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Theorem 3dvds 12518
Description: A rule for divisibility by 3 of a number written in base 10. (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
StepHypRef Expression
1 3nn 9810 . . . 4  |-  3  e.  NN
21nnzi 9979 . . 3  |-  3  e.  ZZ
32a1i 12 . 2  |-  ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  ->  3  e.  ZZ )
4 fzfid 10966 . . 3  |-  ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  ->  ( 0 ... N )  e.  Fin )
5 ffvelrn 5562 . . . . 5  |-  ( ( F : ( 0 ... N ) --> ZZ 
/\  k  e.  ( 0 ... N ) )  ->  ( F `  k )  e.  ZZ )
65adantll 697 . . . 4  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  ( F `  k )  e.  ZZ )
7 10nn 9817 . . . . . 6  |-  10  e.  NN
87nnzi 9979 . . . . 5  |-  10  e.  ZZ
9 elfznn0 10753 . . . . . 6  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
109adantl 454 . . . . 5  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  k  e.  NN0 )
11 zexpcl 11049 . . . . 5  |-  ( ( 10  e.  ZZ  /\  k  e.  NN0 )  -> 
( 10 ^ k
)  e.  ZZ )
128, 10, 11sylancr 647 . . . 4  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  ( 10 ^ k )  e.  ZZ )
136, 12zmulcld 10055 . . 3  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  (
( F `  k
)  x.  ( 10
^ k ) )  e.  ZZ )
144, 13fsumzcl 12138 . 2  |-  ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  ->  sum_ k  e.  ( 0 ... N ) ( ( F `  k )  x.  ( 10 ^ k ) )  e.  ZZ )
154, 6fsumzcl 12138 . 2  |-  ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  ->  sum_ k  e.  ( 0 ... N ) ( F `  k
)  e.  ZZ )
1613, 6zsubcld 10054 . . . 4  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  (
( ( F `  k )  x.  ( 10 ^ k ) )  -  ( F `  k ) )  e.  ZZ )
17 ax-1cn 8728 . . . . . . . . . . . 12  |-  1  e.  CC
187nncni 9689 . . . . . . . . . . . 12  |-  10  e.  CC
1917, 18negsubdi2i 9065 . . . . . . . . . . 11  |-  -u (
1  -  10 )  =  ( 10  - 
1 )
20 df-10 9745 . . . . . . . . . . . 12  |-  10  =  ( 9  +  1 )
2120oveq1i 5767 . . . . . . . . . . 11  |-  ( 10 
-  1 )  =  ( ( 9  +  1 )  -  1 )
22 9nn 9816 . . . . . . . . . . . . 13  |-  9  e.  NN
2322nncni 9689 . . . . . . . . . . . 12  |-  9  e.  CC
24 pncan 8990 . . . . . . . . . . . 12  |-  ( ( 9  e.  CC  /\  1  e.  CC )  ->  ( ( 9  +  1 )  -  1 )  =  9 )
2523, 17, 24mp2an 656 . . . . . . . . . . 11  |-  ( ( 9  +  1 )  -  1 )  =  9
2619, 21, 253eqtri 2280 . . . . . . . . . 10  |-  -u (
1  -  10 )  =  9
27 3t3e9 9805 . . . . . . . . . 10  |-  ( 3  x.  3 )  =  9
2826, 27eqtr4i 2279 . . . . . . . . 9  |-  -u (
1  -  10 )  =  ( 3  x.  3 )
2918a1i 12 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN0  ->  10  e.  CC )
30 1re 8770 . . . . . . . . . . . . . . . . 17  |-  1  e.  RR
31 1lt10 9862 . . . . . . . . . . . . . . . . 17  |-  1  <  10
3230, 31gtneii 8863 . . . . . . . . . . . . . . . 16  |-  10  =/=  1
3332a1i 12 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN0  ->  10  =/=  1 )
34 id 21 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN0  ->  k  e. 
NN0 )
3529, 33, 34geoser 12252 . . . . . . . . . . . . . 14  |-  ( k  e.  NN0  ->  sum_ j  e.  ( 0 ... (
k  -  1 ) ) ( 10 ^
j )  =  ( ( 1  -  ( 10 ^ k ) )  /  ( 1  -  10 ) ) )
36 fzfid 10966 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN0  ->  ( 0 ... ( k  - 
1 ) )  e. 
Fin )
37 elfznn0 10753 . . . . . . . . . . . . . . . . 17  |-  ( j  e.  ( 0 ... ( k  -  1 ) )  ->  j  e.  NN0 )
3837adantl 454 . . . . . . . . . . . . . . . 16  |-  ( ( k  e.  NN0  /\  j  e.  ( 0 ... ( k  - 
1 ) ) )  ->  j  e.  NN0 )
39 zexpcl 11049 . . . . . . . . . . . . . . . 16  |-  ( ( 10  e.  ZZ  /\  j  e.  NN0 )  -> 
( 10 ^ j
)  e.  ZZ )
408, 38, 39sylancr 647 . . . . . . . . . . . . . . 15  |-  ( ( k  e.  NN0  /\  j  e.  ( 0 ... ( k  - 
1 ) ) )  ->  ( 10 ^
j )  e.  ZZ )
4136, 40fsumzcl 12138 . . . . . . . . . . . . . 14  |-  ( k  e.  NN0  ->  sum_ j  e.  ( 0 ... (
k  -  1 ) ) ( 10 ^
j )  e.  ZZ )
4235, 41eqeltrrd 2331 . . . . . . . . . . . . 13  |-  ( k  e.  NN0  ->  ( ( 1  -  ( 10
^ k ) )  /  ( 1  -  10 ) )  e.  ZZ )
43 1z 9985 . . . . . . . . . . . . . . . 16  |-  1  e.  ZZ
44 zsubcl 9993 . . . . . . . . . . . . . . . 16  |-  ( ( 1  e.  ZZ  /\  10  e.  ZZ )  -> 
( 1  -  10 )  e.  ZZ )
4543, 8, 44mp2an 656 . . . . . . . . . . . . . . 15  |-  ( 1  -  10 )  e.  ZZ
4645a1i 12 . . . . . . . . . . . . . 14  |-  ( k  e.  NN0  ->  ( 1  -  10 )  e.  ZZ )
4730, 31ltneii 8864 . . . . . . . . . . . . . . . 16  |-  1  =/=  10
4817, 18subeq0i 9059 . . . . . . . . . . . . . . . . 17  |-  ( ( 1  -  10 )  =  0  <->  1  =  10 )
4948necon3bii 2451 . . . . . . . . . . . . . . . 16  |-  ( ( 1  -  10 )  =/=  0  <->  1  =/=  10 )
5047, 49mpbir 202 . . . . . . . . . . . . . . 15  |-  ( 1  -  10 )  =/=  0
5150a1i 12 . . . . . . . . . . . . . 14  |-  ( k  e.  NN0  ->  ( 1  -  10 )  =/=  0 )
528, 34, 11sylancr 647 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN0  ->  ( 10
^ k )  e.  ZZ )
53 zsubcl 9993 . . . . . . . . . . . . . . 15  |-  ( ( 1  e.  ZZ  /\  ( 10 ^ k )  e.  ZZ )  -> 
( 1  -  ( 10 ^ k ) )  e.  ZZ )
5443, 52, 53sylancr 647 . . . . . . . . . . . . . 14  |-  ( k  e.  NN0  ->  ( 1  -  ( 10 ^
k ) )  e.  ZZ )
55 divides2 12461 . . . . . . . . . . . . . 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 ) )
5646, 51, 54, 55syl3anc 1187 . . . . . . . . . . . . 13  |-  ( k  e.  NN0  ->  ( ( 1  -  10 ) 
||  ( 1  -  ( 10 ^ k
) )  <->  ( (
1  -  ( 10
^ k ) )  /  ( 1  -  10 ) )  e.  ZZ ) )
5742, 56mpbird 225 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  ( 1  -  10 )  ||  ( 1  -  ( 10 ^ k ) ) )
5852zcnd 10050 . . . . . . . . . . . . 13  |-  ( k  e.  NN0  ->  ( 10
^ k )  e.  CC )
59 negsubdi2 9039 . . . . . . . . . . . . 13  |-  ( ( ( 10 ^ k
)  e.  CC  /\  1  e.  CC )  -> 
-u ( ( 10
^ k )  - 
1 )  =  ( 1  -  ( 10
^ k ) ) )
6058, 17, 59sylancl 646 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  -u (
( 10 ^ k
)  -  1 )  =  ( 1  -  ( 10 ^ k
) ) )
6157, 60breqtrrd 3989 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  ( 1  -  10 )  ||  -u ( ( 10 ^
k )  -  1 ) )
62 peano2zm 9994 . . . . . . . . . . . . 13  |-  ( ( 10 ^ k )  e.  ZZ  ->  (
( 10 ^ k
)  -  1 )  e.  ZZ )
6352, 62syl 17 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  ( ( 10 ^ k )  -  1 )  e.  ZZ )
64 dvdsnegb 12473 . . . . . . . . . . . 12  |-  ( ( ( 1  -  10 )  e.  ZZ  /\  (
( 10 ^ k
)  -  1 )  e.  ZZ )  -> 
( ( 1  -  10 )  ||  (
( 10 ^ k
)  -  1 )  <-> 
( 1  -  10 )  ||  -u ( ( 10
^ k )  - 
1 ) ) )
6545, 63, 64sylancr 647 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  ( ( 1  -  10 ) 
||  ( ( 10
^ k )  - 
1 )  <->  ( 1  -  10 )  ||  -u ( ( 10 ^
k )  -  1 ) ) )
6661, 65mpbird 225 . . . . . . . . . 10  |-  ( k  e.  NN0  ->  ( 1  -  10 )  ||  ( ( 10 ^
k )  -  1 ) )
67 negdvdsb 12472 . . . . . . . . . . 11  |-  ( ( ( 1  -  10 )  e.  ZZ  /\  (
( 10 ^ k
)  -  1 )  e.  ZZ )  -> 
( ( 1  -  10 )  ||  (
( 10 ^ k
)  -  1 )  <->  -u ( 1  -  10 )  ||  ( ( 10
^ k )  - 
1 ) ) )
6845, 63, 67sylancr 647 . . . . . . . . . 10  |-  ( k  e.  NN0  ->  ( ( 1  -  10 ) 
||  ( ( 10
^ k )  - 
1 )  <->  -u ( 1  -  10 )  ||  ( ( 10 ^
k )  -  1 ) ) )
6966, 68mpbid 203 . . . . . . . . 9  |-  ( k  e.  NN0  ->  -u (
1  -  10 ) 
||  ( ( 10
^ k )  - 
1 ) )
7028, 69syl5eqbrr 3997 . . . . . . . 8  |-  ( k  e.  NN0  ->  ( 3  x.  3 )  ||  ( ( 10 ^
k )  -  1 ) )
712a1i 12 . . . . . . . . 9  |-  ( k  e.  NN0  ->  3  e.  ZZ )
72 muldvds1 12480 . . . . . . . . 9  |-  ( ( 3  e.  ZZ  /\  3  e.  ZZ  /\  (
( 10 ^ k
)  -  1 )  e.  ZZ )  -> 
( ( 3  x.  3 )  ||  (
( 10 ^ k
)  -  1 )  ->  3  ||  (
( 10 ^ k
)  -  1 ) ) )
7371, 71, 63, 72syl3anc 1187 . . . . . . . 8  |-  ( k  e.  NN0  ->  ( ( 3  x.  3 ) 
||  ( ( 10
^ k )  - 
1 )  ->  3  ||  ( ( 10 ^
k )  -  1 ) ) )
7470, 73mpd 16 . . . . . . 7  |-  ( k  e.  NN0  ->  3  ||  ( ( 10 ^
k )  -  1 ) )
7510, 74syl 17 . . . . . 6  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  3  ||  ( ( 10 ^
k )  -  1 ) )
762a1i 12 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  3  e.  ZZ )
7712, 62syl 17 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  (
( 10 ^ k
)  -  1 )  e.  ZZ )
78 dvdsmultr2 12491 . . . . . . 7  |-  ( ( 3  e.  ZZ  /\  ( F `  k )  e.  ZZ  /\  (
( 10 ^ k
)  -  1 )  e.  ZZ )  -> 
( 3  ||  (
( 10 ^ k
)  -  1 )  ->  3  ||  (
( F `  k
)  x.  ( ( 10 ^ k )  -  1 ) ) ) )
7976, 6, 77, 78syl3anc 1187 . . . . . 6  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  (
3  ||  ( ( 10 ^ k )  - 
1 )  ->  3  ||  ( ( F `  k )  x.  (
( 10 ^ k
)  -  1 ) ) ) )
8075, 79mpd 16 . . . . 5  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  3  ||  ( ( F `  k )  x.  (
( 10 ^ k
)  -  1 ) ) )
816zcnd 10050 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  ( F `  k )  e.  CC )
8212zcnd 10050 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  ( 10 ^ k )  e.  CC )
8317a1i 12 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  1  e.  CC )
8481, 82, 83subdid 9168 . . . . . 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 ) ) )
8581mulid1d 8785 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  (
( F `  k
)  x.  1 )  =  ( F `  k ) )
8685oveq2d 5773 . . . . . 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 )
) )
8784, 86eqtrd 2288 . . . . 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 ) ) )
8880, 87breqtrd 3987 . . . 4  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  3  ||  ( ( ( F `
 k )  x.  ( 10 ^ k
) )  -  ( F `  k )
) )
894, 3, 16, 88fsumdvds 12499 . . 3  |-  ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  ->  3  ||  sum_ k  e.  ( 0 ... N
) ( ( ( F `  k )  x.  ( 10 ^
k ) )  -  ( F `  k ) ) )
9013zcnd 10050 . . . 4  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  (
( F `  k
)  x.  ( 10
^ k ) )  e.  CC )
914, 90, 81fsumsub 12180 . . 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
) ) )
9289, 91breqtrd 3987 . 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
) ) )
93 dvdssub2 12493 . 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 ) ) )
943, 14, 15, 92, 93syl31anc 1190 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 set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2419   class class class wbr 3963   -->wf 4634   ` cfv 4638  (class class class)co 5757   CCcc 8668   0cc0 8670   1c1 8671    + caddc 8673    x. cmul 8675    - cmin 8970   -ucneg 8971    / cdiv 9356   3c3 9729   9c9 9735   10c10 9736   NN0cn0 9897   ZZcz 9956   ...cfz 10713   ^cexp 11035   sum_csu 12088    || cdivides 12458
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-inf2 7275  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-pre-sup 8748
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-oadd 6416  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-sup 7127  df-oi 7158  df-card 7505  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-2 9737  df-3 9738  df-4 9739  df-5 9740  df-6 9741  df-7 9742  df-8 9743  df-9 9744  df-10 9745  df-n0 9898  df-z 9957  df-uz 10163  df-rp 10287  df-fz 10714  df-fzo 10802  df-seq 10978  df-exp 11036  df-hash 11269  df-cj 11514  df-re 11515  df-im 11516  df-sqr 11650  df-abs 11651  df-clim 11892  df-sum 12089  df-divides 12459
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