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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...
StepHypRef Expression
1 nnnn0 6188 . . . 4 |- (k e. NN -> k e. NN0)
2 9re 6048 . . . . . . . 8 |- 9 e. RR
32recni 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))))
53, 4mp3an1 915 . . . . . 6 |- (((10^k) e. CC /\ (10^k) =/= 0) -> (9 / (10^k)) = (9 x. (1 / (10^k))))
6 10re 6049 . . . . . . . 8 |- 10 e. RR
76recni 5379 . . . . . . 7 |- 10 e. CC
8 expcl 6670 . . . . . . 7 |- ((10 e. CC /\ k e. NN0) -> (10^k) e. CC)
97, 8mpan 707 . . . . . 6 |- (k e. NN0 -> (10^k) e. CC)
10 10pos 6059 . . . . . . . 8 |- 0 < 10
116, 10gt0ne0ii 5682 . . . . . . 7 |- 10 =/= 0
12 expne0i 6677 . . . . . . 7 |- ((10 e. CC /\ k e. NN0 /\ 10 =/= 0) -> (10^k) =/= 0)
137, 11, 12mp3an13 919 . . . . . 6 |- (k e. NN0 -> (10^k) =/= 0)
145, 9, 13sylanc 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)))
167, 11, 15mp3an13 919 . . . . . 6 |- (k e. NN0 -> ((1 / 10)^k) = (1 / (10^k)))
1716opreq2d 4034 . . . . 5 |- (k e. NN0 -> (9 x. ((1 / 10)^k)) = (9 x. (1 / (10^k))))
1814, 17eqtr4d 1557 . . . 4 |- (k e. NN0 -> (9 / (10^k)) = (9 x. ((1 / 10)^k)))
191, 18syl 10 . . 3 |- (k e. NN -> (9 / (10^k)) = (9 x. ((1 / 10)^k)))
2019sumeq2i 7078 . 2 |- sum_k e. NN (9 / (10^k)) = sum_k e. NN (9 x. ((1 / 10)^k))
216, 11rereccli 5859 . . . 4 |- (1 / 10) e. RR
2221recni 5379 . . 3 |- (1 / 10) e. CC
23 0re 5505 . . . . . 6 |- 0 e. RR
246, 10recgt0ii 5872 . . . . . 6 |- 0 < (1 / 10)
2523, 21, 24ltleii 5646 . . . . 5 |- 0 <_ (1 / 10)
2621absidi 6951 . . . . 5 |- (0 <_ (1 / 10) -> (abs` (1 / 10)) = (1 / 10))
2725, 26ax-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)))
312, 29, 30mp2an 709 . . . . . . 7 |- (0 < 9 <-> 1 < (9 + 1))
3228, 31mpbi 196 . . . . . 6 |- 1 < (9 + 1)
33 df-10 6039 . . . . . 6 |- 10 = (9 + 1)
3432, 33breqtrri 2695 . . . . 5 |- 1 < 10
35 recgt1 5959 . . . . . 6 |- ((10 e. RR /\ 0 < 10) -> (1 < 10 <-> (1 / 10) < 1))
366, 10, 35mp2an 709 . . . . 5 |- (1 < 10 <-> (1 / 10) < 1)
3734, 36mpbi 196 . . . 4 |- (1 / 10) < 1
3827, 37eqbrtri 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))))
403, 22, 38, 39mp3an 928 . 2 |- sum_k e. NN (9 x. ((1 / 10)^k)) = ((9 x. (1 / 10)) / (1 - (1 / 10)))
413, 7, 11divreci 5799 . . . 4 |- (9 / 10) = (9 x. (1 / 10))
423, 7, 11divcan2i 5784 . . . . . 6 |- (10 x. (9 / 10)) = 9
43 ax1cn 5334 . . . . . . . 8 |- 1 e. CC
447, 43, 22subdii 5494 . . . . . . 7 |- (10 x. (1 - (1 / 10))) = ((10 x. 1) - (10 x. (1 / 10)))
457mulid1i 5397 . . . . . . . 8 |- (10 x. 1) = 10
467, 11recidi 5795 . . . . . . . 8 |- (10 x. (1 / 10)) = 1
4745, 46opreq12i 4031 . . . . . . 7 |- ((10 x. 1) - (10 x. (1 / 10))) = (10 - 1)
4843, 3addcomi 5387 . . . . . . . . 9 |- (1 + 9) = (9 + 1)
4948, 33eqtr4i 1545 . . . . . . . 8 |- (1 + 9) = 10
507, 43, 3, 49subaddrii 5437 . . . . . . 7 |- (10 - 1) = 9
5144, 47, 503eqtrri 1547 . . . . . 6 |- 9 = (10 x. (1 - (1 / 10)))
5242, 51eqtri 1542 . . . . 5 |- (10 x. (9 / 10)) = (10 x. (1 - (1 / 10)))
532, 6, 11redivcli 5856 . . . . . . 7 |- (9 / 10) e. RR
5453recni 5379 . . . . . 6 |- (9 / 10) e. CC
5543, 22subcli 5431 . . . . . 6 |- (1 - (1 / 10)) e. CC
567, 54, 55, 11mulcaniOLD 5752 . . . . 5 |- ((10 x. (9 / 10)) = (10 x. (1 - (1 / 10))) <-> (9 / 10) = (1 - (1 / 10)))
5752, 56mpbi 196 . . . 4 |- (9 / 10) = (1 - (1 / 10))
5841, 57opreq12i 4031 . . 3 |- ((9 / 10) / (9 / 10)) = ((9 x. (1 / 10)) / (1 - (1 / 10)))
592, 6, 28, 10divgt0ii 5918 . . . . 5 |- 0 < (9 / 10)
6053, 59gt0ne0ii 5682 . . . 4 |- (9 / 10) =/= 0
6154, 60dividi 5827 . . 3 |- ((9 / 10) / (9 / 10)) = 1
6258, 61eqtr3i 1544 . 2 |- ((9 x. (1 / 10)) / (1 - (1 / 10))) = 1
6320, 40, 623eqtri 1546 1 |- sum_k e. NN (9 / (10^k)) = 1
Colors of variables: wff set class
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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