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Theorem 0.999... 13915
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 10698 . . . . . . 7  |-  10  e.  RR
21recni 9654 . . . . . 6  |-  10  e.  CC
3 nnnn0 10876 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  NN0 )
4 expcl 12287 . . . . . 6  |-  ( ( 10  e.  CC  /\  k  e.  NN0 )  -> 
( 10 ^ k
)  e.  CC )
52, 3, 4sylancr 667 . . . . 5  |-  ( k  e.  NN  ->  ( 10 ^ k )  e.  CC )
62a1i 11 . . . . . 6  |-  ( k  e.  NN  ->  10  e.  CC )
7 10pos 10712 . . . . . . . 8  |-  0  <  10
81, 7gt0ne0ii 10149 . . . . . . 7  |-  10  =/=  0
98a1i 11 . . . . . 6  |-  ( k  e.  NN  ->  10  =/=  0 )
10 nnz 10959 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  ZZ )
116, 9, 10expne0d 12419 . . . . 5  |-  ( k  e.  NN  ->  ( 10 ^ k )  =/=  0 )
12 9cn 10697 . . . . . 6  |-  9  e.  CC
13 divrec 10285 . . . . . 6  |-  ( ( 9  e.  CC  /\  ( 10 ^ k )  e.  CC  /\  ( 10 ^ k )  =/=  0 )  ->  (
9  /  ( 10
^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
1412, 13mp3an1 1347 . . . . 5  |-  ( ( ( 10 ^ k
)  e.  CC  /\  ( 10 ^ k )  =/=  0 )  -> 
( 9  /  ( 10 ^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
155, 11, 14syl2anc 665 . . . 4  |-  ( k  e.  NN  ->  (
9  /  ( 10
^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
166, 9, 10exprecd 12421 . . . . 5  |-  ( k  e.  NN  ->  (
( 1  /  10 ) ^ k )  =  ( 1  /  ( 10 ^ k ) ) )
1716oveq2d 6321 . . . 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 13743 . 2  |-  sum_ k  e.  NN  ( 9  / 
( 10 ^ k
) )  =  sum_ k  e.  NN  (
9  x.  ( ( 1  /  10 ) ^ k ) )
201, 8rereccli 10371 . . . . 5  |-  ( 1  /  10 )  e.  RR
2120recni 9654 . . . 4  |-  ( 1  /  10 )  e.  CC
22 0re 9642 . . . . . . 7  |-  0  e.  RR
231, 7recgt0ii 10512 . . . . . . 7  |-  0  <  ( 1  /  10 )
2422, 20, 23ltleii 9756 . . . . . 6  |-  0  <_  ( 1  /  10 )
2520absidi 13419 . . . . . 6  |-  ( 0  <_  ( 1  /  10 )  ->  ( abs `  ( 1  /  10 ) )  =  ( 1  /  10 ) )
2624, 25ax-mp 5 . . . . 5  |-  ( abs `  ( 1  /  10 ) )  =  ( 1  /  10 )
27 1lt10 10820 . . . . . 6  |-  1  <  10
28 recgt1 10502 . . . . . . 7  |-  ( ( 10  e.  RR  /\  0  <  10 )  -> 
( 1  <  10  <->  ( 1  /  10 )  <  1 ) )
291, 7, 28mp2an 676 . . . . . 6  |-  ( 1  <  10  <->  ( 1  /  10 )  <  1 )
3027, 29mpbi 211 . . . . 5  |-  ( 1  /  10 )  <  1
3126, 30eqbrtri 4445 . . . 4  |-  ( abs `  ( 1  /  10 ) )  <  1
32 geoisum1c 13914 . . . 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 1360 . . 3  |-  sum_ k  e.  NN  ( 9  x.  ( ( 1  /  10 ) ^ k ) )  =  ( ( 9  x.  ( 1  /  10 ) )  /  ( 1  -  ( 1  /  10 ) ) )
3412, 2, 8divreci 10351 . . . 4  |-  ( 9  /  10 )  =  ( 9  x.  (
1  /  10 ) )
3512, 2, 8divcan2i 10349 . . . . . 6  |-  ( 10  x.  ( 9  /  10 ) )  =  9
36 ax-1cn 9596 . . . . . . . 8  |-  1  e.  CC
372, 36, 21subdii 10066 . . . . . . 7  |-  ( 10  x.  ( 1  -  ( 1  /  10 ) ) )  =  ( ( 10  x.  1 )  -  ( 10  x.  ( 1  /  10 ) ) )
382mulid1i 9644 . . . . . . . 8  |-  ( 10  x.  1 )  =  10
392, 8recidi 10337 . . . . . . . 8  |-  ( 10  x.  ( 1  /  10 ) )  =  1
4038, 39oveq12i 6317 . . . . . . 7  |-  ( ( 10  x.  1 )  -  ( 10  x.  ( 1  /  10 ) ) )  =  ( 10  -  1 )
4136, 12addcomi 9823 . . . . . . . . 9  |-  ( 1  +  9 )  =  ( 9  +  1 )
42 df-10 10676 . . . . . . . . 9  |-  10  =  ( 9  +  1 )
4341, 42eqtr4i 2461 . . . . . . . 8  |-  ( 1  +  9 )  =  10
442, 36, 12, 43subaddrii 9963 . . . . . . 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 10696 . . . . . . . 8  |-  9  e.  RR
4847, 1, 8redivcli 10373 . . . . . . 7  |-  ( 9  /  10 )  e.  RR
4948recni 9654 . . . . . 6  |-  ( 9  /  10 )  e.  CC
5036, 21subcli 9949 . . . . . 6  |-  ( 1  -  ( 1  /  10 ) )  e.  CC
5149, 50, 2, 8mulcani 10250 . . . . 5  |-  ( ( 10  x.  ( 9  /  10 ) )  =  ( 10  x.  ( 1  -  (
1  /  10 ) ) )  <->  ( 9  /  10 )  =  ( 1  -  (
1  /  10 ) ) )
5246, 51mpbi 211 . . . 4  |-  ( 9  /  10 )  =  ( 1  -  (
1  /  10 ) )
5334, 52oveq12i 6317 . . 3  |-  ( ( 9  /  10 )  /  ( 9  /  10 ) )  =  ( ( 9  x.  (
1  /  10 ) )  /  ( 1  -  ( 1  /  10 ) ) )
54 9pos 10711 . . . . . 6  |-  0  <  9
5547, 1, 54, 7divgt0ii 10524 . . . . 5  |-  0  <  ( 9  /  10 )
5648, 55gt0ne0ii 10149 . . . 4  |-  ( 9  /  10 )  =/=  0
5749, 56dividi 10339 . . 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 187    = wceq 1437    e. wcel 1870    =/= wne 2625   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   CCcc 9536   RRcr 9537   0cc0 9538   1c1 9539    + caddc 9541    x. cmul 9543    < clt 9674    <_ cle 9675    - cmin 9859    / cdiv 10268   NNcn 10609   9c9 10666   10c10 10667   NN0cn0 10869   ^cexp 12269   abscabs 13276   sum_csu 13730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-oi 8025  df-card 8372  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11783  df-fzo 11914  df-fl 12025  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-clim 13530  df-rlim 13531  df-sum 13731
This theorem is referenced by: (None)
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