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Theorem 0.999... 9347
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 7769 . . . . . . 7 |- 10 e. RR
21recni 7085 . . . . . 6 |- 10 e. CC
3 nnnn0 7880 . . . . . 6 |- (k e. NN -> k e. NN0)
4 expcl 8639 . . . . . 6 |- ((10 e. CC /\ k e. NN0) -> (10^k) e. CC)
52, 3, 4sylancr 661 . . . . 5 |- (k e. NN -> (10^k) e. CC)
62a1i 10 . . . . . 6 |- (k e. NN -> 10 e. CC)
7 10pos 7780 . . . . . . . 8 |- 0 < 10
81, 7gt0ne0ii 7380 . . . . . . 7 |- 10 =/= 0
98a1i 10 . . . . . 6 |- (k e. NN -> 10 =/= 0)
10 nn0z 7938 . . . . . . 7 |- (k e. NN0 -> k e. ZZ)
113, 10syl 14 . . . . . 6 |- (k e. NN -> k e. ZZ)
12 expne0i 8649 . . . . . 6 |- ((10 e. CC /\ 10 =/= 0 /\ k e. ZZ) -> (10^k) =/= 0)
136, 9, 11, 12syl3anc 1178 . . . . 5 |- (k e. NN -> (10^k) =/= 0)
14 9re 7768 . . . . . . 7 |- 9 e. RR
1514recni 7085 . . . . . 6 |- 9 e. CC
16 divrec 7497 . . . . . 6 |- ((9 e. CC /\ (10^k) e. CC /\ (10^k) =/= 0) -> (9 / (10^k)) = (9 x. (1 / (10^k))))
1715, 16mp3an1 1260 . . . . 5 |- (((10^k) e. CC /\ (10^k) =/= 0) -> (9 / (10^k)) = (9 x. (1 / (10^k))))
185, 13, 17syl2anc 659 . . . 4 |- (k e. NN -> (9 / (10^k)) = (9 x. (1 / (10^k))))
19 exprec 8658 . . . . . 6 |- ((10 e. CC /\ 10 =/= 0 /\ k e. ZZ) -> ((1 / 10)^k) = (1 / (10^k)))
206, 9, 11, 19syl3anc 1178 . . . . 5 |- (k e. NN -> ((1 / 10)^k) = (1 / (10^k)))
2120oveq2d 4944 . . . 4 |- (k e. NN -> (9 x. ((1 / 10)^k)) = (9 x. (1 / (10^k))))
2218, 21eqtr4d 1988 . . 3 |- (k e. NN -> (9 / (10^k)) = (9 x. ((1 / 10)^k)))
2322sumeq2i 9209 . 2 |- sum_k e. NN (9 / (10^k)) = sum_k e. NN (9 x. ((1 / 10)^k))
241, 8rereccli 7556 . . . 4 |- (1 / 10) e. RR
2524recni 7085 . . 3 |- (1 / 10) e. CC
26 0re 7105 . . . . . 6 |- 0 e. RR
271, 7recgt0ii 7570 . . . . . 6 |- 0 < (1 / 10)
2826, 24, 27ltleii 7170 . . . . 5 |- 0 <_ (1 / 10)
2924absidi 9035 . . . . 5 |- (0 <_ (1 / 10) -> (abs` (1 / 10)) = (1 / 10))
3028, 29ax-mp 8 . . . 4 |- (abs` (1 / 10)) = (1 / 10)
31 9pos 7779 . . . . . . 7 |- 0 < 9
32 1re 7104 . . . . . . . 8 |- 1 e. RR
33 ltaddpos2 7411 . . . . . . . 8 |- ((9 e. RR /\ 1 e. RR) -> (0 < 9 <-> 1 < (9 + 1)))
3414, 32, 33mp2an 670 . . . . . . 7 |- (0 < 9 <-> 1 < (9 + 1))
3531, 34mpbi 212 . . . . . 6 |- 1 < (9 + 1)
36 df-10 7758 . . . . . 6 |- 10 = (9 + 1)
3735, 36breqtrri 3398 . . . . 5 |- 1 < 10
38 recgt1 7647 . . . . . 6 |- ((10 e. RR /\ 0 < 10) -> (1 < 10 <-> (1 / 10) < 1))
391, 7, 38mp2an 670 . . . . 5 |- (1 < 10 <-> (1 / 10) < 1)
4037, 39mpbi 212 . . . 4 |- (1 / 10) < 1
4130, 40eqbrtri 3392 . . 3 |- (abs` (1 / 10)) < 1
42 geoisum1c 9346 . . 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 1273 . 2 |- sum_k e. NN (9 x. ((1 / 10)^k)) = ((9 x. (1 / 10)) / (1 - (1 / 10)))
4415, 2, 8divreci 7495 . . . 4 |- (9 / 10) = (9 x. (1 / 10))
4515, 2, 8divcan2i 7480 . . . . . 6 |- (10 x. (9 / 10)) = 9
46 ax-1cn 7053 . . . . . . . 8 |- 1 e. CC
472, 46, 25subdii 7307 . . . . . . 7 |- (10 x. (1 - (1 / 10))) = ((10 x. 1) - (10 x. (1 / 10)))
482mulid1i 7098 . . . . . . . 8 |- (10 x. 1) = 10
492, 8recidi 7491 . . . . . . . 8 |- (10 x. (1 / 10)) = 1
5048, 49oveq12i 4941 . . . . . . 7 |- ((10 x. 1) - (10 x. (1 / 10))) = (10 - 1)
5146, 15addcomi 7219 . . . . . . . . 9 |- (1 + 9) = (9 + 1)
5251, 36eqtr4i 1976 . . . . . . . 8 |- (1 + 9) = 10
532, 46, 15, 52subaddrii 7259 . . . . . . 7 |- (10 - 1) = 9
5447, 50, 533eqtrri 1978 . . . . . 6 |- 9 = (10 x. (1 - (1 / 10)))
5545, 54eqtri 1973 . . . . 5 |- (10 x. (9 / 10)) = (10 x. (1 - (1 / 10)))
5614, 1, 8redivcli 7553 . . . . . . 7 |- (9 / 10) e. RR
5756recni 7085 . . . . . 6 |- (9 / 10) e. CC
5846, 25subcli 7251 . . . . . 6 |- (1 - (1 / 10)) e. CC
5957, 58, 2, 8mulcani 7453 . . . . 5 |- ((10 x. (9 / 10)) = (10 x. (1 - (1 / 10))) <-> (9 / 10) = (1 - (1 / 10)))
6055, 59mpbi 212 . . . 4 |- (9 / 10) = (1 - (1 / 10))
6144, 60oveq12i 4941 . . 3 |- ((9 / 10) / (9 / 10)) = ((9 x. (1 / 10)) / (1 - (1 / 10)))
6214, 1, 31, 7divgt0ii 7612 . . . . 5 |- 0 < (9 / 10)
6356, 62gt0ne0ii 7380 . . . 4 |- (9 / 10) =/= 0
6457, 63dividi 7521 . . 3 |- ((9 / 10) / (9 / 10)) = 1
6561, 64eqtr3i 1975 . 2 |- ((9 x. (1 / 10)) / (1 - (1 / 10))) = 1
6623, 43, 653eqtri 1977 1 |- sum_k e. NN (9 / (10^k)) = 1
Colors of variables: wff set class
Syntax hints:   <-> wb 184   = wceq 1449   e. wcel 1451   =/= wne 2071   class class class wbr 3373  ` cfv 4016  (class class class)co 4931  CCcc 6996  RRcr 6997  0cc0 6998  1c1 6999   + caddc 7001   x. cmul 7003   <_ cle 7106   < clt 7110   - cmin 7221   / cdiv 7223  NNcn 7224  NN0cn0 7225  ZZcz 7226  9c9 7748  10c10 7749  ^cexp 8622  abscabs 8950  sum_csu 9195
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1367  ax-6 1368  ax-7 1369  ax-gen 1370  ax-8 1453  ax-10 1454  ax-11 1455  ax-12 1456  ax-13 1457  ax-14 1458  ax-17 1465  ax-9 1480  ax-4 1486  ax-16 1664  ax-15 1827  ax-ext 1935  ax-rep 3459  ax-sep 3469  ax-nul 3478  ax-pow 3514  ax-pr 3538  ax-un 3808  ax-inf2 6071  ax-resscn 7052  ax-1cn 7053  ax-icn 7054  ax-addcl 7055  ax-addrcl 7056  ax-mulcl 7057  ax-mulrcl 7058  ax-mulcom 7059  ax-addass 7060  ax-mulass 7061  ax-distr 7062  ax-i2m1 7063  ax-1ne0 7064  ax-1rid 7065  ax-rnegex 7066  ax-rrecex 7067  ax-cnre 7068  ax-pre-lttri 7069  ax-pre-lttrn 7070  ax-pre-ltadd 7071  ax-pre-mulgt0 7072  ax-pre-sup 7073
This theorem depends on definitions:  df-bi 185  df-or 378  df-an 379  df-3or 938  df-3an 939  df-tru 1345  df-ex 1372  df-sb 1626  df-eu 1853  df-mo 1854  df-clab 1941  df-cleq 1946  df-clel 1949  df-ne 2073  df-nel 2074  df-ral 2166  df-rex 2167  df-reu 2168  df-rab 2169  df-v 2360  df-sbc 2525  df-csb 2600  df-dif 2660  df-un 2662  df-in 2664  df-ss 2666  df-pss 2668  df-nul 2922  df-if 3023  df-pw 3081  df-sn 3096  df-pr 3097  df-tp 3099  df-op 3100  df-uni 3229  df-int 3263  df-iun 3301  df-br 3374  df-opab 3428  df-tr 3443  df-eprel 3621  df-id 3624  df-po 3629  df-so 3643  df-fr 3662  df-we 3678  df-ord 3694  df-on 3695  df-lim 3696  df-suc 3697  df-om 3971  df-xp 4018  df-rel 4019  df-cnv 4020  df-co 4021  df-dm 4022  df-rn 4023  df-res 4024  df-ima 4025  df-fun 4026  df-fn 4027  df-f 4028  df-f1 4029  df-fo 4030  df-f1o 4031  df-fv 4032  df-iso 4033  df-ov 4933  df-oprab 4934  df-mpt 5068  df-mpt2 5069  df-1st 5166  df-2nd 5167  df-iota 5270  df-rdg 5356  df-1o 5393  df-er 5530  df-map 5618  df-en 5675  df-dom 5676  df-sdom 5677  df-fin 5678  df-riota 5818  df-sup 5992  df-card 6236  df-pnf 7111  df-mnf 7112  df-xr 7113  df-ltxr 7114  df-le 7115  df-sub 7240  df-neg 7242  df-div 7466  df-n 7704  df-2 7750  df-3 7751  df-4 7752  df-5 7753  df-6 7754  df-7 7755  df-8 7756  df-9 7757  df-10 7758  df-n0 7874  df-z 7918  df-uz 8038  df-q 8120  df-rp 8245  df-fz 8389  df-seq 8571  df-exp 8623  df-hash 8802  df-cj 8857  df-re 8858  df-im 8859  df-sqr 8951  df-abs 8952  df-clim 9119  df-sum 9196
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