HomeHome Metamath Proof Explorer < Previous   Next >
Related theorems
Unicode version

Theorem 0.999... 9412
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 7779 . . . . . . 7 |- 10 e. RR
21recni 7094 . . . . . 6 |- 10 e. CC
3 nnnn0 7890 . . . . . 6 |- (k e. NN -> k e. NN0)
4 expcl 8653 . . . . . 6 |- ((10 e. CC /\ k e. NN0) -> (10^k) e. CC)
52, 3, 4sylancr 647 . . . . 5 |- (k e. NN -> (10^k) e. CC)
62a1i 10 . . . . . 6 |- (k e. NN -> 10 e. CC)
7 10pos 7790 . . . . . . . 8 |- 0 < 10
81, 7gt0ne0ii 7389 . . . . . . 7 |- 10 =/= 0
98a1i 10 . . . . . 6 |- (k e. NN -> 10 =/= 0)
10 nnz 7947 . . . . . 6 |- (k e. NN -> k e. ZZ)
11 expne0i 8663 . . . . . 6 |- ((10 e. CC /\ 10 =/= 0 /\ k e. ZZ) -> (10^k) =/= 0)
126, 9, 10, 11syl3anc 1162 . . . . 5 |- (k e. NN -> (10^k) =/= 0)
13 9re 7778 . . . . . . 7 |- 9 e. RR
1413recni 7094 . . . . . 6 |- 9 e. CC
15 divrec 7506 . . . . . 6 |- ((9 e. CC /\ (10^k) e. CC /\ (10^k) =/= 0) -> (9 / (10^k)) = (9 x. (1 / (10^k))))
1614, 15mp3an1 1244 . . . . 5 |- (((10^k) e. CC /\ (10^k) =/= 0) -> (9 / (10^k)) = (9 x. (1 / (10^k))))
175, 12, 16syl2anc 645 . . . 4 |- (k e. NN -> (9 / (10^k)) = (9 x. (1 / (10^k))))
18 exprec 8672 . . . . . 6 |- ((10 e. CC /\ 10 =/= 0 /\ k e. ZZ) -> ((1 / 10)^k) = (1 / (10^k)))
196, 9, 10, 18syl3anc 1162 . . . . 5 |- (k e. NN -> ((1 / 10)^k) = (1 / (10^k)))
2019oveq2d 4940 . . . 4 |- (k e. NN -> (9 x. ((1 / 10)^k)) = (9 x. (1 / (10^k))))
2117, 20eqtr4d 1973 . . 3 |- (k e. NN -> (9 / (10^k)) = (9 x. ((1 / 10)^k)))
2221sumeq2i 9273 . 2 |- sum_k e. NN (9 / (10^k)) = sum_k e. NN (9 x. ((1 / 10)^k))
231, 8rereccli 7565 . . . 4 |- (1 / 10) e. RR
2423recni 7094 . . 3 |- (1 / 10) e. CC
25 0re 7114 . . . . . 6 |- 0 e. RR
261, 7recgt0ii 7580 . . . . . 6 |- 0 < (1 / 10)
2725, 23, 26ltleii 7179 . . . . 5 |- 0 <_ (1 / 10)
2823absidi 9048 . . . . 5 |- (0 <_ (1 / 10) -> (abs` (1 / 10)) = (1 / 10))
2927, 28ax-mp 8 . . . 4 |- (abs` (1 / 10)) = (1 / 10)
30 1lt10 7857 . . . . 5 |- 1 < 10
31 recgt1 7657 . . . . . 6 |- ((10 e. RR /\ 0 < 10) -> (1 < 10 <-> (1 / 10) < 1))
321, 7, 31mp2an 656 . . . . 5 |- (1 < 10 <-> (1 / 10) < 1)
3330, 32mpbi 197 . . . 4 |- (1 / 10) < 1
3429, 33eqbrtri 3382 . . 3 |- (abs` (1 / 10)) < 1
35 geoisum1c 9411 . . 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))))
3614, 24, 34, 35mp3an 1257 . 2 |- sum_k e. NN (9 x. ((1 / 10)^k)) = ((9 x. (1 / 10)) / (1 - (1 / 10)))
3714, 2, 8divreci 7504 . . . 4 |- (9 / 10) = (9 x. (1 / 10))
3814, 2, 8divcan2i 7489 . . . . . 6 |- (10 x. (9 / 10)) = 9
39 ax-1cn 7062 . . . . . . . 8 |- 1 e. CC
402, 39, 24subdii 7316 . . . . . . 7 |- (10 x. (1 - (1 / 10))) = ((10 x. 1) - (10 x. (1 / 10)))
412mulid1i 7107 . . . . . . . 8 |- (10 x. 1) = 10
422, 8recidi 7500 . . . . . . . 8 |- (10 x. (1 / 10)) = 1
4341, 42oveq12i 4937 . . . . . . 7 |- ((10 x. 1) - (10 x. (1 / 10))) = (10 - 1)
4439, 14addcomi 7228 . . . . . . . . 9 |- (1 + 9) = (9 + 1)
45 df-10 7768 . . . . . . . . 9 |- 10 = (9 + 1)
4644, 45eqtr4i 1961 . . . . . . . 8 |- (1 + 9) = 10
472, 39, 14, 46subaddrii 7268 . . . . . . 7 |- (10 - 1) = 9
4840, 43, 473eqtrri 1963 . . . . . 6 |- 9 = (10 x. (1 - (1 / 10)))
4938, 48eqtri 1958 . . . . 5 |- (10 x. (9 / 10)) = (10 x. (1 - (1 / 10)))
5013, 1, 8redivcli 7562 . . . . . . 7 |- (9 / 10) e. RR
5150recni 7094 . . . . . 6 |- (9 / 10) e. CC
5239, 24subcli 7260 . . . . . 6 |- (1 - (1 / 10)) e. CC
5351, 52, 2, 8mulcani 7462 . . . . 5 |- ((10 x. (9 / 10)) = (10 x. (1 - (1 / 10))) <-> (9 / 10) = (1 - (1 / 10)))
5449, 53mpbi 197 . . . 4 |- (9 / 10) = (1 - (1 / 10))
5537, 54oveq12i 4937 . . 3 |- ((9 / 10) / (9 / 10)) = ((9 x. (1 / 10)) / (1 - (1 / 10)))
56 9pos 7789 . . . . . 6 |- 0 < 9
5713, 1, 56, 7divgt0ii 7622 . . . . 5 |- 0 < (9 / 10)
5850, 57gt0ne0ii 7389 . . . 4 |- (9 / 10) =/= 0
5951, 58dividi 7530 . . 3 |- ((9 / 10) / (9 / 10)) = 1
6055, 59eqtr3i 1960 . 2 |- ((9 x. (1 / 10)) / (1 - (1 / 10))) = 1
6122, 36, 603eqtri 1962 1 |- sum_k e. NN (9 / (10^k)) = 1
Colors of variables: wff set class
Syntax hints:   <-> wb 174   = wceq 1434   e. wcel 1436   =/= wne 2056   class class class wbr 3363  ` cfv 4008  (class class class)co 4927  CCcc 7005  RRcr 7006  0cc0 7007  1c1 7008   + caddc 7010   x. cmul 7012   <_ cle 7115   < clt 7119   - cmin 7230   / cdiv 7232  NNcn 7233  NN0cn0 7234  ZZcz 7235  9c9 7758  10c10 7759  ^cexp 8636  abscabs 8964  sum_csu 9259
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1351  ax-6 1352  ax-7 1353  ax-gen 1354  ax-8 1438  ax-10 1439  ax-11 1440  ax-12 1441  ax-13 1442  ax-14 1443  ax-17 1450  ax-9 1465  ax-4 1471  ax-16 1649  ax-15 1812  ax-ext 1920  ax-rep 3449  ax-sep 3459  ax-nul 3468  ax-pow 3504  ax-pr 3528  ax-un 3800  ax-inf2 6079  ax-resscn 7061  ax-1cn 7062  ax-icn 7063  ax-addcl 7064  ax-addrcl 7065  ax-mulcl 7066  ax-mulrcl 7067  ax-mulcom 7068  ax-addass 7069  ax-mulass 7070  ax-distr 7071  ax-i2m1 7072  ax-1ne0 7073  ax-1rid 7074  ax-rnegex 7075  ax-rrecex 7076  ax-cnre 7077  ax-pre-lttri 7078  ax-pre-lttrn 7079  ax-pre-ltadd 7080  ax-pre-mulgt0 7081  ax-pre-sup 7082
This theorem depends on definitions:  df-bi 175  df-or 362  df-an 363  df-3or 922  df-3an 923  df-tru 1329  df-ex 1356  df-sb 1611  df-eu 1838  df-mo 1839  df-clab 1926  df-cleq 1931  df-clel 1934  df-ne 2058  df-nel 2059  df-ral 2151  df-rex 2152  df-reu 2153  df-rab 2154  df-v 2345  df-sbc 2510  df-csb 2585  df-dif 2645  df-un 2647  df-in 2649  df-ss 2651  df-pss 2653  df-nul 2907  df-if 3010  df-pw 3068  df-sn 3085  df-pr 3086  df-tp 3087  df-op 3088  df-uni 3219  df-int 3253  df-iun 3291  df-br 3364  df-opab 3418  df-tr 3433  df-eprel 3613  df-id 3616  df-po 3621  df-so 3635  df-fr 3654  df-we 3670  df-ord 3686  df-on 3687  df-lim 3688  df-suc 3689  df-om 3963  df-xp 4010  df-rel 4011  df-cnv 4012  df-co 4013  df-dm 4014  df-rn 4015  df-res 4016  df-ima 4017  df-fun 4018  df-fn 4019  df-f 4020  df-f1 4021  df-fo 4022  df-f1o 4023  df-fv 4024  df-iso 4025  df-ov 4929  df-oprab 4930  df-mpt 5065  df-mpt2 5066  df-1st 5174  df-2nd 5175  df-iota 5278  df-rdg 5364  df-1o 5401  df-er 5538  df-map 5626  df-pm 5627  df-en 5683  df-dom 5684  df-sdom 5685  df-fin 5686  df-riota 5826  df-sup 6000  df-card 6243  df-pnf 7120  df-mnf 7121  df-xr 7122  df-ltxr 7123  df-le 7124  df-sub 7249  df-neg 7251  df-div 7475  df-n 7714  df-2 7760  df-3 7761  df-4 7762  df-5 7763  df-6 7764  df-7 7765  df-8 7766  df-9 7767  df-10 7768  df-n0 7884  df-z 7928  df-uz 8048  df-q 8130  df-rp 8255  df-fz 8400  df-fl 8493  df-seq 8585  df-exp 8637  df-hash 8816  df-cj 8871  df-re 8872  df-im 8873  df-sqr 8965  df-abs 8966  df-clim 9134  df-rlim 9135  df-sum 9260
Copyright terms: Public domain