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Theorem 0.999... 11537
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. (Contributed by NM, 2-Nov-2007.)
Assertion
Ref Expression
0.999...  |-  sum_ k  e.  NN  ( 9  / 
( 10 ^ k
) )  =  1

Proof of Theorem 0.999...
StepHypRef Expression
1 10re 9213 . . . . . . 7  |-  10  e.  RR
21recni 8275 . . . . . 6  |-  10  e.  CC
3 nnnn0 9356 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  NN0 )
4 expcl 10491 . . . . . 6  |-  ( ( 10  e.  CC  /\  k  e.  NN0 )  -> 
( 10 ^ k
)  e.  CC )
52, 3, 4sylancr 637 . . . . 5  |-  ( k  e.  NN  ->  ( 10 ^ k )  e.  CC )
62a1i 10 . . . . . 6  |-  ( k  e.  NN  ->  10  e.  CC )
7 10pos 9225 . . . . . . . 8  |-  0  <  10
81, 7gt0ne0ii 8716 . . . . . . 7  |-  10  =/=  0
98a1i 10 . . . . . 6  |-  ( k  e.  NN  ->  10  =/=  0 )
10 nnz 9429 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  ZZ )
116, 9, 10expne0d 10620 . . . . 5  |-  ( k  e.  NN  ->  ( 10 ^ k )  =/=  0 )
12 9re 9212 . . . . . . 7  |-  9  e.  RR
1312recni 8275 . . . . . 6  |-  9  e.  CC
14 divrec 8841 . . . . . 6  |-  ( ( 9  e.  CC  /\  ( 10 ^ k )  e.  CC  /\  ( 10 ^ k )  =/=  0 )  ->  (
9  /  ( 10
^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
1513, 14mp3an1 1222 . . . . 5  |-  ( ( ( 10 ^ k
)  e.  CC  /\  ( 10 ^ k )  =/=  0 )  -> 
( 9  /  ( 10 ^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
165, 11, 15syl2anc 635 . . . 4  |-  ( k  e.  NN  ->  (
9  /  ( 10
^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
176, 9, 10exprecd 10622 . . . . 5  |-  ( k  e.  NN  ->  (
( 1  /  10 ) ^ k )  =  ( 1  /  ( 10 ^ k ) ) )
1817oveq2d 5382 . . . 4  |-  ( k  e.  NN  ->  (
9  x.  ( ( 1  /  10 ) ^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
1916, 18eqtr4d 2100 . . 3  |-  ( k  e.  NN  ->  (
9  /  ( 10
^ k ) )  =  ( 9  x.  ( ( 1  /  10 ) ^ k ) ) )
2019sumeq2i 11379 . 2  |-  sum_ k  e.  NN  ( 9  / 
( 10 ^ k
) )  =  sum_ k  e.  NN  (
9  x.  ( ( 1  /  10 ) ^ k ) )
211, 8rereccli 8924 . . . 4  |-  ( 1  /  10 )  e.  RR
2221recni 8275 . . 3  |-  ( 1  /  10 )  e.  CC
23 0re 8264 . . . . . 6  |-  0  e.  RR
241, 7recgt0ii 9050 . . . . . 6  |-  0  <  ( 1  /  10 )
2523, 21, 24ltleii 8366 . . . . 5  |-  0  <_  ( 1  /  10 )
2621absidi 11068 . . . . 5  |-  ( 0  <_  ( 1  /  10 )  ->  ( abs `  ( 1  /  10 ) )  =  ( 1  /  10 ) )
2725, 26ax-mp 8 . . . 4  |-  ( abs `  ( 1  /  10 ) )  =  ( 1  /  10 )
28 1lt10 9316 . . . . 5  |-  1  <  10
29 recgt1 9040 . . . . . 6  |-  ( ( 10  e.  RR  /\  0  <  10 )  -> 
( 1  <  10  <->  ( 1  /  10 )  <  1 ) )
301, 7, 29mp2an 646 . . . . 5  |-  ( 1  <  10  <->  ( 1  /  10 )  <  1 )
3128, 30mpbi 197 . . . 4  |-  ( 1  /  10 )  <  1
3227, 31eqbrtri 3611 . . 3  |-  ( abs `  ( 1  /  10 ) )  <  1
33 geoisum1c 11536 . . 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 ) ) ) )
3413, 22, 32, 33mp3an 1235 . 2  |-  sum_ k  e.  NN  ( 9  x.  ( ( 1  /  10 ) ^ k ) )  =  ( ( 9  x.  ( 1  /  10 ) )  /  ( 1  -  ( 1  /  10 ) ) )
3513, 2, 8divreci 8904 . . . 4  |-  ( 9  /  10 )  =  ( 9  x.  (
1  /  10 ) )
3613, 2, 8divcan2i 8902 . . . . . 6  |-  ( 10  x.  ( 9  /  10 ) )  =  9
37 ax-1cn 8221 . . . . . . . 8  |-  1  e.  CC
382, 37, 22subdii 8635 . . . . . . 7  |-  ( 10  x.  ( 1  -  ( 1  /  10 ) ) )  =  ( ( 10  x.  1 )  -  ( 10  x.  ( 1  /  10 ) ) )
392mulid1i 8265 . . . . . . . 8  |-  ( 10  x.  1 )  =  10
402, 8recidi 8890 . . . . . . . 8  |-  ( 10  x.  ( 1  /  10 ) )  =  1
4139, 40oveq12i 5378 . . . . . . 7  |-  ( ( 10  x.  1 )  -  ( 10  x.  ( 1  /  10 ) ) )  =  ( 10  -  1 )
4237, 13addcomi 8428 . . . . . . . . 9  |-  ( 1  +  9 )  =  ( 9  +  1 )
43 df-10 9199 . . . . . . . . 9  |-  10  =  ( 9  +  1 )
4442, 43eqtr4i 2088 . . . . . . . 8  |-  ( 1  +  9 )  =  10
452, 37, 13, 44subaddrii 8556 . . . . . . 7  |-  ( 10 
-  1 )  =  9
4638, 41, 453eqtrri 2090 . . . . . 6  |-  9  =  ( 10  x.  ( 1  -  (
1  /  10 ) ) )
4736, 46eqtri 2085 . . . . 5  |-  ( 10  x.  ( 9  /  10 ) )  =  ( 10  x.  ( 1  -  ( 1  /  10 ) ) )
4812, 1, 8redivcli 8926 . . . . . . 7  |-  ( 9  /  10 )  e.  RR
4948recni 8275 . . . . . 6  |-  ( 9  /  10 )  e.  CC
5037, 22subcli 8543 . . . . . 6  |-  ( 1  -  ( 1  /  10 ) )  e.  CC
5149, 50, 2, 8mulcani 8810 . . . . 5  |-  ( ( 10  x.  ( 9  /  10 ) )  =  ( 10  x.  ( 1  -  (
1  /  10 ) ) )  <->  ( 9  /  10 )  =  ( 1  -  (
1  /  10 ) ) )
5247, 51mpbi 197 . . . 4  |-  ( 9  /  10 )  =  ( 1  -  (
1  /  10 ) )
5335, 52oveq12i 5378 . . 3  |-  ( ( 9  /  10 )  /  ( 9  /  10 ) )  =  ( ( 9  x.  (
1  /  10 ) )  /  ( 1  -  ( 1  /  10 ) ) )
54 9pos 9224 . . . . . 6  |-  0  <  9
5512, 1, 54, 7divgt0ii 9062 . . . . 5  |-  0  <  ( 9  /  10 )
5648, 55gt0ne0ii 8716 . . . 4  |-  ( 9  /  10 )  =/=  0
5749, 56dividi 8892 . . 3  |-  ( ( 9  /  10 )  /  ( 9  /  10 ) )  =  1
5853, 57eqtr3i 2087 . 2  |-  ( ( 9  x.  ( 1  /  10 ) )  /  ( 1  -  ( 1  /  10 ) ) )  =  1
5920, 34, 583eqtri 2089 1  |-  sum_ k  e.  NN  ( 9  / 
( 10 ^ k
) )  =  1
Colors of variables: wff set class
Syntax hints:    <-> wb 174    = wceq 1520    e. wcel 1522    =/= wne 2183   class class class wbr 3592   ` cfv 4274  (class class class)co 5366   CCcc 8162   RRcr 8163   0cc0 8164   1c1 8165    + caddc 8167    x. cmul 8169    <_ cle 8290    < clt 8294    - cmin 8458    / cdiv 8824   NNcn 9133   9c9 9189   10c10 9190   NN0cn0 9349   ^cexp 10474   abscabs 10929   sum_csu 11365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1442  ax-6 1443  ax-7 1444  ax-gen 1445  ax-8 1524  ax-11 1525  ax-13 1526  ax-14 1527  ax-17 1529  ax-12o 1563  ax-10 1577  ax-9 1583  ax-4 1590  ax-16 1776  ax-ext 2047  ax-rep 3697  ax-sep 3707  ax-nul 3715  ax-pow 3751  ax-pr 3775  ax-un 4067  ax-inf2 6831  ax-cnex 8219  ax-resscn 8220  ax-1cn 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-1ne0 8232  ax-1rid 8233  ax-rnegex 8234  ax-rrecex 8235  ax-cnre 8236  ax-pre-lttri 8237  ax-pre-lttrn 8238  ax-pre-ltadd 8239  ax-pre-mulgt0 8240  ax-pre-sup 8241
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3or 897  df-3an 898  df-tru 1259  df-ex 1447  df-sb 1737  df-eu 1959  df-mo 1960  df-clab 2053  df-cleq 2058  df-clel 2061  df-ne 2185  df-nel 2186  df-ral 2279  df-rex 2280  df-reu 2281  df-rab 2282  df-v 2478  df-sbc 2652  df-csb 2734  df-dif 2797  df-un 2799  df-in 2801  df-ss 2805  df-pss 2807  df-nul 3074  df-if 3183  df-pw 3244  df-sn 3262  df-pr 3263  df-tp 3264  df-op 3265  df-uni 3431  df-int 3465  df-iun 3508  df-br 3593  df-opab 3647  df-mpt 3648  df-tr 3680  df-eprel 3862  df-id 3866  df-po 3871  df-so 3872  df-fr 3909  df-se 3910  df-we 3911  df-ord 3952  df-on 3953  df-lim 3954  df-suc 3955  df-om 4230  df-xp 4276  df-rel 4277  df-cnv 4278  df-co 4279  df-dm 4280  df-rn 4281  df-res 4282  df-ima 4283  df-fun 4284  df-fn 4285  df-f 4286  df-f1 4287  df-fo 4288  df-f1o 4289  df-fv 4290  df-iso 4291  df-ov 5369  df-oprab 5370  df-mpt2 5371  df-1st 5620  df-2nd 5621  df-iota 5776  df-recs 5849  df-rdg 5884  df-1o 5940  df-oadd 5944  df-er 6121  df-pm 6226  df-en 6308  df-dom 6309  df-sdom 6310  df-fin 6311  df-riota 6474  df-sup 6682  df-oi 6714  df-card 7060  df-pnf 8295  df-mnf 8296  df-xr 8297  df-ltxr 8298  df-le 8299  df-sub 8460  df-neg 8461  df-div 8825  df-n 9134  df-2 9191  df-3 9192  df-4 9193  df-5 9194  df-6 9195  df-7 9196  df-8 9197  df-9 9198  df-10 9199  df-n0 9350  df-z 9409  df-uz 9615  df-rp 9739  df-fz 10161  df-fzo 10249  df-fl 10300  df-seq 10417  df-exp 10475  df-hash 10707  df-cj 10794  df-re 10795  df-im 10796  df-sqr 10930  df-abs 10931  df-clim 11169  df-rlim 11170  df-sum 11366
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