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Theorem 3dvds 13717
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 10793 . . 3  |-  3  e.  ZZ
21a1i 11 . 2  |-  ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  ->  3  e.  ZZ )
3 fzfid 11915 . . 3  |-  ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  ->  ( 0 ... N )  e.  Fin )
4 ffvelrn 5953 . . . . 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 10601 . . . . . 6  |-  10  e.  NN
76nnzi 10784 . . . . 5  |-  10  e.  ZZ
8 elfznn0 11601 . . . . . 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 12000 . . . . 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 10867 . . 3  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  (
( F `  k
)  x.  ( 10
^ k ) )  e.  ZZ )
133, 12fsumzcl 13333 . 2  |-  ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  ->  sum_ k  e.  ( 0 ... N ) ( ( F `  k )  x.  ( 10 ^ k ) )  e.  ZZ )
143, 5fsumzcl 13333 . 2  |-  ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  ->  sum_ k  e.  ( 0 ... N ) ( F `  k
)  e.  ZZ )
1512, 5zsubcld 10866 . . . 4  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  (
( ( F `  k )  x.  ( 10 ^ k ) )  -  ( F `  k ) )  e.  ZZ )
16 ax-1cn 9454 . . . . . . . . . . . 12  |-  1  e.  CC
176nncni 10446 . . . . . . . . . . . 12  |-  10  e.  CC
1816, 17negsubdi2i 9808 . . . . . . . . . . 11  |-  -u (
1  -  10 )  =  ( 10  - 
1 )
19 df-10 10502 . . . . . . . . . . . 12  |-  10  =  ( 9  +  1 )
2019oveq1i 6213 . . . . . . . . . . 11  |-  ( 10 
-  1 )  =  ( ( 9  +  1 )  -  1 )
21 9cn 10523 . . . . . . . . . . . 12  |-  9  e.  CC
22 pncan 9730 . . . . . . . . . . . 12  |-  ( ( 9  e.  CC  /\  1  e.  CC )  ->  ( ( 9  +  1 )  -  1 )  =  9 )
2321, 16, 22mp2an 672 . . . . . . . . . . 11  |-  ( ( 9  +  1 )  -  1 )  =  9
2418, 20, 233eqtri 2487 . . . . . . . . . 10  |-  -u (
1  -  10 )  =  9
25 3t3e9 10588 . . . . . . . . . 10  |-  ( 3  x.  3 )  =  9
2624, 25eqtr4i 2486 . . . . . . . . 9  |-  -u (
1  -  10 )  =  ( 3  x.  3 )
2717a1i 11 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN0  ->  10  e.  CC )
28 1re 9499 . . . . . . . . . . . . . . . . 17  |-  1  e.  RR
29 1lt10 10646 . . . . . . . . . . . . . . . . 17  |-  1  <  10
3028, 29gtneii 9600 . . . . . . . . . . . . . . . 16  |-  10  =/=  1
3130a1i 11 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN0  ->  10  =/=  1 )
32 id 22 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN0  ->  k  e. 
NN0 )
3327, 31, 32geoser 13450 . . . . . . . . . . . . . 14  |-  ( k  e.  NN0  ->  sum_ j  e.  ( 0 ... (
k  -  1 ) ) ( 10 ^
j )  =  ( ( 1  -  ( 10 ^ k ) )  /  ( 1  -  10 ) ) )
34 fzfid 11915 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN0  ->  ( 0 ... ( k  - 
1 ) )  e. 
Fin )
35 elfznn0 11601 . . . . . . . . . . . . . . . . 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 12000 . . . . . . . . . . . . . . . 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 13333 . . . . . . . . . . . . . 14  |-  ( k  e.  NN0  ->  sum_ j  e.  ( 0 ... (
k  -  1 ) ) ( 10 ^
j )  e.  ZZ )
4033, 39eqeltrrd 2543 . . . . . . . . . . . . 13  |-  ( k  e.  NN0  ->  ( ( 1  -  ( 10
^ k ) )  /  ( 1  -  10 ) )  e.  ZZ )
41 1z 10790 . . . . . . . . . . . . . . . 16  |-  1  e.  ZZ
42 zsubcl 10801 . . . . . . . . . . . . . . . 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 9601 . . . . . . . . . . . . . . . 16  |-  1  =/=  10
4616, 17subeq0i 9802 . . . . . . . . . . . . . . . . 17  |-  ( ( 1  -  10 )  =  0  <->  1  =  10 )
4746necon3bii 2720 . . . . . . . . . . . . . . . 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 10801 . . . . . . . . . . . . . . 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 13659 . . . . . . . . . . . . . 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 1219 . . . . . . . . . . . . 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 10862 . . . . . . . . . . . . 13  |-  ( k  e.  NN0  ->  ( 10
^ k )  e.  CC )
57 negsubdi2 9782 . . . . . . . . . . . . 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 4429 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  ( 1  -  10 )  ||  -u ( ( 10 ^
k )  -  1 ) )
60 peano2zm 10802 . . . . . . . . . . . . 13  |-  ( ( 10 ^ k )  e.  ZZ  ->  (
( 10 ^ k
)  -  1 )  e.  ZZ )
6150, 60syl 16 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  ( ( 10 ^ k )  -  1 )  e.  ZZ )
62 dvdsnegb 13671 . . . . . . . . . . . 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 13670 . . . . . . . . . . 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 4437 . . . . . . . 8  |-  ( k  e.  NN0  ->  ( 3  x.  3 )  ||  ( ( 10 ^
k )  -  1 ) )
691a1i 11 . . . . . . . . 9  |-  ( k  e.  NN0  ->  3  e.  ZZ )
70 muldvds1 13678 . . . . . . . . 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 1219 . . . . . . . 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 13689 . . . . . . 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 1219 . . . . . 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 10862 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  ( F `  k )  e.  CC )
8011zcnd 10862 . . . . . . 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 9914 . . . . . 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 9517 . . . . . . 7  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  (
( F `  k
)  x.  1 )  =  ( F `  k ) )
8483oveq2d 6219 . . . . . 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 2495 . . . . 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 4427 . . . 4  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  3  ||  ( ( ( F `
 k )  x.  ( 10 ^ k
) )  -  ( F `  k )
) )
873, 2, 15, 86fsumdvds 13697 . . 3  |-  ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  ->  3  ||  sum_ k  e.  ( 0 ... N
) ( ( ( F `  k )  x.  ( 10 ^
k ) )  -  ( F `  k ) ) )
8812zcnd 10862 . . . 4  |-  ( ( ( N  e.  NN0  /\  F : ( 0 ... N ) --> ZZ )  /\  k  e.  ( 0 ... N
) )  ->  (
( F `  k
)  x.  ( 10
^ k ) )  e.  CC )
893, 88, 79fsumsub 13376 . . 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 4427 . 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 13691 . 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 1222 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 1370    e. wcel 1758    =/= wne 2648   class class class wbr 4403   -->wf 5525   ` cfv 5529  (class class class)co 6203   CCcc 9394   0cc0 9396   1c1 9397    + caddc 9399    x. cmul 9401    - cmin 9709   -ucneg 9710    / cdiv 10107   3c3 10486   9c9 10492   10c10 10493   NN0cn0 10693   ZZcz 10760   ...cfz 11557   ^cexp 11985   sum_csu 13284    || cdivides 13656
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 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7961  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-sup 7805  df-oi 7838  df-card 8223  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-2 10494  df-3 10495  df-4 10496  df-5 10497  df-6 10498  df-7 10499  df-8 10500  df-9 10501  df-10 10502  df-n0 10694  df-z 10761  df-uz 10976  df-rp 11106  df-fz 11558  df-fzo 11669  df-seq 11927  df-exp 11986  df-hash 12224  df-cj 12709  df-re 12710  df-im 12711  df-sqr 12845  df-abs 12846  df-clim 13087  df-sum 13285  df-dvds 13657
This theorem is referenced by: (None)
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