Theorem 0.999... 7181

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		nnnn0t 6053	. . . 4 |- (k e. NN -> k e. NN0)
2		9re 5934	. . . . . . . 8 |- 9 e. RR
3	2	recn 5286	. . . . . . 7 |- 9 e. CC
4		divrect 5702	. . . . . . 7 |- ((9 e. CC /\ (10^k) e. CC /\ (10^k) =/= 0) -> (9 / (10^k)) = (9 x. (1 / (10^k))))
5	3, 4	mp3an1 900	. . . . . 6 |- (((10^k) e. CC /\ (10^k) =/= 0) -> (9 / (10^k)) = (9 x. (1 / (10^k))))
6		10re 5935	. . . . . . . 8 |- 10 e. RR
7	6	recn 5286	. . . . . . 7 |- 10 e. CC
8		expclt 6513	. . . . . . 7 |- ((10 e. CC /\ k e. NN0) -> (10^k) e. CC)
9	7, 8	mpan 693	. . . . . 6 |- (k e. NN0 -> (10^k) e. CC)
10		10pos 5945	. . . . . . . 8 |- 0 < 10
11	6, 10	gt0ne0i 5591	. . . . . . 7 |- 10 =/= 0
12		expne0it 6519	. . . . . . 7 |- ((10 e. CC /\ k e. NN0 /\ 10 =/= 0) -> (10^k) =/= 0)
13	7, 11, 12	mp3an13 904	. . . . . 6 |- (k e. NN0 -> (10^k) =/= 0)
14	5, 9, 13	sylanc 471	. . . . 5 |- (k e. NN0 -> (9 / (10^k)) = (9 x. (1 / (10^k))))
15		recexpt 6526	. . . . . . 7 |- ((10 e. CC /\ k e. NN0 /\ 10 =/= 0) -> ((1 / 10)^k) = (1 / (10^k)))
16	7, 11, 15	mp3an13 904	. . . . . 6 |- (k e. NN0 -> ((1 / 10)^k) = (1 / (10^k)))
17	16	opreq2d 3961	. . . . 5 |- (k e. NN0 -> (9 x. ((1 / 10)^k)) = (9 x. (1 / (10^k))))
18	14, 17	eqtr4d 1502	. . . 4 |- (k e. NN0 -> (9 / (10^k)) = (9 x. ((1 / 10)^k)))
19	1, 18	syl 10	. . 3 |- (k e. NN -> (9 / (10^k)) = (9 x. ((1 / 10)^k)))
20	19	sumeq2i 6926	. 2 |- sum_k e. NN (9 / (10^k)) = sum_k e. NN (9 x. ((1 / 10)^k))
21	6, 11	rereccl 5757	. . . 4 |- (1 / 10) e. RR
22	21	recn 5286	. . 3 |- (1 / 10) e. CC
23		0re 5412	. . . . . 6 |- 0 e. RR
24	6, 10	recgt0i 5770	. . . . . 6 |- 0 < (1 / 10)
25	23, 21, 24	ltlei 5554	. . . . 5 |- 0 <_ (1 / 10)
26	21	absid 6796	. . . . 5 |- (0 <_ (1 / 10) -> (abs` (1 / 10)) = (1 / 10))
27	25, 26	ax-mp 7	. . . 4 |- (abs` (1 / 10)) = (1 / 10)
28		9pos 5944	. . . . . . 7 |- 0 < 9
29		1re 5407	. . . . . . . 8 |- 1 e. RR
30		ltaddpos2t 5625	. . . . . . . 8 |- ((9 e. RR /\ 1 e. RR) -> (0 < 9 <-> 1 < (9 + 1)))
31	2, 29, 30	mp2an 695	. . . . . . 7 |- (0 < 9 <-> 1 < (9 + 1))
32	28, 31	mpbi 189	. . . . . 6 |- 1 < (9 + 1)
33		df-10 5925	. . . . . 6 |- 10 = (9 + 1)
34	32, 33	breqtrr 2630	. . . . 5 |- 1 < 10
35		recgt1t 5847	. . . . . 6 |- ((10 e. RR /\ 0 < 10) -> (1 < 10 <-> (1 / 10) < 1))
36	6, 10, 35	mp2an 695	. . . . 5 |- (1 < 10 <-> (1 / 10) < 1)
37	34, 36	mpbi 189	. . . 4 |- (1 / 10) < 1
38	27, 37	eqbrtr 2624	. . 3 |- (abs` (1 / 10)) < 1
39		geoisum1c 7180	. . 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	3, 22, 38, 39	mp3an 913	. 2 |- sum_k e. NN (9 x. ((1 / 10)^k)) = ((9 x. (1 / 10)) / (1 - (1 / 10)))
41	3, 7, 11	divrec 5700	. . . 4 |- (9 / 10) = (9 x. (1 / 10))
42	7, 3, 11	divcan2 5685	. . . . . 6 |- (10 x. (9 / 10)) = 9
43		ax1cn 5241	. . . . . . . 8 |- 1 e. CC
44	7, 43, 22	subdi 5401	. . . . . . 7 |- (10 x. (1 - (1 / 10))) = ((10 x. 1) - (10 x. (1 / 10)))
45	7	mulid1 5304	. . . . . . . 8 |- (10 x. 1) = 10
46	7, 11	recid 5696	. . . . . . . 8 |- (10 x. (1 / 10)) = 1
47	45, 46	opreq12i 3958	. . . . . . 7 |- ((10 x. 1) - (10 x. (1 / 10))) = (10 - 1)
48	43, 3	addcom 5294	. . . . . . . . 9 |- (1 + 9) = (9 + 1)
49	48, 33	eqtr4 1490	. . . . . . . 8 |- (1 + 9) = 10
50	7, 43, 3, 49	subaddri 5344	. . . . . . 7 |- (10 - 1) = 9
51	44, 47, 50	3eqtrr 1492	. . . . . 6 |- 9 = (10 x. (1 - (1 / 10)))
52	42, 51	eqtr 1487	. . . . 5 |- (10 x. (9 / 10)) = (10 x. (1 - (1 / 10)))
53	2, 6, 11	redivcl 5754	. . . . . . 7 |- (9 / 10) e. RR
54	53	recn 5286	. . . . . 6 |- (9 / 10) e. CC
55	43, 22	subcl 5338	. . . . . 6 |- (1 - (1 / 10)) e. CC
56	7, 54, 55, 11	mulcan 5659	. . . . 5 |- ((10 x. (9 / 10)) = (10 x. (1 - (1 / 10))) <-> (9 / 10) = (1 - (1 / 10)))
57	52, 56	mpbi 189	. . . 4 |- (9 / 10) = (1 - (1 / 10))
58	41, 57	opreq12i 3958	. . 3 |- ((9 / 10) / (9 / 10)) = ((9 x. (1 / 10)) / (1 - (1 / 10)))
59	2, 6, 28, 10	divgt0i 5814	. . . . 5 |- 0 < (9 / 10)
60	53, 59	gt0ne0i 5591	. . . 4 |- (9 / 10) =/= 0
61	54, 60	divid 5726	. . 3 |- ((9 / 10) / (9 / 10)) = 1
62	58, 61	eqtr3 1489	. 2 |- ((9 x. (1 / 10)) / (1 - (1 / 10))) = 1
63	20, 40, 62	3eqtr 1491	1 |- sum_k e. NN (9 / (10^k)) = 1

Syntax hints:   <-> wb 146   = wceq 953   e. wcel 955   =/= wne 1577   class class class wbr 2609  ` cfv 3172  (class class class)co 3948  CCcc 5204  RRcr 5205  0cc0 5206  1c1 5207   + caddc 5209   x. cmul 5211   - cmin 5264   / cdiv 5266   <_ cle 5267  NNcn 5268  NN0cn0 5269   < clt 5458  9c9 5915  10c10 5916  ^cexp 6500  abscabs 6681  sum_csu 6917

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-nel 1580  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-en 4351  df-dom 4352  df-sdom 4353  df-sup 4548  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-ltr 5142  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-1 5214  df-i 5215  df-r 5216  df-plus 5217  df-mul 5218  df-lt 5219  df-sub 5328  df-neg 5330  df-pnf 5459  df-mnf 5460  df-xr 5461  df-ltxr 5462  df-le 5463  df-div 5672  df-n 5873  df-2 5917  df-3 5918  df-4 5919  df-5 5920  df-6 5921  df-7 5922  df-8 5923  df-9 5924  df-10 5925  df-n0 6047  df-z 6083  df-fl 6172  df-seq1 6245  df-shft 6278  df-uz 6350  df-fz 6400  df-seqz 6465  df-seq0 6466  df-exp 6501  df-sqr 6600  df-re 6682  df-im 6683  df-cj 6684  df-abs 6685  df-clim 6913  df-sum 6918

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