𝑘 ∈ ℕ → (9 / (;10↑𝑘)) = (9 · ((1 / ;10)↑𝑘)))
17	16	sumeq2i 10749	. 2 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘))
18	3, 9	rerecclapi 8242	. . . . 5 ⊢ (1 / ;10) ∈ ℝ
19	18	recni 7498	. . . 4 ⊢ (1 / ;10) ∈ ℂ
20		0re 7486	. . . . . . 7 ⊢ 0 ∈ ℝ
21	3, 8	recgt0ii 8366	. . . . . . 7 ⊢ 0 < (1 / ;10)
22	20, 18, 21	ltleii 7585	. . . . . 6 ⊢ 0 ≤ (1 / ;10)
23	18	absidi 10555	. . . . . 6 ⊢ (0 ≤ (1 / ;10) → (abs‘(1 / ;10)) = (1 / ;10))
24	22, 23	ax-mp 7	. . . . 5 ⊢ (abs‘(1 / ;10)) = (1 / ;10)
25		1lt10 9013	. . . . . 6 ⊢ 1 < ;10
26		recgt1 8356	. . . . . . 7 ⊢ ((;10 ∈ ℝ ∧ 0 < ;10) → (1 < ;10 ↔ (1 / ;10) < 1))
27	3, 8, 26	mp2an 417	. . . . . 6 ⊢ (1 < ;10 ↔ (1 / ;10) < 1)
28	25, 27	mpbi 143	. . . . 5 ⊢ (1 / ;10) < 1
29	24, 28	eqbrtri 3864	. . . 4 ⊢ (abs‘(1 / ;10)) < 1
30		geoisum1c 10910	. . . 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 1273	. . 3 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = ((9 · (1 / ;10)) / (1 − (1 / ;10)))
32	1, 4, 9	divrecapi 8222	. . . 4 ⊢ (9 / ;10) = (9 · (1 / ;10))
33	1, 4, 9	divcanap2i 8220	. . . . . 6 ⊢ (;10 · (9 / ;10)) = 9
34		ax-1cn 7436	. . . . . . . 8 ⊢ 1 ∈ ℂ
35	4, 34, 19	subdii 7883	. . . . . . 7 ⊢ (;10 · (1 − (1 / ;10))) = ((;10 · 1) − (;10 · (1 / ;10)))
36	4	mulid1i 7488	. . . . . . . 8 ⊢ (;10 · 1) = ;10
37	4, 9	recidapi 8208	. . . . . . . 8 ⊢ (;10 · (1 / ;10)) = 1
38	36, 37	oveq12i 5664	. . . . . . 7 ⊢ ((;10 · 1) − (;10 · (1 / ;10))) = (;10 − 1)
39		10m1e9 8970	. . . . . . 7 ⊢ (;10 − 1) = 9
40	35, 38, 39	3eqtrri 2113	. . . . . 6 ⊢ 9 = (;10 · (1 − (1 / ;10)))
41	33, 40	eqtri 2108	. . . . 5 ⊢ (;10 · (9 / ;10)) = (;10 · (1 − (1 / ;10)))
42		9re 8507	. . . . . . . 8 ⊢ 9 ∈ ℝ
43	42, 3, 9	redivclapi 8244	. . . . . . 7 ⊢ (9 / ;10) ∈ ℝ
44	43	recni 7498	. . . . . 6 ⊢ (9 / ;10) ∈ ℂ
45	34, 19	subcli 7756	. . . . . 6 ⊢ (1 − (1 / ;10)) ∈ ℂ
46	44, 45, 4, 9	mulcanapi 8134	. . . . 5 ⊢ ((;10 · (9 / ;10)) = (;10 · (1 − (1 / ;10))) ↔ (9 / ;10) = (1 − (1 / ;10)))
47	41, 46	mpbi 143	. . . 4 ⊢ (9 / ;10) = (1 − (1 / ;10))
48	32, 47	oveq12i 5664	. . 3 ⊢ ((9 / ;10) / (9 / ;10)) = ((9 · (1 / ;10)) / (1 − (1 / ;10)))
49		9pos 8524	. . . . . 6 ⊢ 0 < 9
50	42, 3, 49, 8	divgt0ii 8378	. . . . 5 ⊢ 0 < (9 / ;10)
51	43, 50	gt0ap0ii 8102	. . . 4 ⊢ (9 / ;10) # 0
52	44, 51	dividapi 8210	. . 3 ⊢ ((9 / ;10) / (9 / ;10)) = 1
53	31, 48, 52	3eqtr2i 2114	. 2 ⊢ Σ𝑘 ∈ ℕ (9 · ((1 / ;10)↑𝑘)) = 1
54	17, 53	eqtri 2108	1 ⊢ Σ𝑘 ∈ ℕ (9 / (;10↑𝑘)) = 1

Colors of variables: wff set class

Syntax hints: ↔ wb 103 = wceq 1289 ∈ wcel 1438 class class class wbr 3845 ‘cfv 5015 (class class class)co 5652 ℂcc 7346 ℝcr 7347 0cc0 7348 1c1 7349 · cmul 7353 < clt 7520 ≤ cle 7521 − cmin 7651 # cap 8056 / cdiv 8137 ℕcn 8420 9c9 8478 cdc 8875 ↑cexp 9950 abscabs 10426 Σcsu 10738

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-coll 3954 ax-sep 3957 ax-nul 3965 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-setind 4353 ax-iinf 4403 ax-cnex 7434 ax-resscn 7435 ax-1cn 7436 ax-1re 7437 ax-icn 7438 ax-addcl 7439 ax-addrcl 7440 ax-mulcl 7441 ax-mulrcl 7442 ax-addcom 7443 ax-mulcom 7444 ax-addass 7445 ax-mulass 7446 ax-distr 7447 ax-i2m1 7448 ax-0lt1 7449 ax-1rid 7450 ax-0id 7451 ax-rnegex 7452 ax-precex 7453 ax-cnre 7454 ax-pre-ltirr 7455 ax-pre-ltwlin 7456 ax-pre-lttrn 7457 ax-pre-apti 7458 ax-pre-ltadd 7459 ax-pre-mulgt0 7460 ax-pre-mulext 7461 ax-arch 7462 ax-caucvg 7463

This theorem depends on definitions: df-bi 115 df-dc 781 df-3or 925 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-nel 2351 df-ral 2364 df-rex 2365 df-reu 2366 df-rmo 2367 df-rab 2368 df-v 2621 df-sbc 2841 df-csb 2934 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-if 3394 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-int 3689 df-iun 3732 df-br 3846 df-opab 3900 df-mpt 3901 df-tr 3937 df-id 4120 df-po 4123 df-iso 4124 df-iord 4193 df-on 4195 df-ilim 4196 df-suc 4198 df-iom 4406 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-res 4450 df-ima 4451 df-iota 4980 df-fun 5017 df-fn 5018 df-f 5019 df-f1 5020 df-fo 5021 df-f1o 5022 df-fv 5023 df-isom 5024 df-riota 5608 df-ov 5655 df-oprab 5656 df-mpt2 5657 df-1st 5911 df-2nd 5912 df-recs 6070 df-irdg 6135 df-frec 6156 df-1o 6181 df-oadd 6185 df-er 6290 df-en 6456 df-dom 6457 df-fin 6458 df-pnf 7522 df-mnf 7523 df-xr 7524 df-ltxr 7525 df-le 7526 df-sub 7653 df-neg 7654 df-reap 8050 df-ap 8057 df-div 8138 df-inn 8421 df-2 8479 df-3 8480 df-4 8481 df-5 8482 df-6 8483 df-7 8484 df-8 8485 df-9 8486 df-n0 8672 df-z 8749 df-dec 8876 df-uz 9018 df-q 9103 df-rp 9133 df-fz 9423 df-fzo 9550 df-iseq 9849 df-seq3 9850 df-exp 9951 df-ihash 10180 df-cj 10272 df-re 10273 df-im 10274 df-rsqrt 10427 df-abs 10428 df-clim 10663 df-isum 10739

This theorem is referenced by: (None)

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