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Theorem 0.999... 11521
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 9201 . . . . . . 7  |-  10  e.  RR
21recni 8267 . . . . . 6  |-  10  e.  CC
3 nnnn0 9341 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  NN0 )
4 expcl 10476 . . . . . 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 9212 . . . . . . . 8  |-  0  <  10
81, 7gt0ne0ii 8708 . . . . . . 7  |-  10  =/=  0
98a1i 10 . . . . . 6  |-  ( k  e.  NN  ->  10  =/=  0 )
10 nnz 9414 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  ZZ )
116, 9, 10expne0d 10605 . . . . 5  |-  ( k  e.  NN  ->  ( 10 ^ k )  =/=  0 )
12 9re 9200 . . . . . . 7  |-  9  e.  RR
1312recni 8267 . . . . . 6  |-  9  e.  CC
14 divrec 8832 . . . . . 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 10607 . . . . 5  |-  ( k  e.  NN  ->  (
( 1  /  10 ) ^ k )  =  ( 1  /  ( 10 ^ k ) ) )
1817oveq2d 5376 . . . 4  |-  ( k  e.  NN  ->  (
9  x.  ( ( 1  /  10 ) ^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
1916, 18eqtr4d 2099 . . 3  |-  ( k  e.  NN  ->  (
9  /  ( 10
^ k ) )  =  ( 9  x.  ( ( 1  /  10 ) ^ k ) ) )
2019sumeq2i 11363 . 2  |-  sum_ k  e.  NN  ( 9  / 
( 10 ^ k
) )  =  sum_ k  e.  NN  (
9  x.  ( ( 1  /  10 ) ^ k ) )
211, 8rereccli 8915 . . . 4  |-  ( 1  /  10 )  e.  RR
2221recni 8267 . . 3  |-  ( 1  /  10 )  e.  CC
23 0re 8256 . . . . . 6  |-  0  e.  RR
241, 7recgt0ii 9041 . . . . . 6  |-  0  <  ( 1  /  10 )
2523, 21, 24ltleii 8358 . . . . 5  |-  0  <_  ( 1  /  10 )
2621absidi 11052 . . . . 5  |-  ( 0  <_  ( 1  /  10 )  ->  ( abs `  ( 1  /  10 ) )  =  ( 1  /  10 ) )
2725, 26ax-mp 8 . . . 4  |-  ( abs `  ( 1  /  10 ) )  =  ( 1  /  10 )
28 1lt10 9301 . . . . 5  |-  1  <  10
29 recgt1 9031 . . . . . 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 3605 . . 3  |-  ( abs `  ( 1  /  10 ) )  <  1
33 geoisum1c 11520 . . 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 8895 . . . 4  |-  ( 9  /  10 )  =  ( 9  x.  (
1  /  10 ) )
3613, 2, 8divcan2i 8893 . . . . . 6  |-  ( 10  x.  ( 9  /  10 ) )  =  9
37 ax-1cn 8215 . . . . . . . 8  |-  1  e.  CC
382, 37, 22subdii 8627 . . . . . . 7  |-  ( 10  x.  ( 1  -  ( 1  /  10 ) ) )  =  ( ( 10  x.  1 )  -  ( 10  x.  ( 1  /  10 ) ) )
392mulid1i 8257 . . . . . . . 8  |-  ( 10  x.  1 )  =  10
402, 8recidi 8881 . . . . . . . 8  |-  ( 10  x.  ( 1  /  10 ) )  =  1
4139, 40oveq12i 5372 . . . . . . 7  |-  ( ( 10  x.  1 )  -  ( 10  x.  ( 1  /  10 ) ) )  =  ( 10  -  1 )
4237, 13addcomi 8420 . . . . . . . . 9  |-  ( 1  +  9 )  =  ( 9  +  1 )
43 df-10 9190 . . . . . . . . 9  |-  10  =  ( 9  +  1 )
4442, 43eqtr4i 2087 . . . . . . . 8  |-  ( 1  +  9 )  =  10
452, 37, 13, 44subaddrii 8548 . . . . . . 7  |-  ( 10 
-  1 )  =  9
4638, 41, 453eqtrri 2089 . . . . . 6  |-  9  =  ( 10  x.  ( 1  -  (
1  /  10 ) ) )
4736, 46eqtri 2084 . . . . 5  |-  ( 10  x.  ( 9  /  10 ) )  =  ( 10  x.  ( 1  -  ( 1  /  10 ) ) )
4812, 1, 8redivcli 8917 . . . . . . 7  |-  ( 9  /  10 )  e.  RR
4948recni 8267 . . . . . 6  |-  ( 9  /  10 )  e.  CC
5037, 22subcli 8535 . . . . . 6  |-  ( 1  -  ( 1  /  10 ) )  e.  CC
5149, 50, 2, 8mulcani 8801 . . . . 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 5372 . . 3  |-  ( ( 9  /  10 )  /  ( 9  /  10 ) )  =  ( ( 9  x.  (
1  /  10 ) )  /  ( 1  -  ( 1  /  10 ) ) )
54 9pos 9211 . . . . . 6  |-  0  <  9
5512, 1, 54, 7divgt0ii 9053 . . . . 5  |-  0  <  ( 9  /  10 )
5648, 55gt0ne0ii 8708 . . . 4  |-  ( 9  /  10 )  =/=  0
5749, 56dividi 8883 . . 3  |-  ( ( 9  /  10 )  /  ( 9  /  10 ) )  =  1
5853, 57eqtr3i 2086 . 2  |-  ( ( 9  x.  ( 1  /  10 ) )  /  ( 1  -  ( 1  /  10 ) ) )  =  1
5920, 34, 583eqtri 2088 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 2182   class class class wbr 3586   ` cfv 4268  (class class class)co 5360   CCcc 8156   RRcr 8157   0cc0 8158   1c1 8159    + caddc 8161    x. cmul 8163    <_ cle 8282    < clt 8286    - cmin 8450    / cdiv 8815   NNcn 9124   9c9 9180   10c10 9181   NN0cn0 9334   ^cexp 10459   abscabs 10914   sum_csu 11349
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 1562  ax-10 1576  ax-9 1582  ax-4 1589  ax-16 1775  ax-ext 2046  ax-rep 3691  ax-sep 3701  ax-nul 3709  ax-pow 3745  ax-pr 3769  ax-un 4061  ax-inf2 6825  ax-cnex 8213  ax-resscn 8214  ax-1cn 8215  ax-icn 8216  ax-addcl 8217  ax-addrcl 8218  ax-mulcl 8219  ax-mulrcl 8220  ax-mulcom 8221  ax-addass 8222  ax-mulass 8223  ax-distr 8224  ax-i2m1 8225  ax-1ne0 8226  ax-1rid 8227  ax-rnegex 8228  ax-rrecex 8229  ax-cnre 8230  ax-pre-lttri 8231  ax-pre-lttrn 8232  ax-pre-ltadd 8233  ax-pre-mulgt0 8234  ax-pre-sup 8235
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 1736  df-eu 1958  df-mo 1959  df-clab 2052  df-cleq 2057  df-clel 2060  df-ne 2184  df-nel 2185  df-ral 2278  df-rex 2279  df-reu 2280  df-rab 2281  df-v 2477  df-sbc 2651  df-csb 2733  df-dif 2796  df-un 2798  df-in 2800  df-ss 2804  df-pss 2806  df-nul 3073  df-if 3182  df-pw 3243  df-sn 3261  df-pr 3262  df-tp 3263  df-op 3264  df-uni 3425  df-int 3459  df-iun 3502  df-br 3587  df-opab 3641  df-mpt 3642  df-tr 3674  df-eprel 3856  df-id 3860  df-po 3865  df-so 3866  df-fr 3903  df-se 3904  df-we 3905  df-ord 3946  df-on 3947  df-lim 3948  df-suc 3949  df-om 4224  df-xp 4270  df-rel 4271  df-cnv 4272  df-co 4273  df-dm 4274  df-rn 4275  df-res 4276  df-ima 4277  df-fun 4278  df-fn 4279  df-f 4280  df-f1 4281  df-fo 4282  df-f1o 4283  df-fv 4284  df-iso 4285  df-ov 5363  df-oprab 5364  df-mpt2 5365  df-1st 5614  df-2nd 5615  df-iota 5770  df-recs 5843  df-rdg 5878  df-1o 5934  df-oadd 5938  df-er 6115  df-pm 6220  df-en 6302  df-dom 6303  df-sdom 6304  df-fin 6305  df-riota 6468  df-sup 6676  df-oi 6708  df-card 7054  df-pnf 8287  df-mnf 8288  df-xr 8289  df-ltxr 8290  df-le 8291  df-sub 8452  df-neg 8453  df-div 8816  df-n 9125  df-2 9182  df-3 9183  df-4 9184  df-5 9185  df-6 9186  df-7 9187  df-8 9188  df-9 9189  df-10 9190  df-n0 9335  df-z 9394  df-uz 9600  df-rp 9724  df-fz 10146  df-fzo 10234  df-fl 10285  df-seq 10402  df-exp 10460  df-hash 10692  df-cj 10779  df-re 10780  df-im 10781  df-sqr 10915  df-abs 10916  df-clim 11153  df-rlim 11154  df-sum 11350
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