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Theorem 0.999... 13333
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 10402 . . . . . . 7  |-  10  e.  RR
21recni 9390 . . . . . 6  |-  10  e.  CC
3 nnnn0 10578 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  NN0 )
4 expcl 11875 . . . . . 6  |-  ( ( 10  e.  CC  /\  k  e.  NN0 )  -> 
( 10 ^ k
)  e.  CC )
52, 3, 4sylancr 663 . . . . 5  |-  ( k  e.  NN  ->  ( 10 ^ k )  e.  CC )
62a1i 11 . . . . . 6  |-  ( k  e.  NN  ->  10  e.  CC )
7 10pos 10416 . . . . . . . 8  |-  0  <  10
81, 7gt0ne0ii 9868 . . . . . . 7  |-  10  =/=  0
98a1i 11 . . . . . 6  |-  ( k  e.  NN  ->  10  =/=  0 )
10 nnz 10660 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  ZZ )
116, 9, 10expne0d 12006 . . . . 5  |-  ( k  e.  NN  ->  ( 10 ^ k )  =/=  0 )
12 9cn 10401 . . . . . 6  |-  9  e.  CC
13 divrec 10002 . . . . . 6  |-  ( ( 9  e.  CC  /\  ( 10 ^ k )  e.  CC  /\  ( 10 ^ k )  =/=  0 )  ->  (
9  /  ( 10
^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
1412, 13mp3an1 1301 . . . . 5  |-  ( ( ( 10 ^ k
)  e.  CC  /\  ( 10 ^ k )  =/=  0 )  -> 
( 9  /  ( 10 ^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
155, 11, 14syl2anc 661 . . . 4  |-  ( k  e.  NN  ->  (
9  /  ( 10
^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
166, 9, 10exprecd 12008 . . . . 5  |-  ( k  e.  NN  ->  (
( 1  /  10 ) ^ k )  =  ( 1  /  ( 10 ^ k ) ) )
1716oveq2d 6102 . . . 4  |-  ( k  e.  NN  ->  (
9  x.  ( ( 1  /  10 ) ^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
1815, 17eqtr4d 2473 . . 3  |-  ( k  e.  NN  ->  (
9  /  ( 10
^ k ) )  =  ( 9  x.  ( ( 1  /  10 ) ^ k ) ) )
1918sumeq2i 13168 . 2  |-  sum_ k  e.  NN  ( 9  / 
( 10 ^ k
) )  =  sum_ k  e.  NN  (
9  x.  ( ( 1  /  10 ) ^ k ) )
201, 8rereccli 10088 . . . . 5  |-  ( 1  /  10 )  e.  RR
2120recni 9390 . . . 4  |-  ( 1  /  10 )  e.  CC
22 0re 9378 . . . . . . 7  |-  0  e.  RR
231, 7recgt0ii 10230 . . . . . . 7  |-  0  <  ( 1  /  10 )
2422, 20, 23ltleii 9489 . . . . . 6  |-  0  <_  ( 1  /  10 )
2520absidi 12857 . . . . . 6  |-  ( 0  <_  ( 1  /  10 )  ->  ( abs `  ( 1  /  10 ) )  =  ( 1  /  10 ) )
2624, 25ax-mp 5 . . . . 5  |-  ( abs `  ( 1  /  10 ) )  =  ( 1  /  10 )
27 1lt10 10524 . . . . . 6  |-  1  <  10
28 recgt1 10220 . . . . . . 7  |-  ( ( 10  e.  RR  /\  0  <  10 )  -> 
( 1  <  10  <->  ( 1  /  10 )  <  1 ) )
291, 7, 28mp2an 672 . . . . . 6  |-  ( 1  <  10  <->  ( 1  /  10 )  <  1 )
3027, 29mpbi 208 . . . . 5  |-  ( 1  /  10 )  <  1
3126, 30eqbrtri 4306 . . . 4  |-  ( abs `  ( 1  /  10 ) )  <  1
32 geoisum1c 13332 . . . 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 ) ) ) )
3312, 21, 31, 32mp3an 1314 . . 3  |-  sum_ k  e.  NN  ( 9  x.  ( ( 1  /  10 ) ^ k ) )  =  ( ( 9  x.  ( 1  /  10 ) )  /  ( 1  -  ( 1  /  10 ) ) )
3412, 2, 8divreci 10068 . . . 4  |-  ( 9  /  10 )  =  ( 9  x.  (
1  /  10 ) )
3512, 2, 8divcan2i 10066 . . . . . 6  |-  ( 10  x.  ( 9  /  10 ) )  =  9
36 ax-1cn 9332 . . . . . . . 8  |-  1  e.  CC
372, 36, 21subdii 9785 . . . . . . 7  |-  ( 10  x.  ( 1  -  ( 1  /  10 ) ) )  =  ( ( 10  x.  1 )  -  ( 10  x.  ( 1  /  10 ) ) )
382mulid1i 9380 . . . . . . . 8  |-  ( 10  x.  1 )  =  10
392, 8recidi 10054 . . . . . . . 8  |-  ( 10  x.  ( 1  /  10 ) )  =  1
4038, 39oveq12i 6098 . . . . . . 7  |-  ( ( 10  x.  1 )  -  ( 10  x.  ( 1  /  10 ) ) )  =  ( 10  -  1 )
4136, 12addcomi 9552 . . . . . . . . 9  |-  ( 1  +  9 )  =  ( 9  +  1 )
42 df-10 10380 . . . . . . . . 9  |-  10  =  ( 9  +  1 )
4341, 42eqtr4i 2461 . . . . . . . 8  |-  ( 1  +  9 )  =  10
442, 36, 12, 43subaddrii 9689 . . . . . . 7  |-  ( 10 
-  1 )  =  9
4537, 40, 443eqtrri 2463 . . . . . 6  |-  9  =  ( 10  x.  ( 1  -  (
1  /  10 ) ) )
4635, 45eqtri 2458 . . . . 5  |-  ( 10  x.  ( 9  /  10 ) )  =  ( 10  x.  ( 1  -  ( 1  /  10 ) ) )
47 9re 10400 . . . . . . . 8  |-  9  e.  RR
4847, 1, 8redivcli 10090 . . . . . . 7  |-  ( 9  /  10 )  e.  RR
4948recni 9390 . . . . . 6  |-  ( 9  /  10 )  e.  CC
5036, 21subcli 9676 . . . . . 6  |-  ( 1  -  ( 1  /  10 ) )  e.  CC
5149, 50, 2, 8mulcani 9967 . . . . 5  |-  ( ( 10  x.  ( 9  /  10 ) )  =  ( 10  x.  ( 1  -  (
1  /  10 ) ) )  <->  ( 9  /  10 )  =  ( 1  -  (
1  /  10 ) ) )
5246, 51mpbi 208 . . . 4  |-  ( 9  /  10 )  =  ( 1  -  (
1  /  10 ) )
5334, 52oveq12i 6098 . . 3  |-  ( ( 9  /  10 )  /  ( 9  /  10 ) )  =  ( ( 9  x.  (
1  /  10 ) )  /  ( 1  -  ( 1  /  10 ) ) )
54 9pos 10415 . . . . . 6  |-  0  <  9
5547, 1, 54, 7divgt0ii 10242 . . . . 5  |-  0  <  ( 9  /  10 )
5648, 55gt0ne0ii 9868 . . . 4  |-  ( 9  /  10 )  =/=  0
5749, 56dividi 10056 . . 3  |-  ( ( 9  /  10 )  /  ( 9  /  10 ) )  =  1
5833, 53, 573eqtr2i 2464 . 2  |-  sum_ k  e.  NN  ( 9  x.  ( ( 1  /  10 ) ^ k ) )  =  1
5919, 58eqtri 2458 1  |-  sum_ k  e.  NN  ( 9  / 
( 10 ^ k
) )  =  1
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1369    e. wcel 1756    =/= wne 2601   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275    + caddc 9277    x. cmul 9279    < clt 9410    <_ cle 9411    - cmin 9587    / cdiv 9985   NNcn 10314   9c9 10370   10c10 10371   NN0cn0 10571   ^cexp 11857   abscabs 12715   sum_csu 13155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-pm 7209  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-oi 7716  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-fz 11430  df-fzo 11541  df-fl 11634  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-rlim 12959  df-sum 13156
This theorem is referenced by: (None)
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