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Theorem 0.999... 11566
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 9242 . . . . . . 7  |-  10  e.  RR
21recni 8304 . . . . . 6  |-  10  e.  CC
3 nnnn0 9385 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  NN0 )
4 expcl 10520 . . . . . 6  |-  ( ( 10  e.  CC  /\  k  e.  NN0 )  -> 
( 10 ^ k
)  e.  CC )
52, 3, 4sylancr 637 . . . . 5  |-  ( k  e.  NN  ->  ( 10 ^ k )  e.  CC )
62a1i 10 . . . . . 6  |-  ( k  e.  NN  ->  10  e.  CC )
7 10pos 9254 . . . . . . . 8  |-  0  <  10
81, 7gt0ne0ii 8745 . . . . . . 7  |-  10  =/=  0
98a1i 10 . . . . . 6  |-  ( k  e.  NN  ->  10  =/=  0 )
10 nnz 9458 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  ZZ )
116, 9, 10expne0d 10649 . . . . 5  |-  ( k  e.  NN  ->  ( 10 ^ k )  =/=  0 )
12 9re 9241 . . . . . . 7  |-  9  e.  RR
1312recni 8304 . . . . . 6  |-  9  e.  CC
14 divrec 8870 . . . . . 6  |-  ( ( 9  e.  CC  /\  ( 10 ^ k )  e.  CC  /\  ( 10 ^ k )  =/=  0 )  ->  (
9  /  ( 10
^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
1513, 14mp3an1 1222 . . . . 5  |-  ( ( ( 10 ^ k
)  e.  CC  /\  ( 10 ^ k )  =/=  0 )  -> 
( 9  /  ( 10 ^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
165, 11, 15syl2anc 635 . . . 4  |-  ( k  e.  NN  ->  (
9  /  ( 10
^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
176, 9, 10exprecd 10651 . . . . 5  |-  ( k  e.  NN  ->  (
( 1  /  10 ) ^ k )  =  ( 1  /  ( 10 ^ k ) ) )
1817oveq2d 5411 . . . 4  |-  ( k  e.  NN  ->  (
9  x.  ( ( 1  /  10 ) ^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
1916, 18eqtr4d 2129 . . 3  |-  ( k  e.  NN  ->  (
9  /  ( 10
^ k ) )  =  ( 9  x.  ( ( 1  /  10 ) ^ k ) ) )
2019sumeq2i 11408 . 2  |-  sum_ k  e.  NN  ( 9  / 
( 10 ^ k
) )  =  sum_ k  e.  NN  (
9  x.  ( ( 1  /  10 ) ^ k ) )
211, 8rereccli 8953 . . . 4  |-  ( 1  /  10 )  e.  RR
2221recni 8304 . . 3  |-  ( 1  /  10 )  e.  CC
23 0re 8293 . . . . . 6  |-  0  e.  RR
241, 7recgt0ii 9079 . . . . . 6  |-  0  <  ( 1  /  10 )
2523, 21, 24ltleii 8395 . . . . 5  |-  0  <_  ( 1  /  10 )
2621absidi 11097 . . . . 5  |-  ( 0  <_  ( 1  /  10 )  ->  ( abs `  ( 1  /  10 ) )  =  ( 1  /  10 ) )
2725, 26ax-mp 8 . . . 4  |-  ( abs `  ( 1  /  10 ) )  =  ( 1  /  10 )
28 1lt10 9345 . . . . 5  |-  1  <  10
29 recgt1 9069 . . . . . 6  |-  ( ( 10  e.  RR  /\  0  <  10 )  -> 
( 1  <  10  <->  ( 1  /  10 )  <  1 ) )
301, 7, 29mp2an 646 . . . . 5  |-  ( 1  <  10  <->  ( 1  /  10 )  <  1 )
3128, 30mpbi 197 . . . 4  |-  ( 1  /  10 )  <  1
3227, 31eqbrtri 3640 . . 3  |-  ( abs `  ( 1  /  10 ) )  <  1
33 geoisum1c 11565 . . 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 ) ) ) )
3413, 22, 32, 33mp3an 1235 . 2  |-  sum_ k  e.  NN  ( 9  x.  ( ( 1  /  10 ) ^ k ) )  =  ( ( 9  x.  ( 1  /  10 ) )  /  ( 1  -  ( 1  /  10 ) ) )
3513, 2, 8divreci 8933 . . . 4  |-  ( 9  /  10 )  =  ( 9  x.  (
1  /  10 ) )
3613, 2, 8divcan2i 8931 . . . . . 6  |-  ( 10  x.  ( 9  /  10 ) )  =  9
37 ax-1cn 8250 . . . . . . . 8  |-  1  e.  CC
382, 37, 22subdii 8664 . . . . . . 7  |-  ( 10  x.  ( 1  -  ( 1  /  10 ) ) )  =  ( ( 10  x.  1 )  -  ( 10  x.  ( 1  /  10 ) ) )
392mulid1i 8294 . . . . . . . 8  |-  ( 10  x.  1 )  =  10
402, 8recidi 8919 . . . . . . . 8  |-  ( 10  x.  ( 1  /  10 ) )  =  1
4139, 40oveq12i 5407 . . . . . . 7  |-  ( ( 10  x.  1 )  -  ( 10  x.  ( 1  /  10 ) ) )  =  ( 10  -  1 )
4237, 13addcomi 8457 . . . . . . . . 9  |-  ( 1  +  9 )  =  ( 9  +  1 )
43 df-10 9228 . . . . . . . . 9  |-  10  =  ( 9  +  1 )
4442, 43eqtr4i 2117 . . . . . . . 8  |-  ( 1  +  9 )  =  10
452, 37, 13, 44subaddrii 8585 . . . . . . 7  |-  ( 10 
-  1 )  =  9
4638, 41, 453eqtrri 2119 . . . . . 6  |-  9  =  ( 10  x.  ( 1  -  (
1  /  10 ) ) )
4736, 46eqtri 2114 . . . . 5  |-  ( 10  x.  ( 9  /  10 ) )  =  ( 10  x.  ( 1  -  ( 1  /  10 ) ) )
4812, 1, 8redivcli 8955 . . . . . . 7  |-  ( 9  /  10 )  e.  RR
4948recni 8304 . . . . . 6  |-  ( 9  /  10 )  e.  CC
5037, 22subcli 8572 . . . . . 6  |-  ( 1  -  ( 1  /  10 ) )  e.  CC
5149, 50, 2, 8mulcani 8839 . . . . 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 5407 . . 3  |-  ( ( 9  /  10 )  /  ( 9  /  10 ) )  =  ( ( 9  x.  (
1  /  10 ) )  /  ( 1  -  ( 1  /  10 ) ) )
54 9pos 9253 . . . . . 6  |-  0  <  9
5512, 1, 54, 7divgt0ii 9091 . . . . 5  |-  0  <  ( 9  /  10 )
5648, 55gt0ne0ii 8745 . . . 4  |-  ( 9  /  10 )  =/=  0
5749, 56dividi 8921 . . 3  |-  ( ( 9  /  10 )  /  ( 9  /  10 ) )  =  1
5853, 57eqtr3i 2116 . 2  |-  ( ( 9  x.  ( 1  /  10 ) )  /  ( 1  -  ( 1  /  10 ) ) )  =  1
5920, 34, 583eqtri 2118 1  |-  sum_ k  e.  NN  ( 9  / 
( 10 ^ k
) )  =  1
Colors of variables: wff set class
Syntax hints:    <-> wb 174    = wceq 1525    e. wcel 1527    =/= wne 2212   class class class wbr 3621   ` cfv 4303  (class class class)co 5395   CCcc 8191   RRcr 8192   0cc0 8193   1c1 8194    + caddc 8196    x. cmul 8198    <_ cle 8319    < clt 8323    - cmin 8487    / cdiv 8853   NNcn 9162   9c9 9218   10c10 9219   NN0cn0 9378   ^cexp 10503   abscabs 10958   sum_csu 11394
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1446  ax-6 1447  ax-7 1448  ax-gen 1449  ax-8 1529  ax-11 1530  ax-13 1531  ax-14 1532  ax-17 1534  ax-12o 1568  ax-10 1582  ax-9 1588  ax-4 1595  ax-16 1781  ax-ext 2076  ax-rep 3726  ax-sep 3736  ax-nul 3744  ax-pow 3780  ax-pr 3804  ax-un 4096  ax-inf2 6860  ax-cnex 8248  ax-resscn 8249  ax-1cn 8250  ax-icn 8251  ax-addcl 8252  ax-addrcl 8253  ax-mulcl 8254  ax-mulrcl 8255  ax-mulcom 8256  ax-addass 8257  ax-mulass 8258  ax-distr 8259  ax-i2m1 8260  ax-1ne0 8261  ax-1rid 8262  ax-rnegex 8263  ax-rrecex 8264  ax-cnre 8265  ax-pre-lttri 8266  ax-pre-lttrn 8267  ax-pre-ltadd 8268  ax-pre-mulgt0 8269  ax-pre-sup 8270
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3or 897  df-3an 898  df-tru 1259  df-ex 1451  df-sb 1742  df-eu 1964  df-mo 1965  df-clab 2082  df-cleq 2087  df-clel 2090  df-ne 2214  df-nel 2215  df-ral 2308  df-rex 2309  df-reu 2310  df-rab 2311  df-v 2507  df-sbc 2681  df-csb 2763  df-dif 2826  df-un 2828  df-in 2830  df-ss 2834  df-pss 2836  df-nul 3103  df-if 3212  df-pw 3273  df-sn 3291  df-pr 3292  df-tp 3293  df-op 3294  df-uni 3460  df-int 3494  df-iun 3537  df-br 3622  df-opab 3676  df-mpt 3677  df-tr 3709  df-eprel 3891  df-id 3895  df-po 3900  df-so 3901  df-fr 3938  df-se 3939  df-we 3940  df-ord 3981  df-on 3982  df-lim 3983  df-suc 3984  df-om 4259  df-xp 4305  df-rel 4306  df-cnv 4307  df-co 4308  df-dm 4309  df-rn 4310  df-res 4311  df-ima 4312  df-fun 4313  df-fn 4314  df-f 4315  df-f1 4316  df-fo 4317  df-f1o 4318  df-fv 4319  df-iso 4320  df-ov 5398  df-oprab 5399  df-mpt2 5400  df-1st 5649  df-2nd 5650  df-iota 5805  df-recs 5878  df-rdg 5913  df-1o 5969  df-oadd 5973  df-er 6150  df-pm 6255  df-en 6337  df-dom 6338  df-sdom 6339  df-fin 6340  df-riota 6503  df-sup 6711  df-oi 6743  df-card 7089  df-pnf 8324  df-mnf 8325  df-xr 8326  df-ltxr 8327  df-le 8328  df-sub 8489  df-neg 8490  df-div 8854  df-n 9163  df-2 9220  df-3 9221  df-4 9222  df-5 9223  df-6 9224  df-7 9225  df-8 9226  df-9 9227  df-10 9228  df-n0 9379  df-z 9438  df-uz 9644  df-rp 9768  df-fz 10190  df-fzo 10278  df-fl 10329  df-seq 10446  df-exp 10504  df-hash 10736  df-cj 10823  df-re 10824  df-im 10825  df-sqr 10959  df-abs 10960  df-clim 11198  df-rlim 11199  df-sum 11395
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