MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  3dvds Unicode version

Theorem 3dvds 12554
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 9846 . . . 4  |-  3  e.  NN
21nnzi 10015 . . 3  |-  3  e.  ZZ
32a1i 12 . 2  |-  ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  ->  3  e.  ZZ )
4 fzfid 11002 . . 3  |-  ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  ->  ( 0 ... N )  e.  Fin )
5 ffvelrn 5597 . . . . 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 9853 . . . . . 6  |-  10  e.  NN
87nnzi 10015 . . . . 5  |-  10  e.  ZZ
9 elfznn0 10789 . . . . . 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 11085 . . . . 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 10091 . . 3  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  (
( F `  k
)  x.  ( 10
^ k ) )  e.  ZZ )
144, 13fsumzcl 12174 . 2  |-  ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  ->  sum_ k  e.  ( 0 ... N ) ( ( F `  k )  x.  ( 10 ^ k ) )  e.  ZZ )
154, 6fsumzcl 12174 . 2  |-  ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  ->  sum_ k  e.  ( 0 ... N ) ( F `  k
)  e.  ZZ )
1613, 6zsubcld 10090 . . . 4  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  (
( ( F `  k )  x.  ( 10 ^ k ) )  -  ( F `  k ) )  e.  ZZ )
17 ax-1cn 8763 . . . . . . . . . . . 12  |-  1  e.  CC
187nncni 9724 . . . . . . . . . . . 12  |-  10  e.  CC
1917, 18negsubdi2i 9100 . . . . . . . . . . 11  |-  -u (
1  -  10 )  =  ( 10  - 
1 )
20 df-10 9780 . . . . . . . . . . . 12  |-  10  =  ( 9  +  1 )
2120oveq1i 5802 . . . . . . . . . . 11  |-  ( 10 
-  1 )  =  ( ( 9  +  1 )  -  1 )
22 9nn 9852 . . . . . . . . . . . . 13  |-  9  e.  NN
2322nncni 9724 . . . . . . . . . . . 12  |-  9  e.  CC
24 pncan 9025 . . . . . . . . . . . 12  |-  ( ( 9  e.  CC  /\  1  e.  CC )  ->  ( ( 9  +  1 )  -  1 )  =  9 )
2523, 17, 24mp2an 656 . . . . . . . . . . 11  |-  ( ( 9  +  1 )  -  1 )  =  9
2619, 21, 253eqtri 2282 . . . . . . . . . 10  |-  -u (
1  -  10 )  =  9
27 3t3e9 9841 . . . . . . . . . 10  |-  ( 3  x.  3 )  =  9
2826, 27eqtr4i 2281 . . . . . . . . 9  |-  -u (
1  -  10 )  =  ( 3  x.  3 )
2918a1i 12 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN0  ->  10  e.  CC )
30 1re 8805 . . . . . . . . . . . . . . . . 17  |-  1  e.  RR
31 1lt10 9898 . . . . . . . . . . . . . . . . 17  |-  1  <  10
3230, 31gtneii 8898 . . . . . . . . . . . . . . . 16  |-  10  =/=  1
3332a1i 12 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN0  ->  10  =/=  1 )
34 id 21 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN0  ->  k  e. 
NN0 )
3529, 33, 34geoser 12288 . . . . . . . . . . . . . 14  |-  ( k  e.  NN0  ->  sum_ j  e.  ( 0 ... (
k  -  1 ) ) ( 10 ^
j )  =  ( ( 1  -  ( 10 ^ k ) )  /  ( 1  -  10 ) ) )
36 fzfid 11002 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN0  ->  ( 0 ... ( k  - 
1 ) )  e. 
Fin )
37 elfznn0 10789 . . . . . . . . . . . . . . . . 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 11085 . . . . . . . . . . . . . . . 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 12174 . . . . . . . . . . . . . 14  |-  ( k  e.  NN0  ->  sum_ j  e.  ( 0 ... (
k  -  1 ) ) ( 10 ^
j )  e.  ZZ )
4235, 41eqeltrrd 2333 . . . . . . . . . . . . 13  |-  ( k  e.  NN0  ->  ( ( 1  -  ( 10
^ k ) )  /  ( 1  -  10 ) )  e.  ZZ )
43 1z 10021 . . . . . . . . . . . . . . . 16  |-  1  e.  ZZ
44 zsubcl 10029 . . . . . . . . . . . . . . . 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 8899 . . . . . . . . . . . . . . . 16  |-  1  =/=  10
4817, 18subeq0i 9094 . . . . . . . . . . . . . . . . 17  |-  ( ( 1  -  10 )  =  0  <->  1  =  10 )
4948necon3bii 2453 . . . . . . . . . . . . . . . 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 10029 . . . . . . . . . . . . . . 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 12497 . . . . . . . . . . . . . 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 10086 . . . . . . . . . . . . 13  |-  ( k  e.  NN0  ->  ( 10
^ k )  e.  CC )
59 negsubdi2 9074 . . . . . . . . . . . . 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 4023 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  ( 1  -  10 )  ||  -u ( ( 10 ^
k )  -  1 ) )
62 peano2zm 10030 . . . . . . . . . . . . 13  |-  ( ( 10 ^ k )  e.  ZZ  ->  (
( 10 ^ k
)  -  1 )  e.  ZZ )
6352, 62syl 17 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  ( ( 10 ^ k )  -  1 )  e.  ZZ )
64 dvdsnegb 12509 . . . . . . . . . . . 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 12508 . . . . . . . . . . 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 4031 . . . . . . . 8  |-  ( k  e.  NN0  ->  ( 3  x.  3 )  ||  ( ( 10 ^
k )  -  1 ) )
712a1i 12 . . . . . . . . 9  |-  ( k  e.  NN0  ->  3  e.  ZZ )
72 muldvds1 12516 . . . . . . . . 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 12527 . . . . . . 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 10086 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  ( F `  k )  e.  CC )
8212zcnd 10086 . . . . . . 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 9203 . . . . . 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 8820 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  (
( F `  k
)  x.  1 )  =  ( F `  k ) )
8685oveq2d 5808 . . . . . 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 2290 . . . . 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 4021 . . . 4  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  3  ||  ( ( ( F `
 k )  x.  ( 10 ^ k
) )  -  ( F `  k )
) )
894, 3, 16, 88fsumdvds 12535 . . 3  |-  ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  ->  3  ||  sum_ k  e.  ( 0 ... N
) ( ( ( F `  k )  x.  ( 10 ^
k ) )  -  ( F `  k ) ) )
9013zcnd 10086 . . . 4  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  (
( F `  k
)  x.  ( 10
^ k ) )  e.  CC )
914, 90, 81fsumsub 12216 . . 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 4021 . 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 12529 . 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 2421   class class class wbr 3997   -->wf 4669   ` cfv 4673  (class class class)co 5792   CCcc 8703   0cc0 8705   1c1 8706    + caddc 8708    x. cmul 8710    - cmin 9005   -ucneg 9006    / cdiv 9391   3c3 9764   9c9 9770   10c10 9771   NN0cn0 9933   ZZcz 9992   ...cfz 10749   ^cexp 11071   sum_csu 12124    || cdivides 12494
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 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-inf2 7310  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-sup 7162  df-oi 7193  df-card 7540  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-4 9774  df-5 9775  df-6 9776  df-7 9777  df-8 9778  df-9 9779  df-10 9780  df-n0 9934  df-z 9993  df-uz 10199  df-rp 10323  df-fz 10750  df-fzo 10838  df-seq 11014  df-exp 11072  df-hash 11305  df-cj 11550  df-re 11551  df-im 11552  df-sqr 11686  df-abs 11687  df-clim 11928  df-sum 12125  df-divides 12495
  Copyright terms: Public domain W3C validator