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Theorem 0.999... 13454
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 10516 . . . . . . 7  |-  10  e.  RR
21recni 9504 . . . . . 6  |-  10  e.  CC
3 nnnn0 10692 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  NN0 )
4 expcl 11995 . . . . . 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 10530 . . . . . . . 8  |-  0  <  10
81, 7gt0ne0ii 9982 . . . . . . 7  |-  10  =/=  0
98a1i 11 . . . . . 6  |-  ( k  e.  NN  ->  10  =/=  0 )
10 nnz 10774 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  ZZ )
116, 9, 10expne0d 12126 . . . . 5  |-  ( k  e.  NN  ->  ( 10 ^ k )  =/=  0 )
12 9cn 10515 . . . . . 6  |-  9  e.  CC
13 divrec 10116 . . . . . 6  |-  ( ( 9  e.  CC  /\  ( 10 ^ k )  e.  CC  /\  ( 10 ^ k )  =/=  0 )  ->  (
9  /  ( 10
^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
1412, 13mp3an1 1302 . . . . 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 12128 . . . . 5  |-  ( k  e.  NN  ->  (
( 1  /  10 ) ^ k )  =  ( 1  /  ( 10 ^ k ) ) )
1716oveq2d 6211 . . . 4  |-  ( k  e.  NN  ->  (
9  x.  ( ( 1  /  10 ) ^ k ) )  =  ( 9  x.  ( 1  /  ( 10 ^ k ) ) ) )
1815, 17eqtr4d 2496 . . 3  |-  ( k  e.  NN  ->  (
9  /  ( 10
^ k ) )  =  ( 9  x.  ( ( 1  /  10 ) ^ k ) ) )
1918sumeq2i 13289 . 2  |-  sum_ k  e.  NN  ( 9  / 
( 10 ^ k
) )  =  sum_ k  e.  NN  (
9  x.  ( ( 1  /  10 ) ^ k ) )
201, 8rereccli 10202 . . . . 5  |-  ( 1  /  10 )  e.  RR
2120recni 9504 . . . 4  |-  ( 1  /  10 )  e.  CC
22 0re 9492 . . . . . . 7  |-  0  e.  RR
231, 7recgt0ii 10344 . . . . . . 7  |-  0  <  ( 1  /  10 )
2422, 20, 23ltleii 9603 . . . . . 6  |-  0  <_  ( 1  /  10 )
2520absidi 12978 . . . . . 6  |-  ( 0  <_  ( 1  /  10 )  ->  ( abs `  ( 1  /  10 ) )  =  ( 1  /  10 ) )
2624, 25ax-mp 5 . . . . 5  |-  ( abs `  ( 1  /  10 ) )  =  ( 1  /  10 )
27 1lt10 10638 . . . . . 6  |-  1  <  10
28 recgt1 10334 . . . . . . 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 4414 . . . 4  |-  ( abs `  ( 1  /  10 ) )  <  1
32 geoisum1c 13453 . . . 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 1315 . . 3  |-  sum_ k  e.  NN  ( 9  x.  ( ( 1  /  10 ) ^ k ) )  =  ( ( 9  x.  ( 1  /  10 ) )  /  ( 1  -  ( 1  /  10 ) ) )
3412, 2, 8divreci 10182 . . . 4  |-  ( 9  /  10 )  =  ( 9  x.  (
1  /  10 ) )
3512, 2, 8divcan2i 10180 . . . . . 6  |-  ( 10  x.  ( 9  /  10 ) )  =  9
36 ax-1cn 9446 . . . . . . . 8  |-  1  e.  CC
372, 36, 21subdii 9899 . . . . . . 7  |-  ( 10  x.  ( 1  -  ( 1  /  10 ) ) )  =  ( ( 10  x.  1 )  -  ( 10  x.  ( 1  /  10 ) ) )
382mulid1i 9494 . . . . . . . 8  |-  ( 10  x.  1 )  =  10
392, 8recidi 10168 . . . . . . . 8  |-  ( 10  x.  ( 1  /  10 ) )  =  1
4038, 39oveq12i 6207 . . . . . . 7  |-  ( ( 10  x.  1 )  -  ( 10  x.  ( 1  /  10 ) ) )  =  ( 10  -  1 )
4136, 12addcomi 9666 . . . . . . . . 9  |-  ( 1  +  9 )  =  ( 9  +  1 )
42 df-10 10494 . . . . . . . . 9  |-  10  =  ( 9  +  1 )
4341, 42eqtr4i 2484 . . . . . . . 8  |-  ( 1  +  9 )  =  10
442, 36, 12, 43subaddrii 9803 . . . . . . 7  |-  ( 10 
-  1 )  =  9
4537, 40, 443eqtrri 2486 . . . . . 6  |-  9  =  ( 10  x.  ( 1  -  (
1  /  10 ) ) )
4635, 45eqtri 2481 . . . . 5  |-  ( 10  x.  ( 9  /  10 ) )  =  ( 10  x.  ( 1  -  ( 1  /  10 ) ) )
47 9re 10514 . . . . . . . 8  |-  9  e.  RR
4847, 1, 8redivcli 10204 . . . . . . 7  |-  ( 9  /  10 )  e.  RR
4948recni 9504 . . . . . 6  |-  ( 9  /  10 )  e.  CC
5036, 21subcli 9790 . . . . . 6  |-  ( 1  -  ( 1  /  10 ) )  e.  CC
5149, 50, 2, 8mulcani 10081 . . . . 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 6207 . . 3  |-  ( ( 9  /  10 )  /  ( 9  /  10 ) )  =  ( ( 9  x.  (
1  /  10 ) )  /  ( 1  -  ( 1  /  10 ) ) )
54 9pos 10529 . . . . . 6  |-  0  <  9
5547, 1, 54, 7divgt0ii 10356 . . . . 5  |-  0  <  ( 9  /  10 )
5648, 55gt0ne0ii 9982 . . . 4  |-  ( 9  /  10 )  =/=  0
5749, 56dividi 10170 . . 3  |-  ( ( 9  /  10 )  /  ( 9  /  10 ) )  =  1
5833, 53, 573eqtr2i 2487 . 2  |-  sum_ k  e.  NN  ( 9  x.  ( ( 1  /  10 ) ^ k ) )  =  1
5919, 58eqtri 2481 1  |-  sum_ k  e.  NN  ( 9  / 
( 10 ^ k
) )  =  1
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1370    e. wcel 1758    =/= wne 2645   class class class wbr 4395   ` cfv 5521  (class class class)co 6195   CCcc 9386   RRcr 9387   0cc0 9388   1c1 9389    + caddc 9391    x. cmul 9393    < clt 9524    <_ cle 9525    - cmin 9701    / cdiv 10099   NNcn 10428   9c9 10484   10c10 10485   NN0cn0 10685   ^cexp 11977   abscabs 12836   sum_csu 13276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-inf2 7953  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-pre-sup 9466
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-isom 5530  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-1o 7025  df-oadd 7029  df-er 7206  df-pm 7322  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-sup 7797  df-oi 7830  df-card 8215  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-nn 10429  df-2 10486  df-3 10487  df-4 10488  df-5 10489  df-6 10490  df-7 10491  df-8 10492  df-9 10493  df-10 10494  df-n0 10686  df-z 10753  df-uz 10968  df-rp 11098  df-fz 11550  df-fzo 11661  df-fl 11754  df-seq 11919  df-exp 11978  df-hash 12216  df-cj 12701  df-re 12702  df-im 12703  df-sqr 12837  df-abs 12838  df-clim 13079  df-rlim 13080  df-sum 13277
This theorem is referenced by: (None)
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