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Theorem 0.999... 10958
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 8945 . . . . . . 7  |-  10  e.  RR
21recni 8242 . . . . . 6  |-  10  e.  CC
3 nnnn0 9078 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  NN0 )
4 expcl 10116 . . . . . 6  |-  ( ( 10  e.  CC  /\  k  e.  NN0 )  -> 
( 10 ^ k
)  e.  CC )
52, 3, 4sylancr 635 . . . . 5  |-  ( k  e.  NN  ->  ( 10 ^ k )  e.  CC )
62a1i 10 . . . . . 6  |-  ( k  e.  NN  ->  10  e.  CC )
7 10pos 8956 . . . . . . . 8  |-  0  <  10
81, 7gt0ne0ii 8547 . . . . . . 7  |-  10  =/=  0
98a1i 10 . . . . . 6  |-  ( k  e.  NN  ->  10  =/=  0 )
10 nnz 9144 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  ZZ )
11 expne0i 10129 . . . . . 6  |-  ( ( 10  e.  CC  /\  10  =/=  0  /\  k  e.  ZZ )  ->  ( 10 ^ k )  =/=  0 )
126, 9, 10, 11syl3anc 1137 . . . . 5  |-  ( k  e.  NN  ->  ( 10 ^ k )  =/=  0 )
13 9re 8944 . . . . . . 7  |-  9  e.  RR
1413recni 8242 . . . . . 6  |-  9  e.  CC
15 divrec 8668 . . . . . 6  |-  ( ( 9  e.  CC  /\  ( 10 ^ k )  e.  CC  /\  ( 10 ^ k )  =/=  0 )  ->  (
9  /  ( 10
^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
1614, 15mp3an1 1219 . . . . 5  |-  ( ( ( 10 ^ k
)  e.  CC  /\  ( 10 ^ k )  =/=  0 )  -> 
( 9  /  ( 10 ^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
175, 12, 16syl2anc 633 . . . 4  |-  ( k  e.  NN  ->  (
9  /  ( 10
^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
18 exprec 10138 . . . . . 6  |-  ( ( 10  e.  CC  /\  10  =/=  0  /\  k  e.  ZZ )  ->  (
( 1  /  10 ) ^ k )  =  ( 1  /  ( 10 ^ k ) ) )
196, 9, 10, 18syl3anc 1137 . . . . 5  |-  ( k  e.  NN  ->  (
( 1  /  10 ) ^ k )  =  ( 1  /  ( 10 ^ k ) ) )
2019oveq2d 5371 . . . 4  |-  ( k  e.  NN  ->  (
9  x.  ( ( 1  /  10 ) ^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
2117, 20eqtr4d 2096 . . 3  |-  ( k  e.  NN  ->  (
9  /  ( 10
^ k ) )  =  ( 9  x.  ( ( 1  /  10 ) ^ k ) ) )
2221sumeq2i 10807 . 2  |-  sum_ k  e.  NN  ( 9  / 
( 10 ^ k
) )  =  sum_ k  e.  NN  (
9  x.  ( ( 1  /  10 ) ^ k ) )
231, 8rereccli 8729 . . . 4  |-  ( 1  /  10 )  e.  RR
2423recni 8242 . . 3  |-  ( 1  /  10 )  e.  CC
25 0re 8263 . . . . . 6  |-  0  e.  RR
261, 7recgt0ii 8744 . . . . . 6  |-  0  <  ( 1  /  10 )
2725, 23, 26ltleii 8333 . . . . 5  |-  0  <_  ( 1  /  10 )
2823absidi 10556 . . . . 5  |-  ( 0  <_  ( 1  /  10 )  ->  ( abs `  ( 1  /  10 ) )  =  ( 1  /  10 ) )
2927, 28ax-mp 8 . . . 4  |-  ( abs `  ( 1  /  10 ) )  =  ( 1  /  10 )
30 1lt10 9045 . . . . 5  |-  1  <  10
31 recgt1 8822 . . . . . 6  |-  ( ( 10  e.  RR  /\  0  <  10 )  -> 
( 1  <  10  <->  ( 1  /  10 )  <  1 ) )
321, 7, 31mp2an 644 . . . . 5  |-  ( 1  <  10  <->  ( 1  /  10 )  <  1 )
3330, 32mpbi 197 . . . 4  |-  ( 1  /  10 )  <  1
3429, 33eqbrtri 3602 . . 3  |-  ( abs `  ( 1  /  10 ) )  <  1
35 geoisum1c 10957 . . 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 ) ) ) )
3614, 24, 34, 35mp3an 1232 . 2  |-  sum_ k  e.  NN  ( 9  x.  ( ( 1  /  10 ) ^ k ) )  =  ( ( 9  x.  ( 1  /  10 ) )  /  ( 1  -  ( 1  /  10 ) ) )
3714, 2, 8divreci 8666 . . . 4  |-  ( 9  /  10 )  =  ( 9  x.  (
1  /  10 ) )
3814, 2, 8divcan2i 8651 . . . . . 6  |-  ( 10  x.  ( 9  /  10 ) )  =  9
39 ax-1cn 8208 . . . . . . . 8  |-  1  e.  CC
402, 39, 24subdii 8474 . . . . . . 7  |-  ( 10  x.  ( 1  -  ( 1  /  10 ) ) )  =  ( ( 10  x.  1 )  -  ( 10  x.  ( 1  /  10 ) ) )
412mulid1i 8256 . . . . . . . 8  |-  ( 10  x.  1 )  =  10
422, 8recidi 8662 . . . . . . . 8  |-  ( 10  x.  ( 1  /  10 ) )  =  1
4341, 42oveq12i 5367 . . . . . . 7  |-  ( ( 10  x.  1 )  -  ( 10  x.  ( 1  /  10 ) ) )  =  ( 10  -  1 )
4439, 14addcomi 8384 . . . . . . . . 9  |-  ( 1  +  9 )  =  ( 9  +  1 )
45 df-10 8934 . . . . . . . . 9  |-  10  =  ( 9  +  1 )
4644, 45eqtr4i 2084 . . . . . . . 8  |-  ( 1  +  9 )  =  10
472, 39, 14, 46subaddrii 8425 . . . . . . 7  |-  ( 10 
-  1 )  =  9
4840, 43, 473eqtrri 2086 . . . . . 6  |-  9  =  ( 10  x.  ( 1  -  (
1  /  10 ) ) )
4938, 48eqtri 2081 . . . . 5  |-  ( 10  x.  ( 9  /  10 ) )  =  ( 10  x.  ( 1  -  ( 1  /  10 ) ) )
5013, 1, 8redivcli 8726 . . . . . . 7  |-  ( 9  /  10 )  e.  RR
5150recni 8242 . . . . . 6  |-  ( 9  /  10 )  e.  CC
5239, 24subcli 8417 . . . . . 6  |-  ( 1  -  ( 1  /  10 ) )  e.  CC
5351, 52, 2, 8mulcani 8624 . . . . 5  |-  ( ( 10  x.  ( 9  /  10 ) )  =  ( 10  x.  ( 1  -  (
1  /  10 ) ) )  <->  ( 9  /  10 )  =  ( 1  -  (
1  /  10 ) ) )
5449, 53mpbi 197 . . . 4  |-  ( 9  /  10 )  =  ( 1  -  (
1  /  10 ) )
5537, 54oveq12i 5367 . . 3  |-  ( ( 9  /  10 )  /  ( 9  /  10 ) )  =  ( ( 9  x.  (
1  /  10 ) )  /  ( 1  -  ( 1  /  10 ) ) )
56 9pos 8955 . . . . . 6  |-  0  <  9
5713, 1, 56, 7divgt0ii 8787 . . . . 5  |-  0  <  ( 9  /  10 )
5850, 57gt0ne0ii 8547 . . . 4  |-  ( 9  /  10 )  =/=  0
5951, 58dividi 8692 . . 3  |-  ( ( 9  /  10 )  /  ( 9  /  10 ) )  =  1
6055, 59eqtr3i 2083 . 2  |-  ( ( 9  x.  ( 1  /  10 ) )  /  ( 1  -  ( 1  /  10 ) ) )  =  1
6122, 36, 603eqtri 2085 1  |-  sum_ k  e.  NN  ( 9  / 
( 10 ^ k
) )  =  1
Colors of variables: wff set class
Syntax hints:    <-> wb 174    = wceq 1517    e. wcel 1519    =/= wne 2179   class class class wbr 3583   ` cfv 4265  (class class class)co 5355   CCcc 8149   RRcr 8150   0cc0 8151   1c1 8152    + caddc 8154    x. cmul 8156    <_ cle 8264    < clt 8268    - cmin 8387    / cdiv 8389   NNcn 8390   NN0cn0 8391   ZZcz 8392   9c9 8924   10c10 8925   ^cexp 10099   abscabs 10472   sum_csu 10793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1439  ax-6 1440  ax-7 1441  ax-gen 1442  ax-8 1521  ax-11 1522  ax-13 1523  ax-14 1524  ax-17 1526  ax-12o 1559  ax-10 1573  ax-9 1579  ax-4 1586  ax-16 1772  ax-ext 2043  ax-rep 3688  ax-sep 3698  ax-nul 3706  ax-pow 3742  ax-pr 3766  ax-un 4058  ax-inf2 6819  ax-cnex 8206  ax-resscn 8207  ax-1cn 8208  ax-icn 8209  ax-addcl 8210  ax-addrcl 8211  ax-mulcl 8212  ax-mulrcl 8213  ax-mulcom 8214  ax-addass 8215  ax-mulass 8216  ax-distr 8217  ax-i2m1 8218  ax-1ne0 8219  ax-1rid 8220  ax-rnegex 8221  ax-rrecex 8222  ax-cnre 8223  ax-pre-lttri 8224  ax-pre-lttrn 8225  ax-pre-ltadd 8226  ax-pre-mulgt0 8227  ax-pre-sup 8228
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3or 894  df-3an 895  df-tru 1256  df-ex 1444  df-sb 1733  df-eu 1955  df-mo 1956  df-clab 2049  df-cleq 2054  df-clel 2057  df-ne 2181  df-nel 2182  df-ral 2275  df-rex 2276  df-reu 2277  df-rab 2278  df-v 2474  df-sbc 2648  df-csb 2730  df-dif 2793  df-un 2795  df-in 2797  df-ss 2801  df-pss 2803  df-nul 3070  df-if 3179  df-pw 3240  df-sn 3258  df-pr 3259  df-tp 3260  df-op 3261  df-uni 3422  df-int 3456  df-iun 3499  df-br 3584  df-opab 3638  df-mpt 3639  df-tr 3671  df-eprel 3853  df-id 3857  df-po 3862  df-so 3863  df-fr 3900  df-se 3901  df-we 3902  df-ord 3943  df-on 3944  df-lim 3945  df-suc 3946  df-om 4221  df-xp 4267  df-rel 4268  df-cnv 4269  df-co 4270  df-dm 4271  df-rn 4272  df-res 4273  df-ima 4274  df-fun 4275  df-fn 4276  df-f 4277  df-f1 4278  df-fo 4279  df-f1o 4280  df-fv 4281  df-iso 4282  df-ov 5358  df-oprab 5359  df-mpt2 5360  df-1st 5609  df-2nd 5610  df-iota 5764  df-recs 5837  df-rdg 5872  df-1o 5928  df-oadd 5932  df-er 6109  df-pm 6214  df-en 6296  df-dom 6297  df-sdom 6298  df-fin 6299  df-riota 6462  df-sup 6670  df-oi 6702  df-card 7048  df-pnf 8269  df-mnf 8270  df-xr 8271  df-ltxr 8272  df-le 8273  df-sub 8406  df-neg 8407  df-div 8637  df-n 8879  df-2 8926  df-3 8927  df-4 8928  df-5 8929  df-6 8930  df-7 8931  df-8 8932  df-9 8933  df-10 8934  df-n0 9072  df-z 9124  df-uz 9321  df-rp 9443  df-fz 9799  df-fzo 9887  df-fl 9938  df-seq 10042  df-exp 10100  df-hash 10288  df-cj 10375  df-re 10376  df-im 10377  df-sqr 10473  df-abs 10474  df-clim 10654  df-rlim 10655  df-sum 10794
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