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Theorem 0.999... 9445
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 8011 . . . 4 |- (k e. NN -> k e. NN0)
2 10re 7835 . . . . . . . 8 |- 10 e. RR
32recni 7193 . . . . . . 7 |- 10 e. CC
4 expcl 8732 . . . . . . 7 |- ((10 e. CC /\ k e. NN0) -> (10^k) e. CC)
53, 4mpan 753 . . . . . 6 |- (k e. NN0 -> (10^k) e. CC)
6 10pos 7846 . . . . . . . 8 |- 0 < 10
72, 6gt0ne0ii 7485 . . . . . . 7 |- 10 =/= 0
8 expne0i 8738 . . . . . . 7 |- ((10 e. CC /\ 10 =/= 0 /\ k e. NN0) -> (10^k) =/= 0)
93, 7, 8mp3an12 1385 . . . . . 6 |- (k e. NN0 -> (10^k) =/= 0)
10 9re 7834 . . . . . . . 8 |- 9 e. RR
1110recni 7193 . . . . . . 7 |- 9 e. CC
12 divrec 7594 . . . . . . 7 |- ((9 e. CC /\ (10^k) e. CC /\ (10^k) =/= 0) -> (9 / (10^k)) = (9 x. (1 / (10^k))))
1311, 12mp3an1 1382 . . . . . 6 |- (((10^k) e. CC /\ (10^k) =/= 0) -> (9 / (10^k)) = (9 x. (1 / (10^k))))
145, 9, 13syl11anc 736 . . . . 5 |- (k e. NN0 -> (9 / (10^k)) = (9 x. (1 / (10^k))))
15 exprec 8745 . . . . . . 7 |- ((10 e. CC /\ 10 =/= 0 /\ k e. NN0) -> ((1 / 10)^k) = (1 / (10^k)))
163, 7, 15mp3an12 1385 . . . . . 6 |- (k e. NN0 -> ((1 / 10)^k) = (1 / (10^k)))
1716opreq2d 5080 . . . . 5 |- (k e. NN0 -> (9 x. ((1 / 10)^k)) = (9 x. (1 / (10^k))))
1814, 17eqtr4d 2127 . . . 4 |- (k e. NN0 -> (9 / (10^k)) = (9 x. ((1 / 10)^k)))
191, 18syl 13 . . 3 |- (k e. NN -> (9 / (10^k)) = (9 x. ((1 / 10)^k)))
2019sumeq2i 9176 . 2 |- sum_k e. NN (9 / (10^k)) = sum_k e. NN (9 x. ((1 / 10)^k))
212, 7rereccli 7652 . . . 4 |- (1 / 10) e. RR
2221recni 7193 . . 3 |- (1 / 10) e. CC
23 0re 7213 . . . . . 6 |- 0 e. RR
242, 6recgt0ii 7666 . . . . . 6 |- 0 < (1 / 10)
2523, 21, 24ltleii 7277 . . . . 5 |- 0 <_ (1 / 10)
2621absidi 9009 . . . . 5 |- (0 <_ (1 / 10) -> (abs` (1 / 10)) = (1 / 10))
2725, 26ax-mp 7 . . . 4 |- (abs` (1 / 10)) = (1 / 10)
28 9pos 7845 . . . . . . 7 |- 0 < 9
29 1re 7212 . . . . . . . 8 |- 1 e. RR
30 ltaddpos2 7516 . . . . . . . 8 |- ((9 e. RR /\ 1 e. RR) -> (0 < 9 <-> 1 < (9 + 1)))
3110, 29, 30mp2an 756 . . . . . . 7 |- (0 < 9 <-> 1 < (9 + 1))
3228, 31mpbi 237 . . . . . 6 |- 1 < (9 + 1)
33 df-10 7824 . . . . . 6 |- 10 = (9 + 1)
3432, 33breqtrri 3534 . . . . 5 |- 1 < 10
35 recgt1 7758 . . . . . 6 |- ((10 e. RR /\ 0 < 10) -> (1 < 10 <-> (1 / 10) < 1))
362, 6, 35mp2an 756 . . . . 5 |- (1 < 10 <-> (1 / 10) < 1)
3734, 36mpbi 237 . . . 4 |- (1 / 10) < 1
3827, 37eqbrtri 3528 . . 3 |- (abs` (1 / 10)) < 1
39 geoisum1c 9444 . . 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))))
4011, 22, 38, 39mp3an 1395 . 2 |- sum_k e. NN (9 x. ((1 / 10)^k)) = ((9 x. (1 / 10)) / (1 - (1 / 10)))
4111, 3, 7divreci 7592 . . . 4 |- (9 / 10) = (9 x. (1 / 10))
4211, 3, 7divcan2i 7577 . . . . . 6 |- (10 x. (9 / 10)) = 9
43 ax-1cn 7161 . . . . . . . 8 |- 1 e. CC
443, 43, 22subdii 7413 . . . . . . 7 |- (10 x. (1 - (1 / 10))) = ((10 x. 1) - (10 x. (1 / 10)))
453mulid1i 7206 . . . . . . . 8 |- (10 x. 1) = 10
463, 7recidi 7588 . . . . . . . 8 |- (10 x. (1 / 10)) = 1
4745, 46opreq12i 5077 . . . . . . 7 |- ((10 x. 1) - (10 x. (1 / 10))) = (10 - 1)
4843, 11addcomi 7326 . . . . . . . . 9 |- (1 + 9) = (9 + 1)
4948, 33eqtr4i 2115 . . . . . . . 8 |- (1 + 9) = 10
503, 43, 11, 49subaddrii 7366 . . . . . . 7 |- (10 - 1) = 9
5144, 47, 503eqtrri 2117 . . . . . 6 |- 9 = (10 x. (1 - (1 / 10)))
5242, 51eqtri 2112 . . . . 5 |- (10 x. (9 / 10)) = (10 x. (1 - (1 / 10)))
5310, 2, 7redivcli 7649 . . . . . . 7 |- (9 / 10) e. RR
5453recni 7193 . . . . . 6 |- (9 / 10) e. CC
5543, 22subcli 7358 . . . . . 6 |- (1 - (1 / 10)) e. CC
5654, 55, 3, 7mulcani 7551 . . . . 5 |- ((10 x. (9 / 10)) = (10 x. (1 - (1 / 10))) <-> (9 / 10) = (1 - (1 / 10)))
5752, 56mpbi 237 . . . 4 |- (9 / 10) = (1 - (1 / 10))
5841, 57opreq12i 5077 . . 3 |- ((9 / 10) / (9 / 10)) = ((9 x. (1 / 10)) / (1 - (1 / 10)))
5910, 2, 28, 6divgt0ii 7718 . . . . 5 |- 0 < (9 / 10)
6053, 59gt0ne0ii 7485 . . . 4 |- (9 / 10) =/= 0
6154, 60dividi 7618 . . 3 |- ((9 / 10) / (9 / 10)) = 1
6258, 61eqtr3i 2114 . 2 |- ((9 x. (1 / 10)) / (1 - (1 / 10))) = 1
6320, 40, 623eqtri 2116 1 |- sum_k e. NN (9 / (10^k)) = 1
Colors of variables: wff set class
Syntax hints:   <-> wb 209   = wceq 1592   e. wcel 1594   =/= wne 2218   class class class wbr 3509  ` cfv 4149  (class class class)co 5067  CCcc 7104  RRcr 7105  0cc0 7106  1c1 7107   + caddc 7109   x. cmul 7111   <_ cle 7214   < clt 7218   - cmin 7328   / cdiv 7330  NNcn 7331  NN0cn0 7332  9c9 7814  10c10 7815  ^cexp 8720  abscabs 8925  sum_csu 9167
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1516  ax-6 1517  ax-7 1518  ax-gen 1519  ax-8 1596  ax-10 1597  ax-11 1598  ax-12 1599  ax-13 1600  ax-14 1601  ax-17 1608  ax-9 1620  ax-4 1626  ax-16 1803  ax-15 1966  ax-ext 2074  ax-rep 3599  ax-sep 3609  ax-nul 3619  ax-pow 3655  ax-pr 3679  ax-un 3947  ax-inf2 6137  ax-resscn 7160  ax-1cn 7161  ax-icn 7162  ax-addcl 7163  ax-addrcl 7164  ax-mulcl 7165  ax-mulrcl 7166  ax-mulcom 7167  ax-addass 7168  ax-mulass 7169  ax-distr 7170  ax-i2m1 7171  ax-1ne0 7172  ax-1rid 7173  ax-rnegex 7174  ax-rrecex 7175  ax-cnre 7176  ax-pre-lttri 7177  ax-pre-lttrn 7178  ax-pre-ltadd 7179  ax-pre-mulgt0 7180  ax-pre-sup 7181  ax-addopr 7182
This theorem depends on definitions:  df-bi 210  df-or 419  df-an 420  df-3or 1038  df-3an 1039  df-tru 1468  df-ex 1521  df-sb 1765  df-eu 1992  df-mo 1993  df-clab 2080  df-cleq 2085  df-clel 2088  df-ne 2220  df-nel 2221  df-ral 2314  df-rex 2315  df-reu 2316  df-rab 2317  df-v 2501  df-sbc 2671  df-csb 2745  df-dif 2804  df-un 2806  df-in 2808  df-ss 2810  df-pss 2812  df-nul 3066  df-if 3166  df-pw 3222  df-sn 3237  df-pr 3238  df-tp 3240  df-op 3241  df-uni 3365  df-int 3399  df-iun 3437  df-br 3510  df-opab 3568  df-tr 3583  df-eprel 3762  df-id 3765  df-po 3770  df-so 3782  df-fr 3800  df-we 3816  df-ord 3832  df-on 3833  df-lim 3834  df-suc 3835  df-om 4104  df-xp 4151  df-rel 4152  df-cnv 4153  df-co 4154  df-dm 4155  df-rn 4156  df-res 4157  df-ima 4158  df-fun 4159  df-fn 4160  df-f 4161  df-f1 4162  df-fo 4163  df-f1o 4164  df-fv 4165  df-opr 5069  df-oprab 5070  df-mpt 5202  df-mpt2 5203  df-1st 5268  df-2nd 5269  df-iota 5374  df-rdg 5460  df-er 5637  df-map 5705  df-en 5752  df-dom 5753  df-sdom 5754  df-riota 5896  df-sup 6059  df-pnf 7219  df-mnf 7220  df-xr 7221  df-ltxr 7222  df-le 7223  df-sub 7347  df-neg 7349  df-div 7564  df-n 7769  df-2 7816  df-3 7817  df-4 7818  df-5 7819  df-6 7820  df-7 7821  df-8 7822  df-9 7823  df-10 7824  df-rp 7941  df-n0 8005  df-z 8044  df-uz 8143  df-q 8217  df-fl 8337  df-fz 8472  df-seq 8586  df-seq1 8631  df-seq0 8632  df-seqz 8633  df-exp 8721  df-cj 8839  df-re 8840  df-im 8841  df-sqr 8926  df-abs 8927  df-clim 9163  df-sum 9168
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