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Theorem 0.999... 9284
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 10re 7728 . . . . . . 7 |- 10 e. RR
21recni 7049 . . . . . 6 |- 10 e. CC
3 nnnn0 7839 . . . . . 6 |- (k e. NN -> k e. NN0)
4 expcl 8594 . . . . . 6 |- ((10 e. CC /\ k e. NN0) -> (10^k) e. CC)
52, 3, 4sylancr 670 . . . . 5 |- (k e. NN -> (10^k) e. CC)
62a1i 10 . . . . . 6 |- (k e. NN -> 10 e. CC)
7 10pos 7739 . . . . . . . 8 |- 0 < 10
81, 7gt0ne0ii 7343 . . . . . . 7 |- 10 =/= 0
98a1i 10 . . . . . 6 |- (k e. NN -> 10 =/= 0)
10 nn0z 7897 . . . . . . 7 |- (k e. NN0 -> k e. ZZ)
113, 10syl 14 . . . . . 6 |- (k e. NN -> k e. ZZ)
12 expne0i 8603 . . . . . 6 |- ((10 e. CC /\ 10 =/= 0 /\ k e. ZZ) -> (10^k) =/= 0)
136, 9, 11, 12syl3anc 1187 . . . . 5 |- (k e. NN -> (10^k) =/= 0)
14 9re 7727 . . . . . . 7 |- 9 e. RR
1514recni 7049 . . . . . 6 |- 9 e. CC
16 divrec 7460 . . . . . 6 |- ((9 e. CC /\ (10^k) e. CC /\ (10^k) =/= 0) -> (9 / (10^k)) = (9 x. (1 / (10^k))))
1715, 16mp3an1 1269 . . . . 5 |- (((10^k) e. CC /\ (10^k) =/= 0) -> (9 / (10^k)) = (9 x. (1 / (10^k))))
185, 13, 17syl2anc 668 . . . 4 |- (k e. NN -> (9 / (10^k)) = (9 x. (1 / (10^k))))
19 exprec 8612 . . . . . 6 |- ((10 e. CC /\ 10 =/= 0 /\ k e. ZZ) -> ((1 / 10)^k) = (1 / (10^k)))
206, 9, 11, 19syl3anc 1187 . . . . 5 |- (k e. NN -> ((1 / 10)^k) = (1 / (10^k)))
2120oveq2d 4939 . . . 4 |- (k e. NN -> (9 x. ((1 / 10)^k)) = (9 x. (1 / (10^k))))
2218, 21eqtr4d 1995 . . 3 |- (k e. NN -> (9 / (10^k)) = (9 x. ((1 / 10)^k)))
2322sumeq2i 9147 . 2 |- sum_k e. NN (9 / (10^k)) = sum_k e. NN (9 x. ((1 / 10)^k))
241, 8rereccli 7518 . . . 4 |- (1 / 10) e. RR
2524recni 7049 . . 3 |- (1 / 10) e. CC
26 0re 7069 . . . . . 6 |- 0 e. RR
271, 7recgt0ii 7532 . . . . . 6 |- 0 < (1 / 10)
2826, 24, 27ltleii 7134 . . . . 5 |- 0 <_ (1 / 10)
2924absidi 8989 . . . . 5 |- (0 <_ (1 / 10) -> (abs` (1 / 10)) = (1 / 10))
3028, 29ax-mp 8 . . . 4 |- (abs` (1 / 10)) = (1 / 10)
31 9pos 7738 . . . . . . 7 |- 0 < 9
32 1re 7068 . . . . . . . 8 |- 1 e. RR
33 ltaddpos2 7374 . . . . . . . 8 |- ((9 e. RR /\ 1 e. RR) -> (0 < 9 <-> 1 < (9 + 1)))
3414, 32, 33mp2an 679 . . . . . . 7 |- (0 < 9 <-> 1 < (9 + 1))
3531, 34mpbi 217 . . . . . 6 |- 1 < (9 + 1)
36 df-10 7717 . . . . . 6 |- 10 = (9 + 1)
3735, 36breqtrri 3401 . . . . 5 |- 1 < 10
38 recgt1 7609 . . . . . 6 |- ((10 e. RR /\ 0 < 10) -> (1 < 10 <-> (1 / 10) < 1))
391, 7, 38mp2an 679 . . . . 5 |- (1 < 10 <-> (1 / 10) < 1)
4037, 39mpbi 217 . . . 4 |- (1 / 10) < 1
4130, 40eqbrtri 3395 . . 3 |- (abs` (1 / 10)) < 1
42 geoisum1c 9283 . . 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))))
4315, 25, 41, 42mp3an 1282 . 2 |- sum_k e. NN (9 x. ((1 / 10)^k)) = ((9 x. (1 / 10)) / (1 - (1 / 10)))
4415, 2, 8divreci 7458 . . . 4 |- (9 / 10) = (9 x. (1 / 10))
4515, 2, 8divcan2i 7443 . . . . . 6 |- (10 x. (9 / 10)) = 9
46 ax-1cn 7017 . . . . . . . 8 |- 1 e. CC
472, 46, 25subdii 7270 . . . . . . 7 |- (10 x. (1 - (1 / 10))) = ((10 x. 1) - (10 x. (1 / 10)))
482mulid1i 7062 . . . . . . . 8 |- (10 x. 1) = 10
492, 8recidi 7454 . . . . . . . 8 |- (10 x. (1 / 10)) = 1
5048, 49oveq12i 4936 . . . . . . 7 |- ((10 x. 1) - (10 x. (1 / 10))) = (10 - 1)
5146, 15addcomi 7182 . . . . . . . . 9 |- (1 + 9) = (9 + 1)
5251, 36eqtr4i 1983 . . . . . . . 8 |- (1 + 9) = 10
532, 46, 15, 52subaddrii 7222 . . . . . . 7 |- (10 - 1) = 9
5447, 50, 533eqtrri 1985 . . . . . 6 |- 9 = (10 x. (1 - (1 / 10)))
5545, 54eqtri 1980 . . . . 5 |- (10 x. (9 / 10)) = (10 x. (1 - (1 / 10)))
5614, 1, 8redivcli 7515 . . . . . . 7 |- (9 / 10) e. RR
5756recni 7049 . . . . . 6 |- (9 / 10) e. CC
5846, 25subcli 7214 . . . . . 6 |- (1 - (1 / 10)) e. CC
5957, 58, 2, 8mulcani 7416 . . . . 5 |- ((10 x. (9 / 10)) = (10 x. (1 - (1 / 10))) <-> (9 / 10) = (1 - (1 / 10)))
6055, 59mpbi 217 . . . 4 |- (9 / 10) = (1 - (1 / 10))
6144, 60oveq12i 4936 . . 3 |- ((9 / 10) / (9 / 10)) = ((9 x. (1 / 10)) / (1 - (1 / 10)))
6214, 1, 31, 7divgt0ii 7574 . . . . 5 |- 0 < (9 / 10)
6356, 62gt0ne0ii 7343 . . . 4 |- (9 / 10) =/= 0
6457, 63dividi 7484 . . 3 |- ((9 / 10) / (9 / 10)) = 1
6561, 64eqtr3i 1982 . 2 |- ((9 x. (1 / 10)) / (1 - (1 / 10))) = 1
6623, 43, 653eqtri 1984 1 |- sum_k e. NN (9 / (10^k)) = 1
Colors of variables: wff set class
Syntax hints:   <-> wb 189   = wceq 1457   e. wcel 1459   =/= wne 2078   class class class wbr 3376  ` cfv 4012  (class class class)co 4926  CCcc 6960  RRcr 6961  0cc0 6962  1c1 6963   + caddc 6965   x. cmul 6967   <_ cle 7070   < clt 7074   - cmin 7184   / cdiv 7186  NNcn 7187  NN0cn0 7188  ZZcz 7189  9c9 7707  10c10 7708  ^cexp 8577  abscabs 8904  sum_csu 9133
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1376  ax-6 1377  ax-7 1378  ax-gen 1379  ax-8 1461  ax-10 1462  ax-11 1463  ax-12 1464  ax-13 1465  ax-14 1466  ax-17 1473  ax-9 1488  ax-4 1494  ax-16 1671  ax-15 1834  ax-ext 1942  ax-rep 3462  ax-sep 3472  ax-nul 3481  ax-pow 3517  ax-pr 3541  ax-un 3811  ax-inf2 6035  ax-resscn 7016  ax-1cn 7017  ax-icn 7018  ax-addcl 7019  ax-addrcl 7020  ax-mulcl 7021  ax-mulrcl 7022  ax-mulcom 7023  ax-addass 7024  ax-mulass 7025  ax-distr 7026  ax-i2m1 7027  ax-1ne0 7028  ax-1rid 7029  ax-rnegex 7030  ax-rrecex 7031  ax-cnre 7032  ax-pre-lttri 7033  ax-pre-lttrn 7034  ax-pre-ltadd 7035  ax-pre-mulgt0 7036  ax-pre-sup 7037
This theorem depends on definitions:  df-bi 190  df-or 383  df-an 384  df-3or 947  df-3an 948  df-tru 1354  df-ex 1381  df-sb 1633  df-eu 1860  df-mo 1861  df-clab 1948  df-cleq 1953  df-clel 1956  df-ne 2080  df-nel 2081  df-ral 2173  df-rex 2174  df-reu 2175  df-rab 2176  df-v 2367  df-sbc 2532  df-csb 2606  df-dif 2665  df-un 2667  df-in 2669  df-ss 2671  df-pss 2673  df-nul 2927  df-if 3028  df-pw 3084  df-sn 3099  df-pr 3100  df-tp 3102  df-op 3103  df-uni 3232  df-int 3266  df-iun 3304  df-br 3377  df-opab 3431  df-tr 3446  df-eprel 3624  df-id 3627  df-po 3632  df-so 3646  df-fr 3665  df-we 3681  df-ord 3697  df-on 3698  df-lim 3699  df-suc 3700  df-om 3967  df-xp 4014  df-rel 4015  df-cnv 4016  df-co 4017  df-dm 4018  df-rn 4019  df-res 4020  df-ima 4021  df-fun 4022  df-fn 4023  df-f 4024  df-f1 4025  df-fo 4026  df-f1o 4027  df-fv 4028  df-iso 4029  df-ov 4928  df-oprab 4929  df-mpt 5063  df-mpt2 5064  df-1st 5132  df-2nd 5133  df-iota 5236  df-rdg 5322  df-1o 5359  df-er 5496  df-map 5584  df-en 5641  df-dom 5642  df-sdom 5643  df-fin 5644  df-riota 5784  df-sup 5957  df-card 6200  df-pnf 7075  df-mnf 7076  df-xr 7077  df-ltxr 7078  df-le 7079  df-sub 7203  df-neg 7205  df-div 7429  df-n 7663  df-2 7709  df-3 7710  df-4 7711  df-5 7712  df-6 7713  df-7 7714  df-8 7715  df-9 7716  df-10 7717  df-n0 7833  df-z 7877  df-uz 7997  df-q 8079  df-rp 8201  df-fz 8344  df-seq 8526  df-exp 8578  df-hash 8756  df-cj 8814  df-re 8815  df-im 8816  df-sqr 8905  df-abs 8906  df-clim 9057  df-sum 9134
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