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Theorem 0.999... 12173
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 9674 . . . . . . 7  |-  10  e.  RR
21recni 8726 . . . . . 6  |-  10  e.  CC
3 nnnn0 9818 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  NN0 )
4 expcl 10964 . . . . . 6  |-  ( ( 10  e.  CC  /\  k  e.  NN0 )  -> 
( 10 ^ k
)  e.  CC )
52, 3, 4sylancr 647 . . . . 5  |-  ( k  e.  NN  ->  ( 10 ^ k )  e.  CC )
62a1i 12 . . . . . 6  |-  ( k  e.  NN  ->  10  e.  CC )
7 10pos 9686 . . . . . . . 8  |-  0  <  10
81, 7gt0ne0ii 9170 . . . . . . 7  |-  10  =/=  0
98a1i 12 . . . . . 6  |-  ( k  e.  NN  ->  10  =/=  0 )
10 nnz 9891 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  ZZ )
116, 9, 10expne0d 11093 . . . . 5  |-  ( k  e.  NN  ->  ( 10 ^ k )  =/=  0 )
12 9re 9673 . . . . . . 7  |-  9  e.  RR
1312recni 8726 . . . . . 6  |-  9  e.  CC
14 divrec 9296 . . . . . 6  |-  ( ( 9  e.  CC  /\  ( 10 ^ k )  e.  CC  /\  ( 10 ^ k )  =/=  0 )  ->  (
9  /  ( 10
^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
1513, 14mp3an1 1269 . . . . 5  |-  ( ( ( 10 ^ k
)  e.  CC  /\  ( 10 ^ k )  =/=  0 )  -> 
( 9  /  ( 10 ^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
165, 11, 15syl2anc 645 . . . 4  |-  ( k  e.  NN  ->  (
9  /  ( 10
^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
176, 9, 10exprecd 11095 . . . . 5  |-  ( k  e.  NN  ->  (
( 1  /  10 ) ^ k )  =  ( 1  /  ( 10 ^ k ) ) )
1817oveq2d 5723 . . . 4  |-  ( k  e.  NN  ->  (
9  x.  ( ( 1  /  10 ) ^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
1916, 18eqtr4d 2288 . . 3  |-  ( k  e.  NN  ->  (
9  /  ( 10
^ k ) )  =  ( 9  x.  ( ( 1  /  10 ) ^ k ) ) )
2019sumeq2i 12011 . 2  |-  sum_ k  e.  NN  ( 9  / 
( 10 ^ k
) )  =  sum_ k  e.  NN  (
9  x.  ( ( 1  /  10 ) ^ k ) )
211, 8rereccli 9379 . . . . 5  |-  ( 1  /  10 )  e.  RR
2221recni 8726 . . . 4  |-  ( 1  /  10 )  e.  CC
23 0re 8715 . . . . . . 7  |-  0  e.  RR
241, 7recgt0ii 9510 . . . . . . 7  |-  0  <  ( 1  /  10 )
2523, 21, 24ltleii 8817 . . . . . 6  |-  0  <_  ( 1  /  10 )
2621absidi 11700 . . . . . 6  |-  ( 0  <_  ( 1  /  10 )  ->  ( abs `  ( 1  /  10 ) )  =  ( 1  /  10 ) )
2725, 26ax-mp 10 . . . . 5  |-  ( abs `  ( 1  /  10 ) )  =  ( 1  /  10 )
28 1lt10 9777 . . . . . 6  |-  1  <  10
29 recgt1 9500 . . . . . . 7  |-  ( ( 10  e.  RR  /\  0  <  10 )  -> 
( 1  <  10  <->  ( 1  /  10 )  <  1 ) )
301, 7, 29mp2an 656 . . . . . 6  |-  ( 1  <  10  <->  ( 1  /  10 )  <  1 )
3128, 30mpbi 201 . . . . 5  |-  ( 1  /  10 )  <  1
3227, 31eqbrtri 3936 . . . 4  |-  ( abs `  ( 1  /  10 ) )  <  1
33 geoisum1c 12172 . . . 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 1282 . . 3  |-  sum_ k  e.  NN  ( 9  x.  ( ( 1  /  10 ) ^ k ) )  =  ( ( 9  x.  ( 1  /  10 ) )  /  ( 1  -  ( 1  /  10 ) ) )
3513, 2, 8divreci 9359 . . . 4  |-  ( 9  /  10 )  =  ( 9  x.  (
1  /  10 ) )
3613, 2, 8divcan2i 9357 . . . . . 6  |-  ( 10  x.  ( 9  /  10 ) )  =  9
37 ax-1cn 8672 . . . . . . . 8  |-  1  e.  CC
382, 37, 22subdii 9089 . . . . . . 7  |-  ( 10  x.  ( 1  -  ( 1  /  10 ) ) )  =  ( ( 10  x.  1 )  -  ( 10  x.  ( 1  /  10 ) ) )
392mulid1i 8716 . . . . . . . 8  |-  ( 10  x.  1 )  =  10
402, 8recidi 9345 . . . . . . . 8  |-  ( 10  x.  ( 1  /  10 ) )  =  1
4139, 40oveq12i 5719 . . . . . . 7  |-  ( ( 10  x.  1 )  -  ( 10  x.  ( 1  /  10 ) ) )  =  ( 10  -  1 )
4237, 13addcomi 8879 . . . . . . . . 9  |-  ( 1  +  9 )  =  ( 9  +  1 )
43 df-10 9660 . . . . . . . . 9  |-  10  =  ( 9  +  1 )
4442, 43eqtr4i 2276 . . . . . . . 8  |-  ( 1  +  9 )  =  10
452, 37, 13, 44subaddrii 9007 . . . . . . 7  |-  ( 10 
-  1 )  =  9
4638, 41, 453eqtrri 2278 . . . . . 6  |-  9  =  ( 10  x.  ( 1  -  (
1  /  10 ) ) )
4736, 46eqtri 2273 . . . . 5  |-  ( 10  x.  ( 9  /  10 ) )  =  ( 10  x.  ( 1  -  ( 1  /  10 ) ) )
4812, 1, 8redivcli 9381 . . . . . . 7  |-  ( 9  /  10 )  e.  RR
4948recni 8726 . . . . . 6  |-  ( 9  /  10 )  e.  CC
5037, 22subcli 8994 . . . . . 6  |-  ( 1  -  ( 1  /  10 ) )  e.  CC
5149, 50, 2, 8mulcani 9265 . . . . 5  |-  ( ( 10  x.  ( 9  /  10 ) )  =  ( 10  x.  ( 1  -  (
1  /  10 ) ) )  <->  ( 9  /  10 )  =  ( 1  -  (
1  /  10 ) ) )
5247, 51mpbi 201 . . . 4  |-  ( 9  /  10 )  =  ( 1  -  (
1  /  10 ) )
5335, 52oveq12i 5719 . . 3  |-  ( ( 9  /  10 )  /  ( 9  /  10 ) )  =  ( ( 9  x.  (
1  /  10 ) )  /  ( 1  -  ( 1  /  10 ) ) )
54 9pos 9685 . . . . . 6  |-  0  <  9
5512, 1, 54, 7divgt0ii 9522 . . . . 5  |-  0  <  ( 9  /  10 )
5648, 55gt0ne0ii 9170 . . . 4  |-  ( 9  /  10 )  =/=  0
5749, 56dividi 9347 . . 3  |-  ( ( 9  /  10 )  /  ( 9  /  10 ) )  =  1
5834, 53, 573eqtr2i 2279 . 2  |-  sum_ k  e.  NN  ( 9  x.  ( ( 1  /  10 ) ^ k ) )  =  1
5920, 58eqtri 2273 1  |-  sum_ k  e.  NN  ( 9  / 
( 10 ^ k
) )  =  1
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1619    e. wcel 1621    =/= wne 2412   class class class wbr 3917   ` cfv 4589  (class class class)co 5707   CCcc 8612   RRcr 8613   0cc0 8614   1c1 8615    + caddc 8617    x. cmul 8619    < clt 8744    <_ cle 8745    - cmin 8909    / cdiv 9279   NNcn 9594   9c9 9650   10c10 9651   NN0cn0 9811   ^cexp 10947   abscabs 11560   sum_csu 11997
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4025  ax-sep 4035  ax-nul 4043  ax-pow 4079  ax-pr 4105  ax-un 4400  ax-inf2 7223  ax-cnex 8670  ax-resscn 8671  ax-1cn 8672  ax-icn 8673  ax-addcl 8674  ax-addrcl 8675  ax-mulcl 8676  ax-mulrcl 8677  ax-mulcom 8678  ax-addass 8679  ax-mulass 8680  ax-distr 8681  ax-i2m1 8682  ax-1ne0 8683  ax-1rid 8684  ax-rnegex 8685  ax-rrecex 8686  ax-cnre 8687  ax-pre-lttri 8688  ax-pre-lttrn 8689  ax-pre-ltadd 8690  ax-pre-mulgt0 8691  ax-pre-sup 8692
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2511  df-rex 2512  df-reu 2513  df-rab 2514  df-v 2727  df-sbc 2920  df-csb 3007  df-dif 3078  df-un 3080  df-in 3082  df-ss 3086  df-pss 3088  df-nul 3360  df-if 3468  df-pw 3529  df-sn 3547  df-pr 3548  df-tp 3549  df-op 3550  df-uni 3725  df-int 3758  df-iun 3802  df-br 3918  df-opab 3972  df-mpt 3973  df-tr 4008  df-eprel 4195  df-id 4199  df-po 4204  df-so 4205  df-fr 4242  df-se 4243  df-we 4244  df-ord 4285  df-on 4286  df-lim 4287  df-suc 4288  df-om 4545  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-fun 4599  df-fn 4600  df-f 4601  df-f1 4602  df-fo 4603  df-f1o 4604  df-fv 4605  df-isom 4606  df-ov 5710  df-oprab 5711  df-mpt2 5712  df-1st 5971  df-2nd 5972  df-iota 6140  df-riota 6187  df-recs 6271  df-rdg 6306  df-1o 6362  df-oadd 6366  df-er 6543  df-pm 6658  df-en 6747  df-dom 6748  df-sdom 6749  df-fin 6750  df-sup 7075  df-oi 7106  df-card 7453  df-pnf 8746  df-mnf 8747  df-xr 8748  df-ltxr 8749  df-le 8750  df-sub 8911  df-neg 8912  df-div 9280  df-n 9595  df-2 9652  df-3 9653  df-4 9654  df-5 9655  df-6 9656  df-7 9657  df-8 9658  df-9 9659  df-10 9660  df-n0 9812  df-z 9871  df-uz 10077  df-rp 10201  df-fz 10626  df-fzo 10714  df-fl 10768  df-seq 10890  df-exp 10948  df-hash 11180  df-cj 11425  df-re 11426  df-im 11427  df-sqr 11561  df-abs 11562  df-clim 11801  df-rlim 11802  df-sum 11998
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