Theorem 0.999... 7451

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 6274	. . . 4 |- (k e. NN -> k e. NN0)
2		9re 6133	. . . . . . . 8 |- 9 e. RR
3	2	recni 5468	. . . . . . 7 |- 9 e. CC
4		divrec 5885	. . . . . . 7 |- ((9 e. CC /\ (10^k) e. CC /\ (10^k) =/= 0) -> (9 / (10^k)) = (9 x. (1 / (10^k))))
5	3, 4	mp3an1 909	. . . . . 6 |- (((10^k) e. CC /\ (10^k) =/= 0) -> (9 / (10^k)) = (9 x. (1 / (10^k))))
6		10re 6134	. . . . . . . 8 |- 10 e. RR
7	6	recni 5468	. . . . . . 7 |- 10 e. CC
8		expcl 6776	. . . . . . 7 |- ((10 e. CC /\ k e. NN0) -> (10^k) e. CC)
9	7, 8	mpan 699	. . . . . 6 |- (k e. NN0 -> (10^k) e. CC)
10		10pos 6144	. . . . . . . 8 |- 0 < 10
11	6, 10	gt0ne0ii 5771	. . . . . . 7 |- 10 =/= 0
12		expne0i 6782	. . . . . . 7 |- ((10 e. CC /\ 10 =/= 0 /\ k e. NN0) -> (10^k) =/= 0)
13	7, 11, 12	mp3an12 912	. . . . . 6 |- (k e. NN0 -> (10^k) =/= 0)
14	5, 9, 13	sylanc 473	. . . . 5 |- (k e. NN0 -> (9 / (10^k)) = (9 x. (1 / (10^k))))
15		exprecOLD 6790	. . . . . . 7 |- ((10 e. CC /\ k e. NN0 /\ 10 =/= 0) -> ((1 / 10)^k) = (1 / (10^k)))
16	7, 11, 15	mp3an13 913	. . . . . 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 1553	. . . 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 7191	. 2 |- sum_k e. NN (9 / (10^k)) = sum_k e. NN (9 x. ((1 / 10)^k))
21	6, 11	rereccli 5941	. . . 4 |- (1 / 10) e. RR
22	21	recni 5468	. . 3 |- (1 / 10) e. CC
23		0re 5594	. . . . . 6 |- 0 e. RR
24	6, 10	recgt0ii 5954	. . . . . 6 |- 0 < (1 / 10)
25	23, 21, 24	ltleii 5735	. . . . 5 |- 0 <_ (1 / 10)
26	21	absidi 7063	. . . . 5 |- (0 <_ (1 / 10) -> (abs` (1 / 10)) = (1 / 10))
27	25, 26	ax-mp 7	. . . 4 |- (abs` (1 / 10)) = (1 / 10)
28		9pos 6143	. . . . . . 7 |- 0 < 9
29		1re 5589	. . . . . . . 8 |- 1 e. RR
30		ltaddpos2 5806	. . . . . . . 8 |- ((9 e. RR /\ 1 e. RR) -> (0 < 9 <-> 1 < (9 + 1)))
31	2, 29, 30	mp2an 701	. . . . . . 7 |- (0 < 9 <-> 1 < (9 + 1))
32	28, 31	mpbi 187	. . . . . 6 |- 1 < (9 + 1)
33		df-10 6124	. . . . . 6 |- 10 = (9 + 1)
34	32, 33	breqtrri 2713	. . . . 5 |- 1 < 10
35		recgt1 6044	. . . . . 6 |- ((10 e. RR /\ 0 < 10) -> (1 < 10 <-> (1 / 10) < 1))
36	6, 10, 35	mp2an 701	. . . . 5 |- (1 < 10 <-> (1 / 10) < 1)
37	34, 36	mpbi 187	. . . 4 |- (1 / 10) < 1
38	27, 37	eqbrtri 2707	. . 3 |- (abs` (1 / 10)) < 1
39		geoisum1c 7450	. . 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 922	. 2 |- sum_k e. NN (9 x. ((1 / 10)^k)) = ((9 x. (1 / 10)) / (1 - (1 / 10)))
41	3, 7, 11	divreci 5883	. . . 4 |- (9 / 10) = (9 x. (1 / 10))
42	3, 7, 11	divcan2i 5868	. . . . . 6 |- (10 x. (9 / 10)) = 9
43		ax1cn 5423	. . . . . . . 8 |- 1 e. CC
44	7, 43, 22	subdii 5583	. . . . . . 7 |- (10 x. (1 - (1 / 10))) = ((10 x. 1) - (10 x. (1 / 10)))
45	7	mulid1i 5486	. . . . . . . 8 |- (10 x. 1) = 10
46	7, 11	recidi 5879	. . . . . . . 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 5476	. . . . . . . . 9 |- (1 + 9) = (9 + 1)
49	48, 33	eqtr4i 1541	. . . . . . . 8 |- (1 + 9) = 10
50	7, 43, 3, 49	subaddrii 5526	. . . . . . 7 |- (10 - 1) = 9
51	44, 47, 50	3eqtrri 1543	. . . . . 6 |- 9 = (10 x. (1 - (1 / 10)))
52	42, 51	eqtri 1538	. . . . 5 |- (10 x. (9 / 10)) = (10 x. (1 - (1 / 10)))
53	2, 6, 11	redivcli 5938	. . . . . . 7 |- (9 / 10) e. RR
54	53	recni 5468	. . . . . 6 |- (9 / 10) e. CC
55	43, 22	subcli 5520	. . . . . 6 |- (1 - (1 / 10)) e. CC
56	54, 55, 7, 11	mulcani 5842	. . . . 5 |- ((10 x. (9 / 10)) = (10 x. (1 - (1 / 10))) <-> (9 / 10) = (1 - (1 / 10)))
57	52, 56	mpbi 187	. . . 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 6004	. . . . 5 |- 0 < (9 / 10)
60	53, 59	gt0ne0ii 5771	. . . 4 |- (9 / 10) =/= 0
61	54, 60	dividi 5909	. . 3 |- ((9 / 10) / (9 / 10)) = 1
62	58, 61	eqtr3i 1540	. 2 |- ((9 x. (1 / 10)) / (1 - (1 / 10))) = 1
63	20, 40, 62	3eqtri 1542	1 |- sum_k e. NN (9 / (10^k)) = 1

Syntax hints:   <-> wb 144   = wceq 992   e. wcel 994   =/= wne 1628   class class class wbr 2692  ` cfv 3263  (class class class)co 4021  CCcc 5386  RRcr 5387  0cc0 5388  1c1 5389   + caddc 5391   x. cmul 5393   - cmin 5446   / cdiv 5448   <_ cle 5449  NNcn 5450  NN0cn0 5451   < clt 5640  9c9 6114  10c10 6115  ^cexp 6763  abscabs 6951  sum_csu 7182

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089  ax-inf2 4770

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-nel 1631  df-ral 1695  df-rex 1696  df-reu 1697  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-pss 2107  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-int 2601  df-iun 2635  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-suc 2981  df-om 3219  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-f1 3276  df-fo 3277  df-f1o 3278  df-fv 3279  df-opr 4023  df-oprab 4024  df-1st 4140  df-2nd 4141  df-rdg 4233  df-1o 4269  df-oadd 4271  df-omul 4272  df-er 4401  df-ec 4403  df-qs 4406  df-en 4509  df-dom 4510  df-sdom 4511  df-sup 4717  df-ni 5154  df-pli 5155  df-mi 5156  df-lti 5157  df-plpq 5189  df-mpq 5190  df-enq 5191  df-nq 5192  df-plq 5193  df-mq 5194  df-rq 5195  df-ltq 5196  df-1q 5197  df-np 5240  df-1p 5241  df-plp 5242  df-mp 5243  df-ltp 5244  df-plpr 5318  df-mpr 5319  df-enr 5320  df-nr 5321  df-plr 5322  df-mr 5323  df-ltr 5324  df-0r 5325  df-1r 5326  df-m1r 5327  df-c 5394  df-0 5395  df-1 5396  df-i 5397  df-r 5398  df-plus 5399  df-mul 5400  df-lt 5401  df-sub 5510  df-neg 5512  df-pnf 5641  df-mnf 5642  df-xr 5643  df-ltxr 5644  df-le 5645  df-div 5855  df-n 6070  df-2 6116  df-3 6117  df-4 6118  df-5 6119  df-6 6120  df-7 6121  df-8 6122  df-9 6123  df-10 6124  df-n0 6268  df-z 6304  df-fl 6422  df-uz 6545  df-fz 6596  df-seq1 6673  df-shft 6706  df-seqz 6728  df-seq0 6729  df-exp 6764  df-sqr 6871  df-re 6952  df-im 6953  df-cj 6954  df-abs 6955  df-clim 7178  df-sum 7183

Copyright terms: Public domain