MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0.999... Unicode version

Theorem 0.999... 11575
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 9251 . . . . . . 7  |-  10  e.  RR
21recni 8313 . . . . . 6  |-  10  e.  CC
3 nnnn0 9394 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  NN0 )
4 expcl 10529 . . . . . 6  |-  ( ( 10  e.  CC  /\  k  e.  NN0 )  -> 
( 10 ^ k
)  e.  CC )
52, 3, 4sylancr 639 . . . . 5  |-  ( k  e.  NN  ->  ( 10 ^ k )  e.  CC )
62a1i 10 . . . . . 6  |-  ( k  e.  NN  ->  10  e.  CC )
7 10pos 9263 . . . . . . . 8  |-  0  <  10
81, 7gt0ne0ii 8754 . . . . . . 7  |-  10  =/=  0
98a1i 10 . . . . . 6  |-  ( k  e.  NN  ->  10  =/=  0 )
10 nnz 9467 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  ZZ )
116, 9, 10expne0d 10658 . . . . 5  |-  ( k  e.  NN  ->  ( 10 ^ k )  =/=  0 )
12 9re 9250 . . . . . . 7  |-  9  e.  RR
1312recni 8313 . . . . . 6  |-  9  e.  CC
14 divrec 8879 . . . . . 6  |-  ( ( 9  e.  CC  /\  ( 10 ^ k )  e.  CC  /\  ( 10 ^ k )  =/=  0 )  ->  (
9  /  ( 10
^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
1513, 14mp3an1 1228 . . . . 5  |-  ( ( ( 10 ^ k
)  e.  CC  /\  ( 10 ^ k )  =/=  0 )  -> 
( 9  /  ( 10 ^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
165, 11, 15syl2anc 637 . . . 4  |-  ( k  e.  NN  ->  (
9  /  ( 10
^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
176, 9, 10exprecd 10660 . . . . 5  |-  ( k  e.  NN  ->  (
( 1  /  10 ) ^ k )  =  ( 1  /  ( 10 ^ k ) ) )
1817oveq2d 5420 . . . 4  |-  ( k  e.  NN  ->  (
9  x.  ( ( 1  /  10 ) ^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
1916, 18eqtr4d 2135 . . 3  |-  ( k  e.  NN  ->  (
9  /  ( 10
^ k ) )  =  ( 9  x.  ( ( 1  /  10 ) ^ k ) ) )
2019sumeq2i 11417 . 2  |-  sum_ k  e.  NN  ( 9  / 
( 10 ^ k
) )  =  sum_ k  e.  NN  (
9  x.  ( ( 1  /  10 ) ^ k ) )
211, 8rereccli 8962 . . . . 5  |-  ( 1  /  10 )  e.  RR
2221recni 8313 . . . 4  |-  ( 1  /  10 )  e.  CC
23 0re 8302 . . . . . . 7  |-  0  e.  RR
241, 7recgt0ii 9088 . . . . . . 7  |-  0  <  ( 1  /  10 )
2523, 21, 24ltleii 8404 . . . . . 6  |-  0  <_  ( 1  /  10 )
2621absidi 11106 . . . . . 6  |-  ( 0  <_  ( 1  /  10 )  ->  ( abs `  ( 1  /  10 ) )  =  ( 1  /  10 ) )
2725, 26ax-mp 8 . . . . 5  |-  ( abs `  ( 1  /  10 ) )  =  ( 1  /  10 )
28 1lt10 9354 . . . . . 6  |-  1  <  10
29 recgt1 9078 . . . . . . 7  |-  ( ( 10  e.  RR  /\  0  <  10 )  -> 
( 1  <  10  <->  ( 1  /  10 )  <  1 ) )
301, 7, 29mp2an 648 . . . . . 6  |-  ( 1  <  10  <->  ( 1  /  10 )  <  1 )
3128, 30mpbi 197 . . . . 5  |-  ( 1  /  10 )  <  1
3227, 31eqbrtri 3649 . . . 4  |-  ( abs `  ( 1  /  10 ) )  <  1
33 geoisum1c 11574 . . . 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 1241 . . 3  |-  sum_ k  e.  NN  ( 9  x.  ( ( 1  /  10 ) ^ k ) )  =  ( ( 9  x.  ( 1  /  10 ) )  /  ( 1  -  ( 1  /  10 ) ) )
3513, 2, 8divreci 8942 . . . 4  |-  ( 9  /  10 )  =  ( 9  x.  (
1  /  10 ) )
3613, 2, 8divcan2i 8940 . . . . . 6  |-  ( 10  x.  ( 9  /  10 ) )  =  9
37 ax-1cn 8259 . . . . . . . 8  |-  1  e.  CC
382, 37, 22subdii 8673 . . . . . . 7  |-  ( 10  x.  ( 1  -  ( 1  /  10 ) ) )  =  ( ( 10  x.  1 )  -  ( 10  x.  ( 1  /  10 ) ) )
392mulid1i 8303 . . . . . . . 8  |-  ( 10  x.  1 )  =  10
402, 8recidi 8928 . . . . . . . 8  |-  ( 10  x.  ( 1  /  10 ) )  =  1
4139, 40oveq12i 5416 . . . . . . 7  |-  ( ( 10  x.  1 )  -  ( 10  x.  ( 1  /  10 ) ) )  =  ( 10  -  1 )
4237, 13addcomi 8466 . . . . . . . . 9  |-  ( 1  +  9 )  =  ( 9  +  1 )
43 df-10 9237 . . . . . . . . 9  |-  10  =  ( 9  +  1 )
4442, 43eqtr4i 2123 . . . . . . . 8  |-  ( 1  +  9 )  =  10
452, 37, 13, 44subaddrii 8594 . . . . . . 7  |-  ( 10 
-  1 )  =  9
4638, 41, 453eqtrri 2125 . . . . . 6  |-  9  =  ( 10  x.  ( 1  -  (
1  /  10 ) ) )
4736, 46eqtri 2120 . . . . 5  |-  ( 10  x.  ( 9  /  10 ) )  =  ( 10  x.  ( 1  -  ( 1  /  10 ) ) )
4812, 1, 8redivcli 8964 . . . . . . 7  |-  ( 9  /  10 )  e.  RR
4948recni 8313 . . . . . 6  |-  ( 9  /  10 )  e.  CC
5037, 22subcli 8581 . . . . . 6  |-  ( 1  -  ( 1  /  10 ) )  e.  CC
5149, 50, 2, 8mulcani 8848 . . . . 5  |-  ( ( 10  x.  ( 9  /  10 ) )  =  ( 10  x.  ( 1  -  (
1  /  10 ) ) )  <->  ( 9  /  10 )  =  ( 1  -  (
1  /  10 ) ) )
5247, 51mpbi 197 . . . 4  |-  ( 9  /  10 )  =  ( 1  -  (
1  /  10 ) )
5335, 52oveq12i 5416 . . 3  |-  ( ( 9  /  10 )  /  ( 9  /  10 ) )  =  ( ( 9  x.  (
1  /  10 ) )  /  ( 1  -  ( 1  /  10 ) ) )
54 9pos 9262 . . . . . 6  |-  0  <  9
5512, 1, 54, 7divgt0ii 9100 . . . . 5  |-  0  <  ( 9  /  10 )
5648, 55gt0ne0ii 8754 . . . 4  |-  ( 9  /  10 )  =/=  0
5749, 56dividi 8930 . . 3  |-  ( ( 9  /  10 )  /  ( 9  /  10 ) )  =  1
5834, 53, 573eqtr2i 2126 . 2  |-  sum_ k  e.  NN  ( 9  x.  ( ( 1  /  10 ) ^ k ) )  =  1
5920, 58eqtri 2120 1  |-  sum_ k  e.  NN  ( 9  / 
( 10 ^ k
) )  =  1
Colors of variables: wff set class
Syntax hints:    <-> wb 174    = wceq 1531    e. wcel 1533    =/= wne 2218   class class class wbr 3630   ` cfv 4312  (class class class)co 5404   CCcc 8200   RRcr 8201   0cc0 8202   1c1 8203    + caddc 8205    x. cmul 8207    <_ cle 8328    < clt 8332    - cmin 8496    / cdiv 8862   NNcn 9171   9c9 9227   10c10 9228   NN0cn0 9387   ^cexp 10512   abscabs 10967   sum_csu 11403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1452  ax-6 1453  ax-7 1454  ax-gen 1455  ax-8 1535  ax-11 1536  ax-13 1537  ax-14 1538  ax-17 1540  ax-12o 1574  ax-10 1588  ax-9 1594  ax-4 1601  ax-16 1787  ax-ext 2082  ax-rep 3735  ax-sep 3745  ax-nul 3753  ax-pow 3789  ax-pr 3813  ax-un 4105  ax-inf2 6869  ax-cnex 8257  ax-resscn 8258  ax-1cn 8259  ax-icn 8260  ax-addcl 8261  ax-addrcl 8262  ax-mulcl 8263  ax-mulrcl 8264  ax-mulcom 8265  ax-addass 8266  ax-mulass 8267  ax-distr 8268  ax-i2m1 8269  ax-1ne0 8270  ax-1rid 8271  ax-rnegex 8272  ax-rrecex 8273  ax-cnre 8274  ax-pre-lttri 8275  ax-pre-lttrn 8276  ax-pre-ltadd 8277  ax-pre-mulgt0 8278  ax-pre-sup 8279
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3or 901  df-3an 902  df-tru 1265  df-ex 1457  df-sb 1748  df-eu 1970  df-mo 1971  df-clab 2088  df-cleq 2093  df-clel 2096  df-ne 2220  df-nel 2221  df-ral 2315  df-rex 2316  df-reu 2317  df-rab 2318  df-v 2514  df-sbc 2688  df-csb 2770  df-dif 2833  df-un 2835  df-in 2837  df-ss 2841  df-pss 2843  df-nul 3111  df-if 3221  df-pw 3282  df-sn 3300  df-pr 3301  df-tp 3302  df-op 3303  df-uni 3469  df-int 3503  df-iun 3546  df-br 3631  df-opab 3685  df-mpt 3686  df-tr 3718  df-eprel 3900  df-id 3904  df-po 3909  df-so 3910  df-fr 3947  df-se 3948  df-we 3949  df-ord 3990  df-on 3991  df-lim 3992  df-suc 3993  df-om 4268  df-xp 4314  df-rel 4315  df-cnv 4316  df-co 4317  df-dm 4318  df-rn 4319  df-res 4320  df-ima 4321  df-fun 4322  df-fn 4323  df-f 4324  df-f1 4325  df-fo 4326  df-f1o 4327  df-fv 4328  df-iso 4329  df-ov 5407  df-oprab 5408  df-mpt2 5409  df-1st 5658  df-2nd 5659  df-iota 5814  df-recs 5887  df-rdg 5922  df-1o 5978  df-oadd 5982  df-er 6159  df-pm 6264  df-en 6346  df-dom 6347  df-sdom 6348  df-fin 6349  df-riota 6512  df-sup 6720  df-oi 6752  df-card 7098  df-pnf 8333  df-mnf 8334  df-xr 8335  df-ltxr 8336  df-le 8337  df-sub 8498  df-neg 8499  df-div 8863  df-n 9172  df-2 9229  df-3 9230  df-4 9231  df-5 9232  df-6 9233  df-7 9234  df-8 9235  df-9 9236  df-10 9237  df-n0 9388  df-z 9447  df-uz 9653  df-rp 9777  df-fz 10199  df-fzo 10287  df-fl 10338  df-seq 10455  df-exp 10513  df-hash 10745  df-cj 10832  df-re 10833  df-im 10834  df-sqr 10968  df-abs 10969  df-clim 11207  df-rlim 11208  df-sum 11404
  Copyright terms: Public domain W3C validator