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Theorem 0.999... 13701
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 10645 . . . . . . 7  |-  10  e.  RR
21recni 9625 . . . . . 6  |-  10  e.  CC
3 nnnn0 10823 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  NN0 )
4 expcl 12186 . . . . . 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 10659 . . . . . . . 8  |-  0  <  10
81, 7gt0ne0ii 10110 . . . . . . 7  |-  10  =/=  0
98a1i 11 . . . . . 6  |-  ( k  e.  NN  ->  10  =/=  0 )
10 nnz 10907 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  ZZ )
116, 9, 10expne0d 12318 . . . . 5  |-  ( k  e.  NN  ->  ( 10 ^ k )  =/=  0 )
12 9cn 10644 . . . . . 6  |-  9  e.  CC
13 divrec 10244 . . . . . 6  |-  ( ( 9  e.  CC  /\  ( 10 ^ k )  e.  CC  /\  ( 10 ^ k )  =/=  0 )  ->  (
9  /  ( 10
^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
1412, 13mp3an1 1311 . . . . 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 12320 . . . . 5  |-  ( k  e.  NN  ->  (
( 1  /  10 ) ^ k )  =  ( 1  /  ( 10 ^ k ) ) )
1716oveq2d 6312 . . . 4  |-  ( k  e.  NN  ->  (
9  x.  ( ( 1  /  10 ) ^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
1815, 17eqtr4d 2501 . . 3  |-  ( k  e.  NN  ->  (
9  /  ( 10
^ k ) )  =  ( 9  x.  ( ( 1  /  10 ) ^ k ) ) )
1918sumeq2i 13532 . 2  |-  sum_ k  e.  NN  ( 9  / 
( 10 ^ k
) )  =  sum_ k  e.  NN  (
9  x.  ( ( 1  /  10 ) ^ k ) )
201, 8rereccli 10330 . . . . 5  |-  ( 1  /  10 )  e.  RR
2120recni 9625 . . . 4  |-  ( 1  /  10 )  e.  CC
22 0re 9613 . . . . . . 7  |-  0  e.  RR
231, 7recgt0ii 10471 . . . . . . 7  |-  0  <  ( 1  /  10 )
2422, 20, 23ltleii 9724 . . . . . 6  |-  0  <_  ( 1  /  10 )
2520absidi 13221 . . . . . 6  |-  ( 0  <_  ( 1  /  10 )  ->  ( abs `  ( 1  /  10 ) )  =  ( 1  /  10 ) )
2624, 25ax-mp 5 . . . . 5  |-  ( abs `  ( 1  /  10 ) )  =  ( 1  /  10 )
27 1lt10 10767 . . . . . 6  |-  1  <  10
28 recgt1 10461 . . . . . . 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 4475 . . . 4  |-  ( abs `  ( 1  /  10 ) )  <  1
32 geoisum1c 13700 . . . 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 1324 . . 3  |-  sum_ k  e.  NN  ( 9  x.  ( ( 1  /  10 ) ^ k ) )  =  ( ( 9  x.  ( 1  /  10 ) )  /  ( 1  -  ( 1  /  10 ) ) )
3412, 2, 8divreci 10310 . . . 4  |-  ( 9  /  10 )  =  ( 9  x.  (
1  /  10 ) )
3512, 2, 8divcan2i 10308 . . . . . 6  |-  ( 10  x.  ( 9  /  10 ) )  =  9
36 ax-1cn 9567 . . . . . . . 8  |-  1  e.  CC
372, 36, 21subdii 10026 . . . . . . 7  |-  ( 10  x.  ( 1  -  ( 1  /  10 ) ) )  =  ( ( 10  x.  1 )  -  ( 10  x.  ( 1  /  10 ) ) )
382mulid1i 9615 . . . . . . . 8  |-  ( 10  x.  1 )  =  10
392, 8recidi 10296 . . . . . . . 8  |-  ( 10  x.  ( 1  /  10 ) )  =  1
4038, 39oveq12i 6308 . . . . . . 7  |-  ( ( 10  x.  1 )  -  ( 10  x.  ( 1  /  10 ) ) )  =  ( 10  -  1 )
4136, 12addcomi 9788 . . . . . . . . 9  |-  ( 1  +  9 )  =  ( 9  +  1 )
42 df-10 10623 . . . . . . . . 9  |-  10  =  ( 9  +  1 )
4341, 42eqtr4i 2489 . . . . . . . 8  |-  ( 1  +  9 )  =  10
442, 36, 12, 43subaddrii 9928 . . . . . . 7  |-  ( 10 
-  1 )  =  9
4537, 40, 443eqtrri 2491 . . . . . 6  |-  9  =  ( 10  x.  ( 1  -  (
1  /  10 ) ) )
4635, 45eqtri 2486 . . . . 5  |-  ( 10  x.  ( 9  /  10 ) )  =  ( 10  x.  ( 1  -  ( 1  /  10 ) ) )
47 9re 10643 . . . . . . . 8  |-  9  e.  RR
4847, 1, 8redivcli 10332 . . . . . . 7  |-  ( 9  /  10 )  e.  RR
4948recni 9625 . . . . . 6  |-  ( 9  /  10 )  e.  CC
5036, 21subcli 9914 . . . . . 6  |-  ( 1  -  ( 1  /  10 ) )  e.  CC
5149, 50, 2, 8mulcani 10209 . . . . 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 6308 . . 3  |-  ( ( 9  /  10 )  /  ( 9  /  10 ) )  =  ( ( 9  x.  (
1  /  10 ) )  /  ( 1  -  ( 1  /  10 ) ) )
54 9pos 10658 . . . . . 6  |-  0  <  9
5547, 1, 54, 7divgt0ii 10483 . . . . 5  |-  0  <  ( 9  /  10 )
5648, 55gt0ne0ii 10110 . . . 4  |-  ( 9  /  10 )  =/=  0
5749, 56dividi 10298 . . 3  |-  ( ( 9  /  10 )  /  ( 9  /  10 ) )  =  1
5833, 53, 573eqtr2i 2492 . 2  |-  sum_ k  e.  NN  ( 9  x.  ( ( 1  /  10 ) ^ k ) )  =  1
5919, 58eqtri 2486 1  |-  sum_ k  e.  NN  ( 9  / 
( 10 ^ k
) )  =  1
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1395    e. wcel 1819    =/= wne 2652   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   CCcc 9507   RRcr 9508   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514    < clt 9645    <_ cle 9646    - cmin 9824    / cdiv 10227   NNcn 10556   9c9 10613   10c10 10614   NN0cn0 10816   ^cexp 12168   abscabs 13078   sum_csu 13519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-fz 11698  df-fzo 11821  df-fl 11931  df-seq 12110  df-exp 12169  df-hash 12408  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-clim 13322  df-rlim 13323  df-sum 13520
This theorem is referenced by: (None)
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