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Theorem 3dvds 13926
Description: A rule for divisibility by 3 of a number written in base 10. This is Metamath 100 proof #85. (Contributed by Mario Carneiro, 14-Jul-2014.) (Revised by Mario Carneiro, 17-Jan-2015.)
Assertion
Ref Expression
3dvds  |-  ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  ->  ( 3  ||  sum_ k  e.  ( 0 ... N ) ( ( F `  k
)  x.  ( 10
^ k ) )  <->  3  ||  sum_ k  e.  ( 0 ... N
) ( F `  k ) ) )
Distinct variable groups:    k, F    k, N

Proof of Theorem 3dvds
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 3z 10909 . . 3  |-  3  e.  ZZ
21a1i 11 . 2  |-  ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  ->  3  e.  ZZ )
3 fzfid 12063 . . 3  |-  ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  ->  ( 0 ... N )  e.  Fin )
4 ffvelrn 6030 . . . . 5  |-  ( ( F : ( 0 ... N ) --> ZZ 
/\  k  e.  ( 0 ... N ) )  ->  ( F `  k )  e.  ZZ )
54adantll 713 . . . 4  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  ( F `  k )  e.  ZZ )
6 10nn 10713 . . . . . 6  |-  10  e.  NN
76nnzi 10900 . . . . 5  |-  10  e.  ZZ
8 elfznn0 11782 . . . . . 6  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
98adantl 466 . . . . 5  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  k  e.  NN0 )
10 zexpcl 12161 . . . . 5  |-  ( ( 10  e.  ZZ  /\  k  e.  NN0 )  -> 
( 10 ^ k
)  e.  ZZ )
117, 9, 10sylancr 663 . . . 4  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  ( 10 ^ k )  e.  ZZ )
125, 11zmulcld 10984 . . 3  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  (
( F `  k
)  x.  ( 10
^ k ) )  e.  ZZ )
133, 12fsumzcl 13537 . 2  |-  ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  ->  sum_ k  e.  ( 0 ... N ) ( ( F `  k )  x.  ( 10 ^ k ) )  e.  ZZ )
143, 5fsumzcl 13537 . 2  |-  ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  ->  sum_ k  e.  ( 0 ... N ) ( F `  k
)  e.  ZZ )
1512, 5zsubcld 10983 . . . 4  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  (
( ( F `  k )  x.  ( 10 ^ k ) )  -  ( F `  k ) )  e.  ZZ )
16 ax-1cn 9562 . . . . . . . . . . . 12  |-  1  e.  CC
176nncni 10558 . . . . . . . . . . . 12  |-  10  e.  CC
1816, 17negsubdi2i 9917 . . . . . . . . . . 11  |-  -u (
1  -  10 )  =  ( 10  - 
1 )
19 df-10 10614 . . . . . . . . . . . 12  |-  10  =  ( 9  +  1 )
2019oveq1i 6305 . . . . . . . . . . 11  |-  ( 10 
-  1 )  =  ( ( 9  +  1 )  -  1 )
21 9cn 10635 . . . . . . . . . . . 12  |-  9  e.  CC
22 pncan 9838 . . . . . . . . . . . 12  |-  ( ( 9  e.  CC  /\  1  e.  CC )  ->  ( ( 9  +  1 )  -  1 )  =  9 )
2321, 16, 22mp2an 672 . . . . . . . . . . 11  |-  ( ( 9  +  1 )  -  1 )  =  9
2418, 20, 233eqtri 2500 . . . . . . . . . 10  |-  -u (
1  -  10 )  =  9
25 3t3e9 10700 . . . . . . . . . 10  |-  ( 3  x.  3 )  =  9
2624, 25eqtr4i 2499 . . . . . . . . 9  |-  -u (
1  -  10 )  =  ( 3  x.  3 )
2717a1i 11 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN0  ->  10  e.  CC )
28 1re 9607 . . . . . . . . . . . . . . . . 17  |-  1  e.  RR
29 1lt10 10758 . . . . . . . . . . . . . . . . 17  |-  1  <  10
3028, 29gtneii 9708 . . . . . . . . . . . . . . . 16  |-  10  =/=  1
3130a1i 11 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN0  ->  10  =/=  1 )
32 id 22 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN0  ->  k  e. 
NN0 )
3327, 31, 32geoser 13658 . . . . . . . . . . . . . 14  |-  ( k  e.  NN0  ->  sum_ j  e.  ( 0 ... (
k  -  1 ) ) ( 10 ^
j )  =  ( ( 1  -  ( 10 ^ k ) )  /  ( 1  -  10 ) ) )
34 fzfid 12063 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN0  ->  ( 0 ... ( k  - 
1 ) )  e. 
Fin )
35 elfznn0 11782 . . . . . . . . . . . . . . . . 17  |-  ( j  e.  ( 0 ... ( k  -  1 ) )  ->  j  e.  NN0 )
3635adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( k  e.  NN0  /\  j  e.  ( 0 ... ( k  - 
1 ) ) )  ->  j  e.  NN0 )
37 zexpcl 12161 . . . . . . . . . . . . . . . 16  |-  ( ( 10  e.  ZZ  /\  j  e.  NN0 )  -> 
( 10 ^ j
)  e.  ZZ )
387, 36, 37sylancr 663 . . . . . . . . . . . . . . 15  |-  ( ( k  e.  NN0  /\  j  e.  ( 0 ... ( k  - 
1 ) ) )  ->  ( 10 ^
j )  e.  ZZ )
3934, 38fsumzcl 13537 . . . . . . . . . . . . . 14  |-  ( k  e.  NN0  ->  sum_ j  e.  ( 0 ... (
k  -  1 ) ) ( 10 ^
j )  e.  ZZ )
4033, 39eqeltrrd 2556 . . . . . . . . . . . . 13  |-  ( k  e.  NN0  ->  ( ( 1  -  ( 10
^ k ) )  /  ( 1  -  10 ) )  e.  ZZ )
41 1z 10906 . . . . . . . . . . . . . . . 16  |-  1  e.  ZZ
42 zsubcl 10917 . . . . . . . . . . . . . . . 16  |-  ( ( 1  e.  ZZ  /\  10  e.  ZZ )  -> 
( 1  -  10 )  e.  ZZ )
4341, 7, 42mp2an 672 . . . . . . . . . . . . . . 15  |-  ( 1  -  10 )  e.  ZZ
4443a1i 11 . . . . . . . . . . . . . 14  |-  ( k  e.  NN0  ->  ( 1  -  10 )  e.  ZZ )
4528, 29ltneii 9709 . . . . . . . . . . . . . . . 16  |-  1  =/=  10
4616, 17subeq0i 9911 . . . . . . . . . . . . . . . . 17  |-  ( ( 1  -  10 )  =  0  <->  1  =  10 )
4746necon3bii 2735 . . . . . . . . . . . . . . . 16  |-  ( ( 1  -  10 )  =/=  0  <->  1  =/=  10 )
4845, 47mpbir 209 . . . . . . . . . . . . . . 15  |-  ( 1  -  10 )  =/=  0
4948a1i 11 . . . . . . . . . . . . . 14  |-  ( k  e.  NN0  ->  ( 1  -  10 )  =/=  0 )
507, 32, 10sylancr 663 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN0  ->  ( 10
^ k )  e.  ZZ )
51 zsubcl 10917 . . . . . . . . . . . . . . 15  |-  ( ( 1  e.  ZZ  /\  ( 10 ^ k )  e.  ZZ )  -> 
( 1  -  ( 10 ^ k ) )  e.  ZZ )
5241, 50, 51sylancr 663 . . . . . . . . . . . . . 14  |-  ( k  e.  NN0  ->  ( 1  -  ( 10 ^
k ) )  e.  ZZ )
53 dvdsval2 13867 . . . . . . . . . . . . . 14  |-  ( ( ( 1  -  10 )  e.  ZZ  /\  (
1  -  10 )  =/=  0  /\  (
1  -  ( 10
^ k ) )  e.  ZZ )  -> 
( ( 1  -  10 )  ||  (
1  -  ( 10
^ k ) )  <-> 
( ( 1  -  ( 10 ^ k
) )  /  (
1  -  10 ) )  e.  ZZ ) )
5444, 49, 52, 53syl3anc 1228 . . . . . . . . . . . . 13  |-  ( k  e.  NN0  ->  ( ( 1  -  10 ) 
||  ( 1  -  ( 10 ^ k
) )  <->  ( (
1  -  ( 10
^ k ) )  /  ( 1  -  10 ) )  e.  ZZ ) )
5540, 54mpbird 232 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  ( 1  -  10 )  ||  ( 1  -  ( 10 ^ k ) ) )
5650zcnd 10979 . . . . . . . . . . . . 13  |-  ( k  e.  NN0  ->  ( 10
^ k )  e.  CC )
57 negsubdi2 9890 . . . . . . . . . . . . 13  |-  ( ( ( 10 ^ k
)  e.  CC  /\  1  e.  CC )  -> 
-u ( ( 10
^ k )  - 
1 )  =  ( 1  -  ( 10
^ k ) ) )
5856, 16, 57sylancl 662 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  -u (
( 10 ^ k
)  -  1 )  =  ( 1  -  ( 10 ^ k
) ) )
5955, 58breqtrrd 4479 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  ( 1  -  10 )  ||  -u ( ( 10 ^
k )  -  1 ) )
60 peano2zm 10918 . . . . . . . . . . . . 13  |-  ( ( 10 ^ k )  e.  ZZ  ->  (
( 10 ^ k
)  -  1 )  e.  ZZ )
6150, 60syl 16 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  ( ( 10 ^ k )  -  1 )  e.  ZZ )
62 dvdsnegb 13879 . . . . . . . . . . . 12  |-  ( ( ( 1  -  10 )  e.  ZZ  /\  (
( 10 ^ k
)  -  1 )  e.  ZZ )  -> 
( ( 1  -  10 )  ||  (
( 10 ^ k
)  -  1 )  <-> 
( 1  -  10 )  ||  -u ( ( 10
^ k )  - 
1 ) ) )
6343, 61, 62sylancr 663 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  ( ( 1  -  10 ) 
||  ( ( 10
^ k )  - 
1 )  <->  ( 1  -  10 )  ||  -u ( ( 10 ^
k )  -  1 ) ) )
6459, 63mpbird 232 . . . . . . . . . 10  |-  ( k  e.  NN0  ->  ( 1  -  10 )  ||  ( ( 10 ^
k )  -  1 ) )
65 negdvdsb 13878 . . . . . . . . . . 11  |-  ( ( ( 1  -  10 )  e.  ZZ  /\  (
( 10 ^ k
)  -  1 )  e.  ZZ )  -> 
( ( 1  -  10 )  ||  (
( 10 ^ k
)  -  1 )  <->  -u ( 1  -  10 )  ||  ( ( 10
^ k )  - 
1 ) ) )
6643, 61, 65sylancr 663 . . . . . . . . . 10  |-  ( k  e.  NN0  ->  ( ( 1  -  10 ) 
||  ( ( 10
^ k )  - 
1 )  <->  -u ( 1  -  10 )  ||  ( ( 10 ^
k )  -  1 ) ) )
6764, 66mpbid 210 . . . . . . . . 9  |-  ( k  e.  NN0  ->  -u (
1  -  10 ) 
||  ( ( 10
^ k )  - 
1 ) )
6826, 67syl5eqbrr 4487 . . . . . . . 8  |-  ( k  e.  NN0  ->  ( 3  x.  3 )  ||  ( ( 10 ^
k )  -  1 ) )
691a1i 11 . . . . . . . . 9  |-  ( k  e.  NN0  ->  3  e.  ZZ )
70 muldvds1 13886 . . . . . . . . 9  |-  ( ( 3  e.  ZZ  /\  3  e.  ZZ  /\  (
( 10 ^ k
)  -  1 )  e.  ZZ )  -> 
( ( 3  x.  3 )  ||  (
( 10 ^ k
)  -  1 )  ->  3  ||  (
( 10 ^ k
)  -  1 ) ) )
7169, 69, 61, 70syl3anc 1228 . . . . . . . 8  |-  ( k  e.  NN0  ->  ( ( 3  x.  3 ) 
||  ( ( 10
^ k )  - 
1 )  ->  3  ||  ( ( 10 ^
k )  -  1 ) ) )
7268, 71mpd 15 . . . . . . 7  |-  ( k  e.  NN0  ->  3  ||  ( ( 10 ^
k )  -  1 ) )
739, 72syl 16 . . . . . 6  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  3  ||  ( ( 10 ^
k )  -  1 ) )
741a1i 11 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  3  e.  ZZ )
7511, 60syl 16 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  (
( 10 ^ k
)  -  1 )  e.  ZZ )
76 dvdsmultr2 13897 . . . . . . 7  |-  ( ( 3  e.  ZZ  /\  ( F `  k )  e.  ZZ  /\  (
( 10 ^ k
)  -  1 )  e.  ZZ )  -> 
( 3  ||  (
( 10 ^ k
)  -  1 )  ->  3  ||  (
( F `  k
)  x.  ( ( 10 ^ k )  -  1 ) ) ) )
7774, 5, 75, 76syl3anc 1228 . . . . . 6  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  (
3  ||  ( ( 10 ^ k )  - 
1 )  ->  3  ||  ( ( F `  k )  x.  (
( 10 ^ k
)  -  1 ) ) ) )
7873, 77mpd 15 . . . . 5  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  3  ||  ( ( F `  k )  x.  (
( 10 ^ k
)  -  1 ) ) )
795zcnd 10979 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  ( F `  k )  e.  CC )
8011zcnd 10979 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  ( 10 ^ k )  e.  CC )
8116a1i 11 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  1  e.  CC )
8279, 80, 81subdid 10024 . . . . . 6  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  (
( F `  k
)  x.  ( ( 10 ^ k )  -  1 ) )  =  ( ( ( F `  k )  x.  ( 10 ^
k ) )  -  ( ( F `  k )  x.  1 ) ) )
8379mulid1d 9625 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  (
( F `  k
)  x.  1 )  =  ( F `  k ) )
8483oveq2d 6311 . . . . . 6  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  (
( ( F `  k )  x.  ( 10 ^ k ) )  -  ( ( F `
 k )  x.  1 ) )  =  ( ( ( F `
 k )  x.  ( 10 ^ k
) )  -  ( F `  k )
) )
8582, 84eqtrd 2508 . . . . 5  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  (
( F `  k
)  x.  ( ( 10 ^ k )  -  1 ) )  =  ( ( ( F `  k )  x.  ( 10 ^
k ) )  -  ( F `  k ) ) )
8678, 85breqtrd 4477 . . . 4  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  3  ||  ( ( ( F `
 k )  x.  ( 10 ^ k
) )  -  ( F `  k )
) )
873, 2, 15, 86fsumdvds 13905 . . 3  |-  ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  ->  3  ||  sum_ k  e.  ( 0 ... N
) ( ( ( F `  k )  x.  ( 10 ^
k ) )  -  ( F `  k ) ) )
8812zcnd 10979 . . . 4  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  (
( F `  k
)  x.  ( 10
^ k ) )  e.  CC )
893, 88, 79fsumsub 13583 . . 3  |-  ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  ->  sum_ k  e.  ( 0 ... N ) ( ( ( F `
 k )  x.  ( 10 ^ k
) )  -  ( F `  k )
)  =  ( sum_ k  e.  ( 0 ... N ) ( ( F `  k
)  x.  ( 10
^ k ) )  -  sum_ k  e.  ( 0 ... N ) ( F `  k
) ) )
9087, 89breqtrd 4477 . 2  |-  ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  ->  3  ||  ( sum_ k  e.  ( 0 ... N ) ( ( F `  k
)  x.  ( 10
^ k ) )  -  sum_ k  e.  ( 0 ... N ) ( F `  k
) ) )
91 dvdssub2 13899 . 2  |-  ( ( ( 3  e.  ZZ  /\ 
sum_ k  e.  ( 0 ... N ) ( ( F `  k )  x.  ( 10 ^ k ) )  e.  ZZ  /\  sum_ k  e.  ( 0 ... N ) ( F `  k )  e.  ZZ )  /\  3  ||  ( sum_ k  e.  ( 0 ... N
) ( ( F `
 k )  x.  ( 10 ^ k
) )  -  sum_ k  e.  ( 0 ... N ) ( F `  k ) ) )  ->  (
3  ||  sum_ k  e.  ( 0 ... N
) ( ( F `
 k )  x.  ( 10 ^ k
) )  <->  3  ||  sum_ k  e.  ( 0 ... N ) ( F `  k ) ) )
922, 13, 14, 90, 91syl31anc 1231 1  |-  ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  ->  ( 3  ||  sum_ k  e.  ( 0 ... N ) ( ( F `  k
)  x.  ( 10
^ k ) )  <->  3  ||  sum_ k  e.  ( 0 ... N
) ( F `  k ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4453   -->wf 5590   ` cfv 5594  (class class class)co 6295   CCcc 9502   0cc0 9504   1c1 9505    + caddc 9507    x. cmul 9509    - cmin 9817   -ucneg 9818    / cdiv 10218   3c3 10598   9c9 10604   10c10 10605   NN0cn0 10807   ZZcz 10876   ...cfz 11684   ^cexp 12146   sum_csu 13488    || cdivides 13864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-fz 11685  df-fzo 11805  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-clim 13291  df-sum 13489  df-dvds 13865
This theorem is referenced by: (None)
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