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Theorem 3dvds 12607
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
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 3nn 9894 . . . 4  |-  3  e.  NN
21nnzi 10063 . . 3  |-  3  e.  ZZ
32a1i 10 . 2  |-  ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  ->  3  e.  ZZ )
4 fzfid 11051 . . 3  |-  ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  ->  ( 0 ... N )  e.  Fin )
5 ffvelrn 5679 . . . . 5  |-  ( ( F : ( 0 ... N ) --> ZZ 
/\  k  e.  ( 0 ... N ) )  ->  ( F `  k )  e.  ZZ )
65adantll 694 . . . 4  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  ( F `  k )  e.  ZZ )
7 10nn 9901 . . . . . 6  |-  10  e.  NN
87nnzi 10063 . . . . 5  |-  10  e.  ZZ
9 elfznn0 10838 . . . . . 6  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
109adantl 452 . . . . 5  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  k  e.  NN0 )
11 zexpcl 11134 . . . . 5  |-  ( ( 10  e.  ZZ  /\  k  e.  NN0 )  -> 
( 10 ^ k
)  e.  ZZ )
128, 10, 11sylancr 644 . . . 4  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  ( 10 ^ k )  e.  ZZ )
136, 12zmulcld 10139 . . 3  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  (
( F `  k
)  x.  ( 10
^ k ) )  e.  ZZ )
144, 13fsumzcl 12224 . 2  |-  ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  ->  sum_ k  e.  ( 0 ... N ) ( ( F `  k )  x.  ( 10 ^ k ) )  e.  ZZ )
154, 6fsumzcl 12224 . 2  |-  ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  ->  sum_ k  e.  ( 0 ... N ) ( F `  k
)  e.  ZZ )
1613, 6zsubcld 10138 . . . 4  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  (
( ( F `  k )  x.  ( 10 ^ k ) )  -  ( F `  k ) )  e.  ZZ )
17 ax-1cn 8811 . . . . . . . . . . . 12  |-  1  e.  CC
187nncni 9772 . . . . . . . . . . . 12  |-  10  e.  CC
1917, 18negsubdi2i 9148 . . . . . . . . . . 11  |-  -u (
1  -  10 )  =  ( 10  - 
1 )
20 df-10 9828 . . . . . . . . . . . 12  |-  10  =  ( 9  +  1 )
2120oveq1i 5884 . . . . . . . . . . 11  |-  ( 10 
-  1 )  =  ( ( 9  +  1 )  -  1 )
22 9nn 9900 . . . . . . . . . . . . 13  |-  9  e.  NN
2322nncni 9772 . . . . . . . . . . . 12  |-  9  e.  CC
24 pncan 9073 . . . . . . . . . . . 12  |-  ( ( 9  e.  CC  /\  1  e.  CC )  ->  ( ( 9  +  1 )  -  1 )  =  9 )
2523, 17, 24mp2an 653 . . . . . . . . . . 11  |-  ( ( 9  +  1 )  -  1 )  =  9
2619, 21, 253eqtri 2320 . . . . . . . . . 10  |-  -u (
1  -  10 )  =  9
27 3t3e9 9889 . . . . . . . . . 10  |-  ( 3  x.  3 )  =  9
2826, 27eqtr4i 2319 . . . . . . . . 9  |-  -u (
1  -  10 )  =  ( 3  x.  3 )
2918a1i 10 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN0  ->  10  e.  CC )
30 1re 8853 . . . . . . . . . . . . . . . . 17  |-  1  e.  RR
31 1lt10 9946 . . . . . . . . . . . . . . . . 17  |-  1  <  10
3230, 31gtneii 8946 . . . . . . . . . . . . . . . 16  |-  10  =/=  1
3332a1i 10 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN0  ->  10  =/=  1 )
34 id 19 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN0  ->  k  e. 
NN0 )
3529, 33, 34geoser 12341 . . . . . . . . . . . . . 14  |-  ( k  e.  NN0  ->  sum_ j  e.  ( 0 ... (
k  -  1 ) ) ( 10 ^
j )  =  ( ( 1  -  ( 10 ^ k ) )  /  ( 1  -  10 ) ) )
36 fzfid 11051 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN0  ->  ( 0 ... ( k  - 
1 ) )  e. 
Fin )
37 elfznn0 10838 . . . . . . . . . . . . . . . . 17  |-  ( j  e.  ( 0 ... ( k  -  1 ) )  ->  j  e.  NN0 )
3837adantl 452 . . . . . . . . . . . . . . . 16  |-  ( ( k  e.  NN0  /\  j  e.  ( 0 ... ( k  - 
1 ) ) )  ->  j  e.  NN0 )
39 zexpcl 11134 . . . . . . . . . . . . . . . 16  |-  ( ( 10  e.  ZZ  /\  j  e.  NN0 )  -> 
( 10 ^ j
)  e.  ZZ )
408, 38, 39sylancr 644 . . . . . . . . . . . . . . 15  |-  ( ( k  e.  NN0  /\  j  e.  ( 0 ... ( k  - 
1 ) ) )  ->  ( 10 ^
j )  e.  ZZ )
4136, 40fsumzcl 12224 . . . . . . . . . . . . . 14  |-  ( k  e.  NN0  ->  sum_ j  e.  ( 0 ... (
k  -  1 ) ) ( 10 ^
j )  e.  ZZ )
4235, 41eqeltrrd 2371 . . . . . . . . . . . . 13  |-  ( k  e.  NN0  ->  ( ( 1  -  ( 10
^ k ) )  /  ( 1  -  10 ) )  e.  ZZ )
43 1z 10069 . . . . . . . . . . . . . . . 16  |-  1  e.  ZZ
44 zsubcl 10077 . . . . . . . . . . . . . . . 16  |-  ( ( 1  e.  ZZ  /\  10  e.  ZZ )  -> 
( 1  -  10 )  e.  ZZ )
4543, 8, 44mp2an 653 . . . . . . . . . . . . . . 15  |-  ( 1  -  10 )  e.  ZZ
4645a1i 10 . . . . . . . . . . . . . 14  |-  ( k  e.  NN0  ->  ( 1  -  10 )  e.  ZZ )
4730, 31ltneii 8947 . . . . . . . . . . . . . . . 16  |-  1  =/=  10
4817, 18subeq0i 9142 . . . . . . . . . . . . . . . . 17  |-  ( ( 1  -  10 )  =  0  <->  1  =  10 )
4948necon3bii 2491 . . . . . . . . . . . . . . . 16  |-  ( ( 1  -  10 )  =/=  0  <->  1  =/=  10 )
5047, 49mpbir 200 . . . . . . . . . . . . . . 15  |-  ( 1  -  10 )  =/=  0
5150a1i 10 . . . . . . . . . . . . . 14  |-  ( k  e.  NN0  ->  ( 1  -  10 )  =/=  0 )
528, 34, 11sylancr 644 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN0  ->  ( 10
^ k )  e.  ZZ )
53 zsubcl 10077 . . . . . . . . . . . . . . 15  |-  ( ( 1  e.  ZZ  /\  ( 10 ^ k )  e.  ZZ )  -> 
( 1  -  ( 10 ^ k ) )  e.  ZZ )
5443, 52, 53sylancr 644 . . . . . . . . . . . . . 14  |-  ( k  e.  NN0  ->  ( 1  -  ( 10 ^
k ) )  e.  ZZ )
55 dvdsval2 12550 . . . . . . . . . . . . . 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 1182 . . . . . . . . . . . . 13  |-  ( k  e.  NN0  ->  ( ( 1  -  10 ) 
||  ( 1  -  ( 10 ^ k
) )  <->  ( (
1  -  ( 10
^ k ) )  /  ( 1  -  10 ) )  e.  ZZ ) )
5742, 56mpbird 223 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  ( 1  -  10 )  ||  ( 1  -  ( 10 ^ k ) ) )
5852zcnd 10134 . . . . . . . . . . . . 13  |-  ( k  e.  NN0  ->  ( 10
^ k )  e.  CC )
59 negsubdi2 9122 . . . . . . . . . . . . 13  |-  ( ( ( 10 ^ k
)  e.  CC  /\  1  e.  CC )  -> 
-u ( ( 10
^ k )  - 
1 )  =  ( 1  -  ( 10
^ k ) ) )
6058, 17, 59sylancl 643 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  -u (
( 10 ^ k
)  -  1 )  =  ( 1  -  ( 10 ^ k
) ) )
6157, 60breqtrrd 4065 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  ( 1  -  10 )  ||  -u ( ( 10 ^
k )  -  1 ) )
62 peano2zm 10078 . . . . . . . . . . . . 13  |-  ( ( 10 ^ k )  e.  ZZ  ->  (
( 10 ^ k
)  -  1 )  e.  ZZ )
6352, 62syl 15 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  ( ( 10 ^ k )  -  1 )  e.  ZZ )
64 dvdsnegb 12562 . . . . . . . . . . . 12  |-  ( ( ( 1  -  10 )  e.  ZZ  /\  (
( 10 ^ k
)  -  1 )  e.  ZZ )  -> 
( ( 1  -  10 )  ||  (
( 10 ^ k
)  -  1 )  <-> 
( 1  -  10 )  ||  -u ( ( 10
^ k )  - 
1 ) ) )
6545, 63, 64sylancr 644 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  ( ( 1  -  10 ) 
||  ( ( 10
^ k )  - 
1 )  <->  ( 1  -  10 )  ||  -u ( ( 10 ^
k )  -  1 ) ) )
6661, 65mpbird 223 . . . . . . . . . 10  |-  ( k  e.  NN0  ->  ( 1  -  10 )  ||  ( ( 10 ^
k )  -  1 ) )
67 negdvdsb 12561 . . . . . . . . . . 11  |-  ( ( ( 1  -  10 )  e.  ZZ  /\  (
( 10 ^ k
)  -  1 )  e.  ZZ )  -> 
( ( 1  -  10 )  ||  (
( 10 ^ k
)  -  1 )  <->  -u ( 1  -  10 )  ||  ( ( 10
^ k )  - 
1 ) ) )
6845, 63, 67sylancr 644 . . . . . . . . . 10  |-  ( k  e.  NN0  ->  ( ( 1  -  10 ) 
||  ( ( 10
^ k )  - 
1 )  <->  -u ( 1  -  10 )  ||  ( ( 10 ^
k )  -  1 ) ) )
6966, 68mpbid 201 . . . . . . . . 9  |-  ( k  e.  NN0  ->  -u (
1  -  10 ) 
||  ( ( 10
^ k )  - 
1 ) )
7028, 69syl5eqbrr 4073 . . . . . . . 8  |-  ( k  e.  NN0  ->  ( 3  x.  3 )  ||  ( ( 10 ^
k )  -  1 ) )
712a1i 10 . . . . . . . . 9  |-  ( k  e.  NN0  ->  3  e.  ZZ )
72 muldvds1 12569 . . . . . . . . 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 1182 . . . . . . . 8  |-  ( k  e.  NN0  ->  ( ( 3  x.  3 ) 
||  ( ( 10
^ k )  - 
1 )  ->  3  ||  ( ( 10 ^
k )  -  1 ) ) )
7470, 73mpd 14 . . . . . . 7  |-  ( k  e.  NN0  ->  3  ||  ( ( 10 ^
k )  -  1 ) )
7510, 74syl 15 . . . . . 6  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  3  ||  ( ( 10 ^
k )  -  1 ) )
762a1i 10 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  3  e.  ZZ )
7712, 62syl 15 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  (
( 10 ^ k
)  -  1 )  e.  ZZ )
78 dvdsmultr2 12580 . . . . . . 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 1182 . . . . . 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 14 . . . . 5  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  3  ||  ( ( F `  k )  x.  (
( 10 ^ k
)  -  1 ) ) )
816zcnd 10134 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  ( F `  k )  e.  CC )
8212zcnd 10134 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  ( 10 ^ k )  e.  CC )
8317a1i 10 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  1  e.  CC )
8481, 82, 83subdid 9251 . . . . . 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 8868 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  (
( F `  k
)  x.  1 )  =  ( F `  k ) )
8685oveq2d 5890 . . . . . 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 2328 . . . . 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 4063 . . . 4  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  3  ||  ( ( ( F `
 k )  x.  ( 10 ^ k
) )  -  ( F `  k )
) )
894, 3, 16, 88fsumdvds 12588 . . 3  |-  ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  ->  3  ||  sum_ k  e.  ( 0 ... N
) ( ( ( F `  k )  x.  ( 10 ^
k ) )  -  ( F `  k ) ) )
9013zcnd 10134 . . . 4  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  (
( F `  k
)  x.  ( 10
^ k ) )  e.  CC )
914, 90, 81fsumsub 12266 . . 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 4063 . 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 12582 . 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 1185 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 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    - cmin 9053   -ucneg 9054    / cdiv 9439   3c3 9812   9c9 9818   10c10 9819   NN0cn0 9981   ZZcz 10040   ...cfz 10798   ^cexp 11120   sum_csu 12174    || cdivides 12547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-dvds 12548
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