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Theorem 0.999... 9397
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 7764 . . . . . . 7 |- 10 e. RR
21recni 7079 . . . . . 6 |- 10 e. CC
3 nnnn0 7875 . . . . . 6 |- (k e. NN -> k e. NN0)
4 expcl 8638 . . . . . 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 7775 . . . . . . . 8 |- 0 < 10
81, 7gt0ne0ii 7374 . . . . . . 7 |- 10 =/= 0
98a1i 10 . . . . . 6 |- (k e. NN -> 10 =/= 0)
10 nnz 7932 . . . . . 6 |- (k e. NN -> k e. ZZ)
11 expne0i 8648 . . . . . 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 7763 . . . . . . 7 |- 9 e. RR
1413recni 7079 . . . . . 6 |- 9 e. CC
15 divrec 7491 . . . . . 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 8657 . . . . . 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 4935 . . . 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 9258 . 2 |- sum_k e. NN (9 / (10^k)) = sum_k e. NN (9 x. ((1 / 10)^k))
231, 8rereccli 7550 . . . 4 |- (1 / 10) e. RR
2423recni 7079 . . 3 |- (1 / 10) e. CC
25 0re 7099 . . . . . 6 |- 0 e. RR
261, 7recgt0ii 7565 . . . . . 6 |- 0 < (1 / 10)
2725, 23, 26ltleii 7164 . . . . 5 |- 0 <_ (1 / 10)
2823absidi 9033 . . . . 5 |- (0 <_ (1 / 10) -> (abs` (1 / 10)) = (1 / 10))
2927, 28ax-mp 8 . . . 4 |- (abs` (1 / 10)) = (1 / 10)
30 1lt10 7842 . . . . 5 |- 1 < 10
31 recgt1 7642 . . . . . 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 3381 . . 3 |- (abs` (1 / 10)) < 1
35 geoisum1c 9396 . . 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 7489 . . . 4 |- (9 / 10) = (9 x. (1 / 10))
3814, 2, 8divcan2i 7474 . . . . . 6 |- (10 x. (9 / 10)) = 9
39 ax-1cn 7047 . . . . . . . 8 |- 1 e. CC
402, 39, 24subdii 7301 . . . . . . 7 |- (10 x. (1 - (1 / 10))) = ((10 x. 1) - (10 x. (1 / 10)))
412mulid1i 7092 . . . . . . . 8 |- (10 x. 1) = 10
422, 8recidi 7485 . . . . . . . 8 |- (10 x. (1 / 10)) = 1
4341, 42oveq12i 4932 . . . . . . 7 |- ((10 x. 1) - (10 x. (1 / 10))) = (10 - 1)
4439, 14addcomi 7213 . . . . . . . . 9 |- (1 + 9) = (9 + 1)
45 df-10 7753 . . . . . . . . 9 |- 10 = (9 + 1)
4644, 45eqtr4i 1961 . . . . . . . 8 |- (1 + 9) = 10
472, 39, 14, 46subaddrii 7253 . . . . . . 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 7547 . . . . . . 7 |- (9 / 10) e. RR
5150recni 7079 . . . . . 6 |- (9 / 10) e. CC
5239, 24subcli 7245 . . . . . 6 |- (1 - (1 / 10)) e. CC
5351, 52, 2, 8mulcani 7447 . . . . 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 4932 . . 3 |- ((9 / 10) / (9 / 10)) = ((9 x. (1 / 10)) / (1 - (1 / 10)))
56 9pos 7774 . . . . . 6 |- 0 < 9
5713, 1, 56, 7divgt0ii 7607 . . . . 5 |- 0 < (9 / 10)
5850, 57gt0ne0ii 7374 . . . 4 |- (9 / 10) =/= 0
5951, 58dividi 7515 . . 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 3362  ` cfv 4007  (class class class)co 4922  CCcc 6990  RRcr 6991  0cc0 6992  1c1 6993   + caddc 6995   x. cmul 6997   <_ cle 7100   < clt 7104   - cmin 7215   / cdiv 7217  NNcn 7218  NN0cn0 7219  ZZcz 7220  9c9 7743  10c10 7744  ^cexp 8621  abscabs 8949  sum_csu 9244
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 3448  ax-sep 3458  ax-nul 3467  ax-pow 3503  ax-pr 3527  ax-un 3799  ax-inf2 6064  ax-resscn 7046  ax-1cn 7047  ax-icn 7048  ax-addcl 7049  ax-addrcl 7050  ax-mulcl 7051  ax-mulrcl 7052  ax-mulcom 7053  ax-addass 7054  ax-mulass 7055  ax-distr 7056  ax-i2m1 7057  ax-1ne0 7058  ax-1rid 7059  ax-rnegex 7060  ax-rrecex 7061  ax-cnre 7062  ax-pre-lttri 7063  ax-pre-lttrn 7064  ax-pre-ltadd 7065  ax-pre-mulgt0 7066  ax-pre-sup 7067
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 3009  df-pw 3067  df-sn 3084  df-pr 3085  df-tp 3086  df-op 3087  df-uni 3218  df-int 3252  df-iun 3290  df-br 3363  df-opab 3417  df-tr 3432  df-eprel 3612  df-id 3615  df-po 3620  df-so 3634  df-fr 3653  df-we 3669  df-ord 3685  df-on 3686  df-lim 3687  df-suc 3688  df-om 3962  df-xp 4009  df-rel 4010  df-cnv 4011  df-co 4012  df-dm 4013  df-rn 4014  df-res 4015  df-ima 4016  df-fun 4017  df-fn 4018  df-f 4019  df-f1 4020  df-fo 4021  df-f1o 4022  df-fv 4023  df-iso 4024  df-ov 4924  df-oprab 4925  df-mpt 5059  df-mpt2 5060  df-1st 5158  df-2nd 5159  df-iota 5263  df-rdg 5349  df-1o 5386  df-er 5523  df-map 5611  df-pm 5612  df-en 5668  df-dom 5669  df-sdom 5670  df-fin 5671  df-riota 5811  df-sup 5985  df-card 6228  df-pnf 7105  df-mnf 7106  df-xr 7107  df-ltxr 7108  df-le 7109  df-sub 7234  df-neg 7236  df-div 7460  df-n 7699  df-2 7745  df-3 7746  df-4 7747  df-5 7748  df-6 7749  df-7 7750  df-8 7751  df-9 7752  df-10 7753  df-n0 7869  df-z 7913  df-uz 8033  df-q 8115  df-rp 8240  df-fz 8385  df-fl 8478  df-seq 8570  df-exp 8622  df-hash 8801  df-cj 8856  df-re 8857  df-im 8858  df-sqr 8950  df-abs 8951  df-clim 9119  df-rlim 9120  df-sum 9245
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