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Theorem 0.999... 10998
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 8949 . . . . . . 7  |-  10  e.  RR
21recni 8244 . . . . . 6  |-  10  e.  CC
3 nnnn0 9082 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  NN0 )
4 expcl 10128 . . . . . 6  |-  ( ( 10  e.  CC  /\  k  e.  NN0 )  -> 
( 10 ^ k
)  e.  CC )
52, 3, 4sylancr 636 . . . . 5  |-  ( k  e.  NN  ->  ( 10 ^ k )  e.  CC )
62a1i 10 . . . . . 6  |-  ( k  e.  NN  ->  10  e.  CC )
7 10pos 8960 . . . . . . . 8  |-  0  <  10
81, 7gt0ne0ii 8549 . . . . . . 7  |-  10  =/=  0
98a1i 10 . . . . . 6  |-  ( k  e.  NN  ->  10  =/=  0 )
10 nnz 9148 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  ZZ )
11 expne0i 10141 . . . . . 6  |-  ( ( 10  e.  CC  /\  10  =/=  0  /\  k  e.  ZZ )  ->  ( 10 ^ k )  =/=  0 )
126, 9, 10, 11syl3anc 1138 . . . . 5  |-  ( k  e.  NN  ->  ( 10 ^ k )  =/=  0 )
13 9re 8948 . . . . . . 7  |-  9  e.  RR
1413recni 8244 . . . . . 6  |-  9  e.  CC
15 divrec 8671 . . . . . 6  |-  ( ( 9  e.  CC  /\  ( 10 ^ k )  e.  CC  /\  ( 10 ^ k )  =/=  0 )  ->  (
9  /  ( 10
^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
1614, 15mp3an1 1220 . . . . 5  |-  ( ( ( 10 ^ k
)  e.  CC  /\  ( 10 ^ k )  =/=  0 )  -> 
( 9  /  ( 10 ^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
175, 12, 16syl2anc 634 . . . 4  |-  ( k  e.  NN  ->  (
9  /  ( 10
^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
18 exprec 10150 . . . . . 6  |-  ( ( 10  e.  CC  /\  10  =/=  0  /\  k  e.  ZZ )  ->  (
( 1  /  10 ) ^ k )  =  ( 1  /  ( 10 ^ k ) ) )
196, 9, 10, 18syl3anc 1138 . . . . 5  |-  ( k  e.  NN  ->  (
( 1  /  10 ) ^ k )  =  ( 1  /  ( 10 ^ k ) ) )
2019oveq2d 5372 . . . 4  |-  ( k  e.  NN  ->  (
9  x.  ( ( 1  /  10 ) ^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
2117, 20eqtr4d 2097 . . 3  |-  ( k  e.  NN  ->  (
9  /  ( 10
^ k ) )  =  ( 9  x.  ( ( 1  /  10 ) ^ k ) ) )
2221sumeq2i 10843 . 2  |-  sum_ k  e.  NN  ( 9  / 
( 10 ^ k
) )  =  sum_ k  e.  NN  (
9  x.  ( ( 1  /  10 ) ^ k ) )
231, 8rereccli 8733 . . . 4  |-  ( 1  /  10 )  e.  RR
2423recni 8244 . . 3  |-  ( 1  /  10 )  e.  CC
25 0re 8265 . . . . . 6  |-  0  e.  RR
261, 7recgt0ii 8748 . . . . . 6  |-  0  <  ( 1  /  10 )
2725, 23, 26ltleii 8335 . . . . 5  |-  0  <_  ( 1  /  10 )
2823absidi 10570 . . . . 5  |-  ( 0  <_  ( 1  /  10 )  ->  ( abs `  ( 1  /  10 ) )  =  ( 1  /  10 ) )
2927, 28ax-mp 8 . . . 4  |-  ( abs `  ( 1  /  10 ) )  =  ( 1  /  10 )
30 1lt10 9049 . . . . 5  |-  1  <  10
31 recgt1 8826 . . . . . 6  |-  ( ( 10  e.  RR  /\  0  <  10 )  -> 
( 1  <  10  <->  ( 1  /  10 )  <  1 ) )
321, 7, 31mp2an 645 . . . . 5  |-  ( 1  <  10  <->  ( 1  /  10 )  <  1 )
3330, 32mpbi 197 . . . 4  |-  ( 1  /  10 )  <  1
3429, 33eqbrtri 3603 . . 3  |-  ( abs `  ( 1  /  10 ) )  <  1
35 geoisum1c 10997 . . 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 1233 . 2  |-  sum_ k  e.  NN  ( 9  x.  ( ( 1  /  10 ) ^ k ) )  =  ( ( 9  x.  ( 1  /  10 ) )  /  ( 1  -  ( 1  /  10 ) ) )
3714, 2, 8divreci 8669 . . . 4  |-  ( 9  /  10 )  =  ( 9  x.  (
1  /  10 ) )
3814, 2, 8divcan2i 8654 . . . . . 6  |-  ( 10  x.  ( 9  /  10 ) )  =  9
39 ax-1cn 8210 . . . . . . . 8  |-  1  e.  CC
402, 39, 24subdii 8476 . . . . . . 7  |-  ( 10  x.  ( 1  -  ( 1  /  10 ) ) )  =  ( ( 10  x.  1 )  -  ( 10  x.  ( 1  /  10 ) ) )
412mulid1i 8258 . . . . . . . 8  |-  ( 10  x.  1 )  =  10
422, 8recidi 8665 . . . . . . . 8  |-  ( 10  x.  ( 1  /  10 ) )  =  1
4341, 42oveq12i 5368 . . . . . . 7  |-  ( ( 10  x.  1 )  -  ( 10  x.  ( 1  /  10 ) ) )  =  ( 10  -  1 )
4439, 14addcomi 8386 . . . . . . . . 9  |-  ( 1  +  9 )  =  ( 9  +  1 )
45 df-10 8938 . . . . . . . . 9  |-  10  =  ( 9  +  1 )
4644, 45eqtr4i 2085 . . . . . . . 8  |-  ( 1  +  9 )  =  10
472, 39, 14, 46subaddrii 8427 . . . . . . 7  |-  ( 10 
-  1 )  =  9
4840, 43, 473eqtrri 2087 . . . . . 6  |-  9  =  ( 10  x.  ( 1  -  (
1  /  10 ) ) )
4938, 48eqtri 2082 . . . . 5  |-  ( 10  x.  ( 9  /  10 ) )  =  ( 10  x.  ( 1  -  ( 1  /  10 ) ) )
5013, 1, 8redivcli 8730 . . . . . . 7  |-  ( 9  /  10 )  e.  RR
5150recni 8244 . . . . . 6  |-  ( 9  /  10 )  e.  CC
5239, 24subcli 8419 . . . . . 6  |-  ( 1  -  ( 1  /  10 ) )  e.  CC
5351, 52, 2, 8mulcani 8627 . . . . 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 5368 . . 3  |-  ( ( 9  /  10 )  /  ( 9  /  10 ) )  =  ( ( 9  x.  (
1  /  10 ) )  /  ( 1  -  ( 1  /  10 ) ) )
56 9pos 8959 . . . . . 6  |-  0  <  9
5713, 1, 56, 7divgt0ii 8791 . . . . 5  |-  0  <  ( 9  /  10 )
5850, 57gt0ne0ii 8549 . . . 4  |-  ( 9  /  10 )  =/=  0
5951, 58dividi 8695 . . 3  |-  ( ( 9  /  10 )  /  ( 9  /  10 ) )  =  1
6055, 59eqtr3i 2084 . 2  |-  ( ( 9  x.  ( 1  /  10 ) )  /  ( 1  -  ( 1  /  10 ) ) )  =  1
6122, 36, 603eqtri 2086 1  |-  sum_ k  e.  NN  ( 9  / 
( 10 ^ k
) )  =  1
Colors of variables: wff set class
Syntax hints:    <-> wb 174    = wceq 1518    e. wcel 1520    =/= wne 2180   class class class wbr 3584   ` cfv 4266  (class class class)co 5356   CCcc 8151   RRcr 8152   0cc0 8153   1c1 8154    + caddc 8156    x. cmul 8158    <_ cle 8266    < clt 8270    - cmin 8389    / cdiv 8391   NNcn 8392   NN0cn0 8393   ZZcz 8394   9c9 8928   10c10 8929   ^cexp 10111   abscabs 10485   sum_csu 10829
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1440  ax-6 1441  ax-7 1442  ax-gen 1443  ax-8 1522  ax-11 1523  ax-13 1524  ax-14 1525  ax-17 1527  ax-12o 1560  ax-10 1574  ax-9 1580  ax-4 1587  ax-16 1773  ax-ext 2044  ax-rep 3689  ax-sep 3699  ax-nul 3707  ax-pow 3743  ax-pr 3767  ax-un 4059  ax-inf2 6821  ax-cnex 8208  ax-resscn 8209  ax-1cn 8210  ax-icn 8211  ax-addcl 8212  ax-addrcl 8213  ax-mulcl 8214  ax-mulrcl 8215  ax-mulcom 8216  ax-addass 8217  ax-mulass 8218  ax-distr 8219  ax-i2m1 8220  ax-1ne0 8221  ax-1rid 8222  ax-rnegex 8223  ax-rrecex 8224  ax-cnre 8225  ax-pre-lttri 8226  ax-pre-lttrn 8227  ax-pre-ltadd 8228  ax-pre-mulgt0 8229  ax-pre-sup 8230
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3or 895  df-3an 896  df-tru 1257  df-ex 1445  df-sb 1734  df-eu 1956  df-mo 1957  df-clab 2050  df-cleq 2055  df-clel 2058  df-ne 2182  df-nel 2183  df-ral 2276  df-rex 2277  df-reu 2278  df-rab 2279  df-v 2475  df-sbc 2649  df-csb 2731  df-dif 2794  df-un 2796  df-in 2798  df-ss 2802  df-pss 2804  df-nul 3071  df-if 3180  df-pw 3241  df-sn 3259  df-pr 3260  df-tp 3261  df-op 3262  df-uni 3423  df-int 3457  df-iun 3500  df-br 3585  df-opab 3639  df-mpt 3640  df-tr 3672  df-eprel 3854  df-id 3858  df-po 3863  df-so 3864  df-fr 3901  df-se 3902  df-we 3903  df-ord 3944  df-on 3945  df-lim 3946  df-suc 3947  df-om 4222  df-xp 4268  df-rel 4269  df-cnv 4270  df-co 4271  df-dm 4272  df-rn 4273  df-res 4274  df-ima 4275  df-fun 4276  df-fn 4277  df-f 4278  df-f1 4279  df-fo 4280  df-f1o 4281  df-fv 4282  df-iso 4283  df-ov 5359  df-oprab 5360  df-mpt2 5361  df-1st 5610  df-2nd 5611  df-iota 5766  df-recs 5839  df-rdg 5874  df-1o 5930  df-oadd 5934  df-er 6111  df-pm 6216  df-en 6298  df-dom 6299  df-sdom 6300  df-fin 6301  df-riota 6464  df-sup 6672  df-oi 6704  df-card 7050  df-pnf 8271  df-mnf 8272  df-xr 8273  df-ltxr 8274  df-le 8275  df-sub 8408  df-neg 8409  df-div 8640  df-n 8883  df-2 8930  df-3 8931  df-4 8932  df-5 8933  df-6 8934  df-7 8935  df-8 8936  df-9 8937  df-10 8938  df-n0 9076  df-z 9128  df-uz 9326  df-rp 9448  df-fz 9804  df-fzo 9892  df-fl 9943  df-seq 10054  df-exp 10112  df-hash 10301  df-cj 10388  df-re 10389  df-im 10390  df-sqr 10486  df-abs 10487  df-clim 10671  df-rlim 10672  df-sum 10830
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