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Theorem 0.999... 12663
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 10085 . . . . . . 7  |-  10  e.  RR
21recni 9107 . . . . . 6  |-  10  e.  CC
3 nnnn0 10233 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  NN0 )
4 expcl 11404 . . . . . 6  |-  ( ( 10  e.  CC  /\  k  e.  NN0 )  -> 
( 10 ^ k
)  e.  CC )
52, 3, 4sylancr 646 . . . . 5  |-  ( k  e.  NN  ->  ( 10 ^ k )  e.  CC )
62a1i 11 . . . . . 6  |-  ( k  e.  NN  ->  10  e.  CC )
7 10pos 10097 . . . . . . . 8  |-  0  <  10
81, 7gt0ne0ii 9568 . . . . . . 7  |-  10  =/=  0
98a1i 11 . . . . . 6  |-  ( k  e.  NN  ->  10  =/=  0 )
10 nnz 10308 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  ZZ )
116, 9, 10expne0d 11534 . . . . 5  |-  ( k  e.  NN  ->  ( 10 ^ k )  =/=  0 )
12 9re 10084 . . . . . . 7  |-  9  e.  RR
1312recni 9107 . . . . . 6  |-  9  e.  CC
14 divrec 9699 . . . . . 6  |-  ( ( 9  e.  CC  /\  ( 10 ^ k )  e.  CC  /\  ( 10 ^ k )  =/=  0 )  ->  (
9  /  ( 10
^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
1513, 14mp3an1 1267 . . . . 5  |-  ( ( ( 10 ^ k
)  e.  CC  /\  ( 10 ^ k )  =/=  0 )  -> 
( 9  /  ( 10 ^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
165, 11, 15syl2anc 644 . . . 4  |-  ( k  e.  NN  ->  (
9  /  ( 10
^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
176, 9, 10exprecd 11536 . . . . 5  |-  ( k  e.  NN  ->  (
( 1  /  10 ) ^ k )  =  ( 1  /  ( 10 ^ k ) ) )
1817oveq2d 6100 . . . 4  |-  ( k  e.  NN  ->  (
9  x.  ( ( 1  /  10 ) ^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
1916, 18eqtr4d 2473 . . 3  |-  ( k  e.  NN  ->  (
9  /  ( 10
^ k ) )  =  ( 9  x.  ( ( 1  /  10 ) ^ k ) ) )
2019sumeq2i 12498 . 2  |-  sum_ k  e.  NN  ( 9  / 
( 10 ^ k
) )  =  sum_ k  e.  NN  (
9  x.  ( ( 1  /  10 ) ^ k ) )
211, 8rereccli 9784 . . . . 5  |-  ( 1  /  10 )  e.  RR
2221recni 9107 . . . 4  |-  ( 1  /  10 )  e.  CC
23 0re 9096 . . . . . . 7  |-  0  e.  RR
241, 7recgt0ii 9921 . . . . . . 7  |-  0  <  ( 1  /  10 )
2523, 21, 24ltleii 9201 . . . . . 6  |-  0  <_  ( 1  /  10 )
2621absidi 12186 . . . . . 6  |-  ( 0  <_  ( 1  /  10 )  ->  ( abs `  ( 1  /  10 ) )  =  ( 1  /  10 ) )
2725, 26ax-mp 5 . . . . 5  |-  ( abs `  ( 1  /  10 ) )  =  ( 1  /  10 )
28 1lt10 10191 . . . . . 6  |-  1  <  10
29 recgt1 9911 . . . . . . 7  |-  ( ( 10  e.  RR  /\  0  <  10 )  -> 
( 1  <  10  <->  ( 1  /  10 )  <  1 ) )
301, 7, 29mp2an 655 . . . . . 6  |-  ( 1  <  10  <->  ( 1  /  10 )  <  1 )
3128, 30mpbi 201 . . . . 5  |-  ( 1  /  10 )  <  1
3227, 31eqbrtri 4234 . . . 4  |-  ( abs `  ( 1  /  10 ) )  <  1
33 geoisum1c 12662 . . . 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 1280 . . 3  |-  sum_ k  e.  NN  ( 9  x.  ( ( 1  /  10 ) ^ k ) )  =  ( ( 9  x.  ( 1  /  10 ) )  /  ( 1  -  ( 1  /  10 ) ) )
3513, 2, 8divreci 9764 . . . 4  |-  ( 9  /  10 )  =  ( 9  x.  (
1  /  10 ) )
3613, 2, 8divcan2i 9762 . . . . . 6  |-  ( 10  x.  ( 9  /  10 ) )  =  9
37 ax-1cn 9053 . . . . . . . 8  |-  1  e.  CC
382, 37, 22subdii 9487 . . . . . . 7  |-  ( 10  x.  ( 1  -  ( 1  /  10 ) ) )  =  ( ( 10  x.  1 )  -  ( 10  x.  ( 1  /  10 ) ) )
392mulid1i 9097 . . . . . . . 8  |-  ( 10  x.  1 )  =  10
402, 8recidi 9750 . . . . . . . 8  |-  ( 10  x.  ( 1  /  10 ) )  =  1
4139, 40oveq12i 6096 . . . . . . 7  |-  ( ( 10  x.  1 )  -  ( 10  x.  ( 1  /  10 ) ) )  =  ( 10  -  1 )
4237, 13addcomi 9262 . . . . . . . . 9  |-  ( 1  +  9 )  =  ( 9  +  1 )
43 df-10 10071 . . . . . . . . 9  |-  10  =  ( 9  +  1 )
4442, 43eqtr4i 2461 . . . . . . . 8  |-  ( 1  +  9 )  =  10
452, 37, 13, 44subaddrii 9394 . . . . . . 7  |-  ( 10 
-  1 )  =  9
4638, 41, 453eqtrri 2463 . . . . . 6  |-  9  =  ( 10  x.  ( 1  -  (
1  /  10 ) ) )
4736, 46eqtri 2458 . . . . 5  |-  ( 10  x.  ( 9  /  10 ) )  =  ( 10  x.  ( 1  -  ( 1  /  10 ) ) )
4812, 1, 8redivcli 9786 . . . . . . 7  |-  ( 9  /  10 )  e.  RR
4948recni 9107 . . . . . 6  |-  ( 9  /  10 )  e.  CC
5037, 22subcli 9381 . . . . . 6  |-  ( 1  -  ( 1  /  10 ) )  e.  CC
5149, 50, 2, 8mulcani 9666 . . . . 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 6096 . . 3  |-  ( ( 9  /  10 )  /  ( 9  /  10 ) )  =  ( ( 9  x.  (
1  /  10 ) )  /  ( 1  -  ( 1  /  10 ) ) )
54 9pos 10096 . . . . . 6  |-  0  <  9
5512, 1, 54, 7divgt0ii 9933 . . . . 5  |-  0  <  ( 9  /  10 )
5648, 55gt0ne0ii 9568 . . . 4  |-  ( 9  /  10 )  =/=  0
5749, 56dividi 9752 . . 3  |-  ( ( 9  /  10 )  /  ( 9  /  10 ) )  =  1
5834, 53, 573eqtr2i 2464 . 2  |-  sum_ k  e.  NN  ( 9  x.  ( ( 1  /  10 ) ^ k ) )  =  1
5920, 58eqtri 2458 1  |-  sum_ k  e.  NN  ( 9  / 
( 10 ^ k
) )  =  1
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   CCcc 8993   RRcr 8994   0cc0 8995   1c1 8996    + caddc 8998    x. cmul 9000    < clt 9125    <_ cle 9126    - cmin 9296    / cdiv 9682   NNcn 10005   9c9 10061   10c10 10062   NN0cn0 10226   ^cexp 11387   abscabs 12044   sum_csu 12484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-pm 7024  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-fz 11049  df-fzo 11141  df-fl 11207  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287  df-rlim 12288  df-sum 12485
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