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Theorem 0.999... 10911
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 8929 . . . . . . 7  |-  10  e.  RR
21recni 8228 . . . . . 6  |-  10  e.  CC
3 nnnn0 9062 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  NN0 )
4 expcl 10092 . . . . . 6  |-  ( ( 10  e.  CC  /\  k  e.  NN0 )  ->  ( 10 ^ k )  e.  CC )
52, 3, 4sylancr 640 . . . . 5  |-  ( k  e.  NN  ->  ( 10 ^ k )  e.  CC )
62a1i 10 . . . . . 6  |-  ( k  e.  NN  ->  10  e.  CC )
7 10pos 8940 . . . . . . . 8  |-  0  <  10
81, 7gt0ne0ii 8533 . . . . . . 7  |-  10  =/=  0
98a1i 10 . . . . . 6  |-  ( k  e.  NN  ->  10  =/=  0 )
10 nnz 9128 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  ZZ )
11 expne0i 10105 . . . . . 6  |-  ( ( 10  e.  CC  /\  10  =/=  0  /\  k  e.  ZZ )  ->  ( 10 ^ k )  =/=  0 )
126, 9, 10, 11syl3anc 1149 . . . . 5  |-  ( k  e.  NN  ->  ( 10 ^ k )  =/=  0
)
13 9re 8928 . . . . . . 7  |-  9  e.  RR
1413recni 8228 . . . . . 6  |-  9  e.  CC
15 divrec 8652 . . . . . 6  |-  ( (
9  e.  CC  /\  ( 10 ^ k )  e.  CC  /\  ( 10 ^ k )  =/=  0 )  ->  (
9  /  ( 10
^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
1614, 15mp3an1 1231 . . . . 5  |-  ( (
( 10 ^ k
)  e.  CC  /\  ( 10 ^ k )  =/=  0 )  -> 
( 9  /  ( 10 ^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
175, 12, 16syl2anc 638 . . . 4  |-  ( k  e.  NN  ->  ( 9  /  ( 10 ^
k ) )  =  ( 9  x.  (
1  /  ( 10
^ k ) ) ) )
18 exprec 10114 . . . . . 6  |-  ( ( 10  e.  CC  /\  10  =/=  0  /\  k  e.  ZZ )  ->  (
( 1  /  10 ) ^ k )  =  ( 1  /  ( 10 ^ k ) ) )
196, 9, 10, 18syl3anc 1149 . . . . 5  |-  ( k  e.  NN  ->  ( (
1  /  10 ) ^ k )  =  ( 1  /  ( 10 ^ k ) ) )
2019oveq2d 5369 . . . 4  |-  ( k  e.  NN  ->  ( 9  x.  ( ( 1  /  10 ) ^
k ) )  =  ( 9  x.  (
1  /  ( 10
^ k ) ) ) )
2117, 20eqtr4d 2117 . . 3  |-  ( k  e.  NN  ->  ( 9  /  ( 10 ^
k ) )  =  ( 9  x.  (
( 1  /  10 ) ^ k ) ) )
2221sumeq2i 10762 . 2  |-  sum_ k  e.  NN  ( 9  / 
( 10 ^ k
) )  =  sum_ k  e.  NN  (
9  x.  ( ( 1  /  10 ) ^ k ) )
231, 8rereccli 8713 . . . 4  |-  ( 1  /  10 )  e.  RR
2423recni 8228 . . 3  |-  ( 1  /  10 )  e.  CC
25 0re 8249 . . . . . 6  |-  0  e.  RR
261, 7recgt0ii 8728 . . . . . 6  |-  0  <  ( 1  /  10 )
2725, 23, 26ltleii 8319 . . . . 5  |-  0  <_  ( 1  /  10 )
2823absidi 10516 . . . . 5  |-  ( 0  <_  ( 1  /  10 )  ->  ( abs `  ( 1  /  10 ) )  =  ( 1  /  10 ) )
2927, 28ax-mp 8 . . . 4  |-  ( abs `  ( 1  /  10 ) )  =  ( 1  /  10 )
30 1lt10 9029 . . . . 5  |-  1  <  10
31 recgt1 8806 . . . . . 6  |-  ( ( 10  e.  RR  /\  0  <  10 )  ->  (
1  <  10  <->  ( 1  /  10 )  <  1 ) )
321, 7, 31mp2an 649 . . . . 5  |-  ( 1  <  10  <->  ( 1  /  10 )  <  1 )
3330, 32mpbi 197 . . . 4  |-  ( 1  /  10 )  <  1
3429, 33eqbrtri 3619 . . 3  |-  ( abs `  ( 1  /  10 ) )  <  1
35 geoisum1c 10910 . . 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 1244 . 2  |-  sum_ k  e.  NN  ( 9  x.  ( ( 1  /  10 ) ^ k ) )  =  ( ( 9  x.  ( 1  /  10 ) )  /  ( 1  -  ( 1  /  10 ) ) )
3714, 2, 8divreci 8650 . . . 4  |-  ( 9  /  10 )  =  ( 9  x.  (
1  /  10 ) )
3814, 2, 8divcan2i 8635 . . . . . 6  |-  ( 10  x.  ( 9  /  10 ) )  =  9
39 ax-1cn 8194 . . . . . . . 8  |-  1  e.  CC
402, 39, 24subdii 8460 . . . . . . 7  |-  ( 10  x.  ( 1  -  (
1  /  10 ) ) )  =  ( ( 10  x.  1 )  -  ( 10  x.  ( 1  /  10 ) ) )
412mulid1i 8242 . . . . . . . 8  |-  ( 10  x.  1 )  =  10
422, 8recidi 8646 . . . . . . . 8  |-  ( 10  x.  ( 1  /  10 ) )  =  1
4341, 42oveq12i 5366 . . . . . . 7  |-  ( ( 10  x.  1 )  -  ( 10  x.  (
1  /  10 ) ) )  =  ( 10  -  1 )
4439, 14addcomi 8370 . . . . . . . . 9  |-  ( 1  +  9 )  =  ( 9  +  1 )
45 df-10 8918 . . . . . . . . 9  |-  10  =  ( 9  +  1 )
4644, 45eqtr4i 2105 . . . . . . . 8  |-  ( 1  +  9 )  =  10
472, 39, 14, 46subaddrii 8411 . . . . . . 7  |-  ( 10  -  1 )  =  9
4840, 43, 473eqtrri 2107 . . . . . 6  |-  9  =  ( 10  x.  (
1  -  ( 1  /  10 ) ) )
4938, 48eqtri 2102 . . . . 5  |-  ( 10  x.  ( 9  /  10 ) )  =  ( 10  x.  ( 1  -  ( 1  /  10 ) ) )
5013, 1, 8redivcli 8710 . . . . . . 7  |-  ( 9  /  10 )  e.  RR
5150recni 8228 . . . . . 6  |-  ( 9  /  10 )  e.  CC
5239, 24subcli 8403 . . . . . 6  |-  ( 1  -  ( 1  /  10 ) )  e.  CC
5351, 52, 2, 8mulcani 8608 . . . . 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 5366 . . 3  |-  ( (
9  /  10 )  /  ( 9  /  10 ) )  =  ( ( 9  x.  (
1  /  10 ) )  /  ( 1  -  ( 1  /  10 ) ) )
56 9pos 8939 . . . . . 6  |-  0  <  9
5713, 1, 56, 7divgt0ii 8771 . . . . 5  |-  0  <  ( 9  /  10 )
5850, 57gt0ne0ii 8533 . . . 4  |-  ( 9  /  10 )  =/=  0
5951, 58dividi 8676 . . 3  |-  ( (
9  /  10 )  /  ( 9  /  10 ) )  =  1
6055, 59eqtr3i 2104 . 2  |-  ( (
9  x.  ( 1  /  10 ) )  /  ( 1  -  ( 1  /  10 ) ) )  =  1
6122, 36, 603eqtri 2106 1  |-  sum_ k  e.  NN  ( 9  / 
( 10 ^ k
) )  =  1
Colors of variables: wff set class
Syntax hints:    <-> wb 174    = wceq 1536    e. wcel 1538    =/= wne 2199   class class class wbr 3600   ` cfv 4287  (class class class)co 5354   CCcc 8135   RRcr 8136   0cc0 8137   1c1 8138    + caddc 8140    x. cmul 8142    <_ cle 8250    < clt 8254    - cmin 8373    / cdiv 8375   NNcn 8376   NN0cn0 8377   ZZcz 8378   9c9 8908   10c10 8909   ^cexp 10075   abscabs 10432   sum_csu 10748
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1451  ax-6 1452  ax-7 1453  ax-gen 1454  ax-8 1540  ax-11 1541  ax-13 1542  ax-14 1543  ax-17 1545  ax-12o 1578  ax-10 1592  ax-9 1598  ax-4 1606  ax-16 1793  ax-ext 2064  ax-rep 3705  ax-sep 3715  ax-nul 3723  ax-pow 3759  ax-pr 3783  ax-un 4075  ax-inf2 6805  ax-cnex 8192  ax-resscn 8193  ax-1cn 8194  ax-icn 8195  ax-addcl 8196  ax-addrcl 8197  ax-mulcl 8198  ax-mulrcl 8199  ax-mulcom 8200  ax-addass 8201  ax-mulass 8202  ax-distr 8203  ax-i2m1 8204  ax-1ne0 8205  ax-1rid 8206  ax-rnegex 8207  ax-rrecex 8208  ax-cnre 8209  ax-pre-lttri 8210  ax-pre-lttrn 8211  ax-pre-ltadd 8212  ax-pre-mulgt0 8213  ax-pre-sup 8214
This theorem depends on definitions:  df-bi 175  df-or 358  df-an 359  df-3or 904  df-3an 905  df-tru 1268  df-ex 1456  df-sb 1754  df-eu 1976  df-mo 1977  df-clab 2070  df-cleq 2075  df-clel 2078  df-ne 2201  df-nel 2202  df-ral 2295  df-rex 2296  df-reu 2297  df-rab 2298  df-v 2494  df-sbc 2668  df-csb 2750  df-dif 2813  df-un 2815  df-in 2817  df-ss 2821  df-pss 2823  df-nul 3089  df-if 3199  df-pw 3260  df-sn 3278  df-pr 3279  df-tp 3280  df-op 3281  df-uni 3439  df-int 3473  df-iun 3516  df-br 3601  df-opab 3655  df-mpt 3656  df-tr 3688  df-eprel 3870  df-id 3874  df-po 3879  df-so 3880  df-fr 3917  df-se 3918  df-we 3919  df-ord 3960  df-on 3961  df-lim 3962  df-suc 3963  df-om 4243  df-xp 4289  df-rel 4290  df-cnv 4291  df-co 4292  df-dm 4293  df-rn 4294  df-res 4295  df-ima 4296  df-fun 4297  df-fn 4298  df-f 4299  df-f1 4300  df-fo 4301  df-f1o 4302  df-fv 4303  df-iso 4304  df-ov 5357  df-oprab 5358  df-mpt2 5359  df-1st 5608  df-2nd 5609  df-iota 5762  df-recs 5835  df-rdg 5870  df-1o 5926  df-oadd 5930  df-er 6102  df-pm 6208  df-en 6289  df-dom 6290  df-sdom 6291  df-fin 6292  df-riota 6455  df-sup 6656  df-oi 6688  df-card 7034  df-pnf 8255  df-mnf 8256  df-xr 8257  df-ltxr 8258  df-le 8259  df-sub 8392  df-neg 8393  df-div 8621  df-n 8863  df-2 8910  df-3 8911  df-4 8912  df-5 8913  df-6 8914  df-7 8915  df-8 8916  df-9 8917  df-10 8918  df-n0 9056  df-z 9108  df-uz 9305  df-rp 9427  df-fz 9780  df-fl 9917  df-seq 10021  df-exp 10076  df-hash 10261  df-cj 10335  df-re 10336  df-im 10337  df-sqr 10433  df-abs 10434  df-clim 10613  df-rlim 10614  df-sum 10749
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