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Theorem 0.999... 12145
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 9646 . . . . . . 7  |-  10  e.  RR
21recni 8703 . . . . . 6  |-  10  e.  CC
3 nnnn0 9790 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  NN0 )
4 expcl 10936 . . . . . 6  |-  ( ( 10  e.  CC  /\  k  e.  NN0 )  -> 
( 10 ^ k
)  e.  CC )
52, 3, 4sylancr 641 . . . . 5  |-  ( k  e.  NN  ->  ( 10 ^ k )  e.  CC )
62a1i 11 . . . . . 6  |-  ( k  e.  NN  ->  10  e.  CC )
7 10pos 9658 . . . . . . . 8  |-  0  <  10
81, 7gt0ne0ii 9146 . . . . . . 7  |-  10  =/=  0
98a1i 11 . . . . . 6  |-  ( k  e.  NN  ->  10  =/=  0 )
10 nnz 9863 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  ZZ )
116, 9, 10expne0d 11065 . . . . 5  |-  ( k  e.  NN  ->  ( 10 ^ k )  =/=  0 )
12 9re 9645 . . . . . . 7  |-  9  e.  RR
1312recni 8703 . . . . . 6  |-  9  e.  CC
14 divrec 9272 . . . . . 6  |-  ( ( 9  e.  CC  /\  ( 10 ^ k )  e.  CC  /\  ( 10 ^ k )  =/=  0 )  ->  (
9  /  ( 10
^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
1513, 14mp3an1 1263 . . . . 5  |-  ( ( ( 10 ^ k
)  e.  CC  /\  ( 10 ^ k )  =/=  0 )  -> 
( 9  /  ( 10 ^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
165, 11, 15syl2anc 639 . . . 4  |-  ( k  e.  NN  ->  (
9  /  ( 10
^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
176, 9, 10exprecd 11067 . . . . 5  |-  ( k  e.  NN  ->  (
( 1  /  10 ) ^ k )  =  ( 1  /  ( 10 ^ k ) ) )
1817oveq2d 5701 . . . 4  |-  ( k  e.  NN  ->  (
9  x.  ( ( 1  /  10 ) ^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
1916, 18eqtr4d 2276 . . 3  |-  ( k  e.  NN  ->  (
9  /  ( 10
^ k ) )  =  ( 9  x.  ( ( 1  /  10 ) ^ k ) ) )
2019sumeq2i 11983 . 2  |-  sum_ k  e.  NN  ( 9  / 
( 10 ^ k
) )  =  sum_ k  e.  NN  (
9  x.  ( ( 1  /  10 ) ^ k ) )
211, 8rereccli 9355 . . . . 5  |-  ( 1  /  10 )  e.  RR
2221recni 8703 . . . 4  |-  ( 1  /  10 )  e.  CC
23 0re 8692 . . . . . . 7  |-  0  e.  RR
241, 7recgt0ii 9482 . . . . . . 7  |-  0  <  ( 1  /  10 )
2523, 21, 24ltleii 8794 . . . . . 6  |-  0  <_  ( 1  /  10 )
2621absidi 11672 . . . . . 6  |-  ( 0  <_  ( 1  /  10 )  ->  ( abs `  ( 1  /  10 ) )  =  ( 1  /  10 ) )
2725, 26ax-mp 9 . . . . 5  |-  ( abs `  ( 1  /  10 ) )  =  ( 1  /  10 )
28 1lt10 9749 . . . . . 6  |-  1  <  10
29 recgt1 9472 . . . . . . 7  |-  ( ( 10  e.  RR  /\  0  <  10 )  -> 
( 1  <  10  <->  ( 1  /  10 )  <  1 ) )
301, 7, 29mp2an 650 . . . . . 6  |-  ( 1  <  10  <->  ( 1  /  10 )  <  1 )
3128, 30mpbi 198 . . . . 5  |-  ( 1  /  10 )  <  1
3227, 31eqbrtri 3919 . . . 4  |-  ( abs `  ( 1  /  10 ) )  <  1
33 geoisum1c 12144 . . . 4  |-  ( ( 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 1276 . . 3  |-  sum_ k  e.  NN  ( 9  x.  ( ( 1  /  10 ) ^ k ) )  =  ( ( 9  x.  ( 1  /  10 ) )  /  ( 1  -  ( 1  /  10 ) ) )
3513, 2, 8divreci 9335 . . . 4  |-  ( 9  /  10 )  =  ( 9  x.  (
1  /  10 ) )
3613, 2, 8divcan2i 9333 . . . . . 6  |-  ( 10  x.  ( 9  /  10 ) )  =  9
37 ax-1cn 8649 . . . . . . . 8  |-  1  e.  CC
382, 37, 22subdii 9065 . . . . . . 7  |-  ( 10  x.  ( 1  -  ( 1  /  10 ) ) )  =  ( ( 10  x.  1 )  -  ( 10  x.  ( 1  /  10 ) ) )
392mulid1i 8693 . . . . . . . 8  |-  ( 10  x.  1 )  =  10
402, 8recidi 9321 . . . . . . . 8  |-  ( 10  x.  ( 1  /  10 ) )  =  1
4139, 40oveq12i 5697 . . . . . . 7  |-  ( ( 10  x.  1 )  -  ( 10  x.  ( 1  /  10 ) ) )  =  ( 10  -  1 )
4237, 13addcomi 8856 . . . . . . . . 9  |-  ( 1  +  9 )  =  ( 9  +  1 )
43 df-10 9632 . . . . . . . . 9  |-  10  =  ( 9  +  1 )
4442, 43eqtr4i 2264 . . . . . . . 8  |-  ( 1  +  9 )  =  10
452, 37, 13, 44subaddrii 8984 . . . . . . 7  |-  ( 10 
-  1 )  =  9
4638, 41, 453eqtrri 2266 . . . . . 6  |-  9  =  ( 10  x.  ( 1  -  (
1  /  10 ) ) )
4736, 46eqtri 2261 . . . . 5  |-  ( 10  x.  ( 9  /  10 ) )  =  ( 10  x.  ( 1  -  ( 1  /  10 ) ) )
4812, 1, 8redivcli 9357 . . . . . . 7  |-  ( 9  /  10 )  e.  RR
4948recni 8703 . . . . . 6  |-  ( 9  /  10 )  e.  CC
5037, 22subcli 8971 . . . . . 6  |-  ( 1  -  ( 1  /  10 ) )  e.  CC
5149, 50, 2, 8mulcani 9241 . . . . 5  |-  ( ( 10  x.  ( 9  /  10 ) )  =  ( 10  x.  ( 1  -  (
1  /  10 ) ) )  <->  ( 9  /  10 )  =  ( 1  -  (
1  /  10 ) ) )
5247, 51mpbi 198 . . . 4  |-  ( 9  /  10 )  =  ( 1  -  (
1  /  10 ) )
5335, 52oveq12i 5697 . . 3  |-  ( ( 9  /  10 )  /  ( 9  /  10 ) )  =  ( ( 9  x.  (
1  /  10 ) )  /  ( 1  -  ( 1  /  10 ) ) )
54 9pos 9657 . . . . . 6  |-  0  <  9
5512, 1, 54, 7divgt0ii 9494 . . . . 5  |-  0  <  ( 9  /  10 )
5648, 55gt0ne0ii 9146 . . . 4  |-  ( 9  /  10 )  =/=  0
5749, 56dividi 9323 . . 3  |-  ( ( 9  /  10 )  /  ( 9  /  10 ) )  =  1
5834, 53, 573eqtr2i 2267 . 2  |-  sum_ k  e.  NN  ( 9  x.  ( ( 1  /  10 ) ^ k ) )  =  1
5920, 58eqtri 2261 1  |-  sum_ k  e.  NN  ( 9  / 
( 10 ^ k
) )  =  1
Colors of variables: wff set class
Syntax hints:    <-> wb 175    = wceq 1608    e. wcel 1610    =/= wne 2400   class class class wbr 3900   ` cfv 4571  (class class class)co 5685   CCcc 8589   RRcr 8590   0cc0 8591   1c1 8592    + caddc 8594    x. cmul 8596    < clt 8721    <_ cle 8722    - cmin 8886    / cdiv 9255   NNcn 9566   9c9 9622   10c10 9623   NN0cn0 9783   ^cexp 10919   abscabs 11532   sum_csu 11969
This theorem was proved from axioms:  ax-1 6  ax-2 7  ax-3 8  ax-mp 9  ax-5 1522  ax-6 1523  ax-7 1524  ax-gen 1525  ax-8 1612  ax-11 1613  ax-13 1614  ax-14 1615  ax-17 1617  ax-12o 1653  ax-10 1667  ax-9 1673  ax-4 1681  ax-16 1915  ax-ext 2222  ax-rep 4007  ax-sep 4017  ax-nul 4025  ax-pow 4061  ax-pr 4087  ax-un 4382  ax-inf2 7200  ax-cnex 8647  ax-resscn 8648  ax-1cn 8649  ax-icn 8650  ax-addcl 8651  ax-addrcl 8652  ax-mulcl 8653  ax-mulrcl 8654  ax-mulcom 8655  ax-addass 8656  ax-mulass 8657  ax-distr 8658  ax-i2m1 8659  ax-1ne0 8660  ax-1rid 8661  ax-rnegex 8662  ax-rrecex 8663  ax-cnre 8664  ax-pre-lttri 8665  ax-pre-lttrn 8666  ax-pre-ltadd 8667  ax-pre-mulgt0 8668  ax-pre-sup 8669
This theorem depends on definitions:  df-bi 176  df-or 358  df-an 359  df-3or 934  df-3an 935  df-tru 1309  df-ex 1527  df-nf 1529  df-sb 1872  df-eu 2106  df-mo 2107  df-clab 2228  df-cleq 2234  df-clel 2237  df-nfc 2362  df-ne 2402  df-nel 2403  df-ral 2499  df-rex 2500  df-reu 2501  df-rab 2502  df-v 2714  df-sbc 2907  df-csb 2990  df-dif 3061  df-un 3063  df-in 3065  df-ss 3069  df-pss 3071  df-nul 3343  df-if 3451  df-pw 3512  df-sn 3530  df-pr 3531  df-tp 3532  df-op 3533  df-uni 3708  df-int 3741  df-iun 3785  df-br 3901  df-opab 3955  df-mpt 3956  df-tr 3990  df-eprel 4177  df-id 4181  df-po 4186  df-so 4187  df-fr 4224  df-se 4225  df-we 4226  df-ord 4267  df-on 4268  df-lim 4269  df-suc 4270  df-om 4527  df-xp 4573  df-rel 4574  df-cnv 4575  df-co 4576  df-dm 4577  df-rn 4578  df-res 4579  df-ima 4580  df-fun 4581  df-fn 4582  df-f 4583  df-f1 4584  df-fo 4585  df-f1o 4586  df-fv 4587  df-isom 4588  df-ov 5688  df-oprab 5689  df-mpt2 5690  df-1st 5948  df-2nd 5949  df-iota 6117  df-riota 6164  df-recs 6248  df-rdg 6283  df-1o 6339  df-oadd 6343  df-er 6520  df-pm 6635  df-en 6724  df-dom 6725  df-sdom 6726  df-fin 6727  df-sup 7052  df-oi 7083  df-card 7430  df-pnf 8723  df-mnf 8724  df-xr 8725  df-ltxr 8726  df-le 8727  df-sub 8888  df-neg 8889  df-div 9256  df-n 9567  df-2 9624  df-3 9625  df-4 9626  df-5 9627  df-6 9628  df-7 9629  df-8 9630  df-9 9631  df-10 9632  df-n0 9784  df-z 9843  df-uz 10049  df-rp 10173  df-fz 10598  df-fzo 10686  df-fl 10740  df-seq 10862  df-exp 10920  df-hash 11152  df-cj 11397  df-re 11398  df-im 11399  df-sqr 11533  df-abs 11534  df-clim 11773  df-rlim 11774  df-sum 11970
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