Theorem 0.999... 7336

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 6188	. . . 4 |- (k e. NN -> k e. NN0)
2		9re 6048	. . . . . . . 8 |- 9 e. RR
3	2	recni 5379	. . . . . . 7 |- 9 e. CC
4		divrec 5801	. . . . . . 7 |- ((9 e. CC /\ (10^k) e. CC /\ (10^k) =/= 0) -> (9 / (10^k)) = (9 x. (1 / (10^k))))
5	3, 4	mp3an1 915	. . . . . 6 |- (((10^k) e. CC /\ (10^k) =/= 0) -> (9 / (10^k)) = (9 x. (1 / (10^k))))
6		10re 6049	. . . . . . . 8 |- 10 e. RR
7	6	recni 5379	. . . . . . 7 |- 10 e. CC
8		expcl 6670	. . . . . . 7 |- ((10 e. CC /\ k e. NN0) -> (10^k) e. CC)
9	7, 8	mpan 707	. . . . . 6 |- (k e. NN0 -> (10^k) e. CC)
10		10pos 6059	. . . . . . . 8 |- 0 < 10
11	6, 10	gt0ne0ii 5682	. . . . . . 7 |- 10 =/= 0
12		expne0i 6677	. . . . . . 7 |- ((10 e. CC /\ k e. NN0 /\ 10 =/= 0) -> (10^k) =/= 0)
13	7, 11, 12	mp3an13 919	. . . . . 6 |- (k e. NN0 -> (10^k) =/= 0)
14	5, 9, 13	sylanc 482	. . . . 5 |- (k e. NN0 -> (9 / (10^k)) = (9 x. (1 / (10^k))))
15		recexp 6684	. . . . . . 7 |- ((10 e. CC /\ k e. NN0 /\ 10 =/= 0) -> ((1 / 10)^k) = (1 / (10^k)))
16	7, 11, 15	mp3an13 919	. . . . . 6 |- (k e. NN0 -> ((1 / 10)^k) = (1 / (10^k)))
17	16	opreq2d 4034	. . . . 5 |- (k e. NN0 -> (9 x. ((1 / 10)^k)) = (9 x. (1 / (10^k))))
18	14, 17	eqtr4d 1557	. . . 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 7078	. 2 |- sum_k e. NN (9 / (10^k)) = sum_k e. NN (9 x. ((1 / 10)^k))
21	6, 11	rereccli 5859	. . . 4 |- (1 / 10) e. RR
22	21	recni 5379	. . 3 |- (1 / 10) e. CC
23		0re 5505	. . . . . 6 |- 0 e. RR
24	6, 10	recgt0ii 5872	. . . . . 6 |- 0 < (1 / 10)
25	23, 21, 24	ltleii 5646	. . . . 5 |- 0 <_ (1 / 10)
26	21	absidi 6951	. . . . 5 |- (0 <_ (1 / 10) -> (abs` (1 / 10)) = (1 / 10))
27	25, 26	ax-mp 7	. . . 4 |- (abs` (1 / 10)) = (1 / 10)
28		9pos 6058	. . . . . . 7 |- 0 < 9
29		1re 5500	. . . . . . . 8 |- 1 e. RR
30		ltaddpos2 5717	. . . . . . . 8 |- ((9 e. RR /\ 1 e. RR) -> (0 < 9 <-> 1 < (9 + 1)))
31	2, 29, 30	mp2an 709	. . . . . . 7 |- (0 < 9 <-> 1 < (9 + 1))
32	28, 31	mpbi 196	. . . . . 6 |- 1 < (9 + 1)
33		df-10 6039	. . . . . 6 |- 10 = (9 + 1)
34	32, 33	breqtrri 2695	. . . . 5 |- 1 < 10
35		recgt1 5959	. . . . . 6 |- ((10 e. RR /\ 0 < 10) -> (1 < 10 <-> (1 / 10) < 1))
36	6, 10, 35	mp2an 709	. . . . 5 |- (1 < 10 <-> (1 / 10) < 1)
37	34, 36	mpbi 196	. . . 4 |- (1 / 10) < 1
38	27, 37	eqbrtri 2689	. . 3 |- (abs` (1 / 10)) < 1
39		geoisum1c 7335	. . 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 928	. 2 |- sum_k e. NN (9 x. ((1 / 10)^k)) = ((9 x. (1 / 10)) / (1 - (1 / 10)))
41	3, 7, 11	divreci 5799	. . . 4 |- (9 / 10) = (9 x. (1 / 10))
42	3, 7, 11	divcan2i 5784	. . . . . 6 |- (10 x. (9 / 10)) = 9
43		ax1cn 5334	. . . . . . . 8 |- 1 e. CC
44	7, 43, 22	subdii 5494	. . . . . . 7 |- (10 x. (1 - (1 / 10))) = ((10 x. 1) - (10 x. (1 / 10)))
45	7	mulid1i 5397	. . . . . . . 8 |- (10 x. 1) = 10
46	7, 11	recidi 5795	. . . . . . . 8 |- (10 x. (1 / 10)) = 1
47	45, 46	opreq12i 4031	. . . . . . 7 |- ((10 x. 1) - (10 x. (1 / 10))) = (10 - 1)
48	43, 3	addcomi 5387	. . . . . . . . 9 |- (1 + 9) = (9 + 1)
49	48, 33	eqtr4i 1545	. . . . . . . 8 |- (1 + 9) = 10
50	7, 43, 3, 49	subaddrii 5437	. . . . . . 7 |- (10 - 1) = 9
51	44, 47, 50	3eqtrri 1547	. . . . . 6 |- 9 = (10 x. (1 - (1 / 10)))
52	42, 51	eqtri 1542	. . . . 5 |- (10 x. (9 / 10)) = (10 x. (1 - (1 / 10)))
53	2, 6, 11	redivcli 5856	. . . . . . 7 |- (9 / 10) e. RR
54	53	recni 5379	. . . . . 6 |- (9 / 10) e. CC
55	43, 22	subcli 5431	. . . . . 6 |- (1 - (1 / 10)) e. CC
56	7, 54, 55, 11	mulcaniOLD 5752	. . . . 5 |- ((10 x. (9 / 10)) = (10 x. (1 - (1 / 10))) <-> (9 / 10) = (1 - (1 / 10)))
57	52, 56	mpbi 196	. . . 4 |- (9 / 10) = (1 - (1 / 10))
58	41, 57	opreq12i 4031	. . 3 |- ((9 / 10) / (9 / 10)) = ((9 x. (1 / 10)) / (1 - (1 / 10)))
59	2, 6, 28, 10	divgt0ii 5918	. . . . 5 |- 0 < (9 / 10)
60	53, 59	gt0ne0ii 5682	. . . 4 |- (9 / 10) =/= 0
61	54, 60	dividi 5827	. . 3 |- ((9 / 10) / (9 / 10)) = 1
62	58, 61	eqtr3i 1544	. 2 |- ((9 x. (1 / 10)) / (1 - (1 / 10))) = 1
63	20, 40, 62	3eqtri 1546	1 |- sum_k e. NN (9 / (10^k)) = 1

Syntax hints:   <-> wb 153   = wceq 997   e. wcel 999   =/= wne 1632   class class class wbr 2674  ` cfv 3239  (class class class)co 4021  CCcc 5297  RRcr 5298  0cc0 5299  1c1 5300   + caddc 5302   x. cmul 5304   - cmin 5357   / cdiv 5359   <_ cle 5360  NNcn 5361  NN0cn0 5362   < clt 5551  9c9 6029  10c10 6030  ^cexp 6657  abscabs 6840  sum_csu 7069

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-9 1006  ax-10 1007  ax-11 1008  ax-12 1009  ax-13 1010  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-rep 2748  ax-sep 2758  ax-nul 2765  ax-pow 2798  ax-pr 2835  ax-un 2922  ax-inf2 4687

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-3or 788  df-3an 789  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-nel 1635  df-ral 1696  df-rex 1697  df-reu 1698  df-rab 1699  df-v 1859  df-sbc 1989  df-csb 2052  df-dif 2100  df-un 2101  df-in 2102  df-ss 2104  df-pss 2106  df-nul 2332  df-if 2414  df-pw 2454  df-sn 2464  df-pr 2465  df-tp 2467  df-op 2468  df-uni 2558  df-int 2588  df-iun 2622  df-br 2675  df-opab 2722  df-tr 2736  df-eprel 2888  df-id 2891  df-po 2896  df-so 2906  df-fr 2974  df-we 2991  df-ord 3008  df-on 3009  df-lim 3010  df-suc 3011  df-om 3189  df-xp 3241  df-rel 3242  df-cnv 3243  df-co 3244  df-dm 3245  df-rn 3246  df-res 3247  df-ima 3248  df-fun 3249  df-fn 3250  df-f 3251  df-f1 3252  df-fo 3253  df-f1o 3254  df-fv 3255  df-rdg 3990  df-opr 4023  df-oprab 4024  df-1st 4137  df-2nd 4138  df-1o 4191  df-oadd 4193  df-omul 4194  df-er 4319  df-ec 4321  df-qs 4324  df-en 4429  df-dom 4430  df-sdom 4431  df-sup 4634  df-ni 5065  df-pli 5066  df-mi 5067  df-lti 5068  df-plpq 5100  df-mpq 5101  df-enq 5102  df-nq 5103  df-plq 5104  df-mq 5105  df-rq 5106  df-ltq 5107  df-1q 5108  df-np 5151  df-1p 5152  df-plp 5153  df-mp 5154  df-ltp 5155  df-plpr 5229  df-mpr 5230  df-enr 5231  df-nr 5232  df-plr 5233  df-mr 5234  df-ltr 5235  df-0r 5236  df-1r 5237  df-m1r 5238  df-c 5305  df-0 5306  df-1 5307  df-i 5308  df-r 5309  df-plus 5310  df-mul 5311  df-lt 5312  df-sub 5421  df-neg 5423  df-pnf 5552  df-mnf 5553  df-xr 5554  df-ltxr 5555  df-le 5556  df-div 5768  df-n 5985  df-2 6031  df-3 6032  df-4 6033  df-5 6034  df-6 6035  df-7 6036  df-8 6037  df-9 6038  df-10 6039  df-n0 6182  df-z 6218  df-fl 6335  df-uz 6444  df-fz 6494  df-seq1 6567  df-shft 6600  df-seqz 6622  df-seq0 6623  df-exp 6658  df-sqr 6760  df-re 6841  df-im 6842  df-cj 6843  df-abs 6844  df-clim 7065  df-sum 7070

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