Theorem 0.999... 8508

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.

Assertion

Ref	Expression
0.999...	|- sum_k e. NN (9 / (10^k)) = 1

Proof of Theorem 0.999...

Step	Hyp	Ref	Expression
1		nnnn0 7315	. . . 4 |- (k e. NN -> k e. NN0)
2		10re 7172	. . . . . . . 8 |- 10 e. RR
3	2	recni 6467	. . . . . . 7 |- 10 e. CC
4		expcl 7824	. . . . . . 7 |- ((10 e. CC /\ k e. NN0) -> (10^k) e. CC)
5	3, 4	mpan 759	. . . . . 6 |- (k e. NN0 -> (10^k) e. CC)
6		10pos 7182	. . . . . . . 8 |- 0 < 10
7	2, 6	gt0ne0ii 6799	. . . . . . 7 |- 10 =/= 0
8		expne0i 7830	. . . . . . 7 |- ((10 e. CC /\ 10 =/= 0 /\ k e. NN0) -> (10^k) =/= 0)
9	3, 7, 8	mp3an12 1181	. . . . . 6 |- (k e. NN0 -> (10^k) =/= 0)
10		9re 7171	. . . . . . . 8 |- 9 e. RR
11	10	recni 6467	. . . . . . 7 |- 9 e. CC
12		divrec 6922	. . . . . . 7 |- ((9 e. CC /\ (10^k) e. CC /\ (10^k) =/= 0) -> (9 / (10^k)) = (9 x. (1 / (10^k))))
13	11, 12	mp3an1 1178	. . . . . 6 |- (((10^k) e. CC /\ (10^k) =/= 0) -> (9 / (10^k)) = (9 x. (1 / (10^k))))
14	5, 9, 13	syl11anc 524	. . . . 5 |- (k e. NN0 -> (9 / (10^k)) = (9 x. (1 / (10^k))))
15		exprecOLD 7838	. . . . . . 7 |- ((10 e. CC /\ k e. NN0 /\ 10 =/= 0) -> ((1 / 10)^k) = (1 / (10^k)))
16	3, 7, 15	mp3an13 1182	. . . . . 6 |- (k e. NN0 -> ((1 / 10)^k) = (1 / (10^k)))
17	16	opreq2d 4898	. . . . 5 |- (k e. NN0 -> (9 x. ((1 / 10)^k)) = (9 x. (1 / (10^k))))
18	14, 17	eqtr4d 1928	. . . 4 |- (k e. NN0 -> (9 / (10^k)) = (9 x. ((1 / 10)^k)))
19	1, 18	syl 12	. . 3 |- (k e. NN -> (9 / (10^k)) = (9 x. ((1 / 10)^k)))
20	19	sumeq2i 8248	. 2 |- sum_k e. NN (9 / (10^k)) = sum_k e. NN (9 x. ((1 / 10)^k))
21	2, 7	rereccli 6979	. . . 4 |- (1 / 10) e. RR
22	21	recni 6467	. . 3 |- (1 / 10) e. CC
23		0re 6603	. . . . . 6 |- 0 e. RR
24	2, 6	recgt0ii 6992	. . . . . 6 |- 0 < (1 / 10)
25	23, 21, 24	ltleii 6756	. . . . 5 |- 0 <_ (1 / 10)
26	21	absidi 8112	. . . . 5 |- (0 <_ (1 / 10) -> (abs` (1 / 10)) = (1 / 10))
27	25, 26	ax-mp 7	. . . 4 |- (abs` (1 / 10)) = (1 / 10)
28		9pos 7181	. . . . . . 7 |- 0 < 9
29		1re 6598	. . . . . . . 8 |- 1 e. RR
30		ltaddpos2 6840	. . . . . . . 8 |- ((9 e. RR /\ 1 e. RR) -> (0 < 9 <-> 1 < (9 + 1)))
31	10, 29, 30	mp2an 761	. . . . . . 7 |- (0 < 9 <-> 1 < (9 + 1))
32	28, 31	mpbi 206	. . . . . 6 |- 1 < (9 + 1)
33		df-10 7162	. . . . . 6 |- 10 = (9 + 1)
34	32, 33	breqtrri 3362	. . . . 5 |- 1 < 10
35		recgt1 7082	. . . . . 6 |- ((10 e. RR /\ 0 < 10) -> (1 < 10 <-> (1 / 10) < 1))
36	2, 6, 35	mp2an 761	. . . . 5 |- (1 < 10 <-> (1 / 10) < 1)
37	34, 36	mpbi 206	. . . 4 |- (1 / 10) < 1
38	27, 37	eqbrtri 3356	. . 3 |- (abs` (1 / 10)) < 1
39		geoisum1c 8507	. . 3 |- ((9 e. CC /\ (1 / 10) e. CC /\ (abs` (1 / 10)) < 1) -> sum_k e. NN (9 x. ((1 / 10)^k)) = ((9 x. (1 / 10)) / (1 - (1 / 10))))
40	11, 22, 38, 39	mp3an 1191	. 2 |- sum_k e. NN (9 x. ((1 / 10)^k)) = ((9 x. (1 / 10)) / (1 - (1 / 10)))
41	11, 3, 7	divreci 6920	. . . 4 |- (9 / 10) = (9 x. (1 / 10))
42	11, 3, 7	divcan2i 6905	. . . . . 6 |- (10 x. (9 / 10)) = 9
43		ax1cn 6422	. . . . . . . 8 |- 1 e. CC
44	3, 43, 22	subdii 6592	. . . . . . 7 |- (10 x. (1 - (1 / 10))) = ((10 x. 1) - (10 x. (1 / 10)))
45	3	mulid1i 6485	. . . . . . . 8 |- (10 x. 1) = 10
46	3, 7	recidi 6916	. . . . . . . 8 |- (10 x. (1 / 10)) = 1
47	45, 46	opreq12i 4894	. . . . . . 7 |- ((10 x. 1) - (10 x. (1 / 10))) = (10 - 1)
48	43, 11	addcomi 6475	. . . . . . . . 9 |- (1 + 9) = (9 + 1)
49	48, 33	eqtr4i 1911	. . . . . . . 8 |- (1 + 9) = 10
50	3, 43, 11, 49	subaddrii 6529	. . . . . . 7 |- (10 - 1) = 9
51	44, 47, 50	3eqtrri 1913	. . . . . 6 |- 9 = (10 x. (1 - (1 / 10)))
52	42, 51	eqtri 1908	. . . . 5 |- (10 x. (9 / 10)) = (10 x. (1 - (1 / 10)))
53	10, 2, 7	redivcli 6976	. . . . . . 7 |- (9 / 10) e. RR
54	53	recni 6467	. . . . . 6 |- (9 / 10) e. CC
55	43, 22	subcli 6523	. . . . . 6 |- (1 - (1 / 10)) e. CC
56	54, 55, 3, 7	mulcani 6878	. . . . 5 |- ((10 x. (9 / 10)) = (10 x. (1 - (1 / 10))) <-> (9 / 10) = (1 - (1 / 10)))
57	52, 56	mpbi 206	. . . 4 |- (9 / 10) = (1 - (1 / 10))
58	41, 57	opreq12i 4894	. . 3 |- ((9 / 10) / (9 / 10)) = ((9 x. (1 / 10)) / (1 - (1 / 10)))
59	10, 2, 28, 6	divgt0ii 7042	. . . . 5 |- 0 < (9 / 10)
60	53, 59	gt0ne0ii 6799	. . . 4 |- (9 / 10) =/= 0
61	54, 60	dividi 6946	. . 3 |- ((9 / 10) / (9 / 10)) = 1
62	58, 61	eqtr3i 1910	. 2 |- ((9 x. (1 / 10)) / (1 - (1 / 10))) = 1
63	20, 40, 62	3eqtri 1912	1 |- sum_k e. NN (9 / (10^k)) = 1

Syntax hints:   <-> wb 163   = wceq 1298   e. wcel 1300   =/= wne 2017   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387   + caddc 6389   x. cmul 6391   - cmin 6445   / cdiv 6447   <_ cle 6448  NNcn 6449  NN0cn0 6450   < clt 6653  9c9 7152  10c10 7153  ^cexp 7811  abscabs 8000  sum_csu 8239

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-3 7155  df-4 7156  df-5 7157  df-6 7158  df-7 7159  df-8 7160  df-9 7161  df-10 7162  df-n0 7309  df-z 7345  df-fl 7463  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-seq0 7777  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-clim 8235  df-sum 8240

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