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Theorem 0.999... 9331
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 7086 . . . . . 6 |- 10 e. CC
3 nnnn0 7880 . . . . . 6 |- (k e. NN -> k e. NN0)
4 expcl 8637 . . . . . 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 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 8646 . . . . . 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 7768 . . . . . . 7 |- 9 e. RR
1514recni 7086 . . . . . 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 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 8655 . . . . . 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 4948 . . . 4 |- (k e. NN -> (9 x. ((1 / 10)^k)) = (9 x. (1 / (10^k))))
2218, 21eqtr4d 1996 . . 3 |- (k e. NN -> (9 / (10^k)) = (9 x. ((1 / 10)^k)))
2322sumeq2i 9193 . 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 7086 . . 3 |- (1 / 10) e. CC
26 0re 7106 . . . . . 6 |- 0 e. RR
271, 7recgt0ii 7570 . . . . . 6 |- 0 < (1 / 10)
2826, 24, 27ltleii 7171 . . . . 5 |- 0 <_ (1 / 10)
2924absidi 9034 . . . . 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 7105 . . . . . . . 8 |- 1 e. RR
33 ltaddpos2 7411 . . . . . . . 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 7758 . . . . . 6 |- 10 = (9 + 1)
3735, 36breqtrri 3404 . . . . 5 |- 1 < 10
38 recgt1 7647 . . . . . 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 3398 . . 3 |- (abs` (1 / 10)) < 1
42 geoisum1c 9330 . . 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 7495 . . . 4 |- (9 / 10) = (9 x. (1 / 10))
4515, 2, 8divcan2i 7480 . . . . . 6 |- (10 x. (9 / 10)) = 9
46 ax-1cn 7054 . . . . . . . 8 |- 1 e. CC
472, 46, 25subdii 7307 . . . . . . 7 |- (10 x. (1 - (1 / 10))) = ((10 x. 1) - (10 x. (1 / 10)))
482mulid1i 7099 . . . . . . . 8 |- (10 x. 1) = 10
492, 8recidi 7491 . . . . . . . 8 |- (10 x. (1 / 10)) = 1
5048, 49oveq12i 4945 . . . . . . 7 |- ((10 x. 1) - (10 x. (1 / 10))) = (10 - 1)
5146, 15addcomi 7219 . . . . . . . . 9 |- (1 + 9) = (9 + 1)
5251, 36eqtr4i 1984 . . . . . . . 8 |- (1 + 9) = 10
532, 46, 15, 52subaddrii 7259 . . . . . . 7 |- (10 - 1) = 9
5447, 50, 533eqtrri 1986 . . . . . 6 |- 9 = (10 x. (1 - (1 / 10)))
5545, 54eqtri 1981 . . . . 5 |- (10 x. (9 / 10)) = (10 x. (1 - (1 / 10)))
5614, 1, 8redivcli 7553 . . . . . . 7 |- (9 / 10) e. RR
5756recni 7086 . . . . . 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 217 . . . 4 |- (9 / 10) = (1 - (1 / 10))
6144, 60oveq12i 4945 . . 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 1983 . 2 |- ((9 x. (1 / 10)) / (1 - (1 / 10))) = 1
6623, 43, 653eqtri 1985 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 2079   class class class wbr 3379  ` cfv 4020  (class class class)co 4935  CCcc 6997  RRcr 6998  0cc0 6999  1c1 7000   + caddc 7002   x. cmul 7004   <_ cle 7107   < clt 7111   - cmin 7221   / cdiv 7223  NNcn 7224  NN0cn0 7225  ZZcz 7226  9c9 7748  10c10 7749  ^cexp 8620  abscabs 8949  sum_csu 9179
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 1672  ax-15 1835  ax-ext 1943  ax-rep 3465  ax-sep 3475  ax-nul 3484  ax-pow 3520  ax-pr 3544  ax-un 3814  ax-inf2 6072  ax-resscn 7053  ax-1cn 7054  ax-icn 7055  ax-addcl 7056  ax-addrcl 7057  ax-mulcl 7058  ax-mulrcl 7059  ax-mulcom 7060  ax-addass 7061  ax-mulass 7062  ax-distr 7063  ax-i2m1 7064  ax-1ne0 7065  ax-1rid 7066  ax-rnegex 7067  ax-rrecex 7068  ax-cnre 7069  ax-pre-lttri 7070  ax-pre-lttrn 7071  ax-pre-ltadd 7072  ax-pre-mulgt0 7073  ax-pre-sup 7074
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 1634  df-eu 1861  df-mo 1862  df-clab 1949  df-cleq 1954  df-clel 1957  df-ne 2081  df-nel 2082  df-ral 2174  df-rex 2175  df-reu 2176  df-rab 2177  df-v 2368  df-sbc 2533  df-csb 2607  df-dif 2666  df-un 2668  df-in 2670  df-ss 2672  df-pss 2674  df-nul 2928  df-if 3029  df-pw 3087  df-sn 3102  df-pr 3103  df-tp 3105  df-op 3106  df-uni 3235  df-int 3269  df-iun 3307  df-br 3380  df-opab 3434  df-tr 3449  df-eprel 3627  df-id 3630  df-po 3635  df-so 3649  df-fr 3668  df-we 3684  df-ord 3700  df-on 3701  df-lim 3702  df-suc 3703  df-om 3975  df-xp 4022  df-rel 4023  df-cnv 4024  df-co 4025  df-dm 4026  df-rn 4027  df-res 4028  df-ima 4029  df-fun 4030  df-fn 4031  df-f 4032  df-f1 4033  df-fo 4034  df-f1o 4035  df-fv 4036  df-iso 4037  df-ov 4937  df-oprab 4938  df-mpt 5072  df-mpt2 5073  df-1st 5169  df-2nd 5170  df-iota 5273  df-rdg 5359  df-1o 5396  df-er 5533  df-map 5621  df-en 5678  df-dom 5679  df-sdom 5680  df-fin 5681  df-riota 5821  df-sup 5994  df-card 6237  df-pnf 7112  df-mnf 7113  df-xr 7114  df-ltxr 7115  df-le 7116  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 8243  df-fz 8387  df-seq 8569  df-exp 8621  df-hash 8799  df-cj 8857  df-re 8858  df-im 8859  df-sqr 8950  df-abs 8951  df-clim 9103  df-sum 9180
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