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Theorem bitsres 12940
Description: Restrict the bits of a number to an upper integer set. (Contributed by Mario Carneiro, 5-Sep-2016.)
Assertion
Ref Expression
bitsres  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (bits `  A
)  i^i  ( ZZ>= `  N ) )  =  (bits `  ( ( |_ `  ( A  / 
( 2 ^ N
) ) )  x.  ( 2 ^ N
) ) ) )

Proof of Theorem bitsres
StepHypRef Expression
1 simpl 444 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  A  e.  ZZ )
2 2nn 10089 . . . . . . . 8  |-  2  e.  NN
32a1i 11 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
2  e.  NN )
4 simpr 448 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  N  e.  NN0 )
53, 4nnexpcld 11499 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( 2 ^ N
)  e.  NN )
61, 5zmodcld 11222 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( A  mod  (
2 ^ N ) )  e.  NN0 )
76nn0zd 10329 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( A  mod  (
2 ^ N ) )  e.  ZZ )
87znegcld 10333 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  -u ( A  mod  (
2 ^ N ) )  e.  ZZ )
9 sadadd 12934 . . 3  |-  ( (
-u ( A  mod  ( 2 ^ N
) )  e.  ZZ  /\  A  e.  ZZ )  ->  ( (bits `  -u ( A  mod  (
2 ^ N ) ) ) sadd  (bits `  A ) )  =  (bits `  ( -u ( A  mod  ( 2 ^ N ) )  +  A ) ) )
108, 1, 9syl2anc 643 . 2  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (bits `  -u ( A  mod  ( 2 ^ N ) ) ) sadd  (bits `  A )
)  =  (bits `  ( -u ( A  mod  ( 2 ^ N
) )  +  A
) ) )
11 sadadd 12934 . . . . . 6  |-  ( (
-u ( A  mod  ( 2 ^ N
) )  e.  ZZ  /\  ( A  mod  (
2 ^ N ) )  e.  ZZ )  ->  ( (bits `  -u ( A  mod  (
2 ^ N ) ) ) sadd  (bits `  ( A  mod  ( 2 ^ N ) ) ) )  =  (bits `  ( -u ( A  mod  ( 2 ^ N ) )  +  ( A  mod  (
2 ^ N ) ) ) ) )
128, 7, 11syl2anc 643 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (bits `  -u ( A  mod  ( 2 ^ N ) ) ) sadd  (bits `  ( A  mod  ( 2 ^ N
) ) ) )  =  (bits `  ( -u ( A  mod  (
2 ^ N ) )  +  ( A  mod  ( 2 ^ N ) ) ) ) )
138zcnd 10332 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  -u ( A  mod  (
2 ^ N ) )  e.  CC )
147zcnd 10332 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( A  mod  (
2 ^ N ) )  e.  CC )
1513, 14addcomd 9224 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( -u ( A  mod  ( 2 ^ N
) )  +  ( A  mod  ( 2 ^ N ) ) )  =  ( ( A  mod  ( 2 ^ N ) )  +  -u ( A  mod  ( 2 ^ N
) ) ) )
1614negidd 9357 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( A  mod  ( 2 ^ N
) )  +  -u ( A  mod  ( 2 ^ N ) ) )  =  0 )
1715, 16eqtrd 2436 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( -u ( A  mod  ( 2 ^ N
) )  +  ( A  mod  ( 2 ^ N ) ) )  =  0 )
1817fveq2d 5691 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
(bits `  ( -u ( A  mod  ( 2 ^ N ) )  +  ( A  mod  (
2 ^ N ) ) ) )  =  (bits `  0 )
)
19 0bits 12906 . . . . . 6  |-  (bits ` 
0 )  =  (/)
2018, 19syl6eq 2452 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
(bits `  ( -u ( A  mod  ( 2 ^ N ) )  +  ( A  mod  (
2 ^ N ) ) ) )  =  (/) )
2112, 20eqtrd 2436 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (bits `  -u ( A  mod  ( 2 ^ N ) ) ) sadd  (bits `  ( A  mod  ( 2 ^ N
) ) ) )  =  (/) )
2221oveq1d 6055 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( (bits `  -u ( A  mod  (
2 ^ N ) ) ) sadd  (bits `  ( A  mod  ( 2 ^ N ) ) ) ) sadd  ( (bits `  A )  i^i  ( ZZ>=
`  N ) ) )  =  ( (/) sadd  ( (bits `  A )  i^i  ( ZZ>= `  N )
) ) )
23 bitsss 12893 . . . . . 6  |-  (bits `  -u ( A  mod  (
2 ^ N ) ) )  C_  NN0
2423a1i 11 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
(bits `  -u ( A  mod  ( 2 ^ N ) ) ) 
C_  NN0 )
25 bitsss 12893 . . . . . 6  |-  (bits `  ( A  mod  ( 2 ^ N ) ) )  C_  NN0
2625a1i 11 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
(bits `  ( A  mod  ( 2 ^ N
) ) )  C_  NN0 )
27 inss1 3521 . . . . . 6  |-  ( (bits `  A )  i^i  ( ZZ>=
`  N ) ) 
C_  (bits `  A
)
28 bitsss 12893 . . . . . . 7  |-  (bits `  A )  C_  NN0
2928a1i 11 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
(bits `  A )  C_ 
NN0 )
3027, 29syl5ss 3319 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (bits `  A
)  i^i  ( ZZ>= `  N ) )  C_  NN0 )
31 sadass 12938 . . . . 5  |-  ( ( (bits `  -u ( A  mod  ( 2 ^ N ) ) ) 
C_  NN0  /\  (bits `  ( A  mod  (
2 ^ N ) ) )  C_  NN0  /\  ( (bits `  A )  i^i  ( ZZ>= `  N )
)  C_  NN0 )  -> 
( ( (bits `  -u ( A  mod  (
2 ^ N ) ) ) sadd  (bits `  ( A  mod  ( 2 ^ N ) ) ) ) sadd  ( (bits `  A )  i^i  ( ZZ>=
`  N ) ) )  =  ( (bits `  -u ( A  mod  ( 2 ^ N
) ) ) sadd  (
(bits `  ( A  mod  ( 2 ^ N
) ) ) sadd  (
(bits `  A )  i^i  ( ZZ>= `  N )
) ) ) )
3224, 26, 30, 31syl3anc 1184 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( (bits `  -u ( A  mod  (
2 ^ N ) ) ) sadd  (bits `  ( A  mod  ( 2 ^ N ) ) ) ) sadd  ( (bits `  A )  i^i  ( ZZ>=
`  N ) ) )  =  ( (bits `  -u ( A  mod  ( 2 ^ N
) ) ) sadd  (
(bits `  ( A  mod  ( 2 ^ N
) ) ) sadd  (
(bits `  A )  i^i  ( ZZ>= `  N )
) ) ) )
33 bitsmod 12903 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
(bits `  ( A  mod  ( 2 ^ N
) ) )  =  ( (bits `  A
)  i^i  ( 0..^ N ) ) )
3433oveq1d 6055 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (bits `  ( A  mod  ( 2 ^ N ) ) ) sadd  ( (bits `  A
)  i^i  ( ZZ>= `  N ) ) )  =  ( ( (bits `  A )  i^i  (
0..^ N ) ) sadd  ( (bits `  A
)  i^i  ( ZZ>= `  N ) ) ) )
35 inss1 3521 . . . . . . . . 9  |-  ( (bits `  A )  i^i  (
0..^ N ) ) 
C_  (bits `  A
)
3635, 29syl5ss 3319 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (bits `  A
)  i^i  ( 0..^ N ) )  C_  NN0 )
37 fzouzdisj 11124 . . . . . . . . . . 11  |-  ( ( 0..^ N )  i^i  ( ZZ>= `  N )
)  =  (/)
3837ineq2i 3499 . . . . . . . . . 10  |-  ( (bits `  A )  i^i  (
( 0..^ N )  i^i  ( ZZ>= `  N
) ) )  =  ( (bits `  A
)  i^i  (/) )
39 inindi 3518 . . . . . . . . . 10  |-  ( (bits `  A )  i^i  (
( 0..^ N )  i^i  ( ZZ>= `  N
) ) )  =  ( ( (bits `  A )  i^i  (
0..^ N ) )  i^i  ( (bits `  A )  i^i  ( ZZ>=
`  N ) ) )
40 in0 3613 . . . . . . . . . 10  |-  ( (bits `  A )  i^i  (/) )  =  (/)
4138, 39, 403eqtr3i 2432 . . . . . . . . 9  |-  ( ( (bits `  A )  i^i  ( 0..^ N ) )  i^i  ( (bits `  A )  i^i  ( ZZ>=
`  N ) ) )  =  (/)
4241a1i 11 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( (bits `  A )  i^i  (
0..^ N ) )  i^i  ( (bits `  A )  i^i  ( ZZ>=
`  N ) ) )  =  (/) )
4336, 30, 42saddisj 12932 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( (bits `  A )  i^i  (
0..^ N ) ) sadd  ( (bits `  A
)  i^i  ( ZZ>= `  N ) ) )  =  ( ( (bits `  A )  i^i  (
0..^ N ) )  u.  ( (bits `  A )  i^i  ( ZZ>=
`  N ) ) ) )
44 indi 3547 . . . . . . 7  |-  ( (bits `  A )  i^i  (
( 0..^ N )  u.  ( ZZ>= `  N
) ) )  =  ( ( (bits `  A )  i^i  (
0..^ N ) )  u.  ( (bits `  A )  i^i  ( ZZ>=
`  N ) ) )
4543, 44syl6eqr 2454 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( (bits `  A )  i^i  (
0..^ N ) ) sadd  ( (bits `  A
)  i^i  ( ZZ>= `  N ) ) )  =  ( (bits `  A )  i^i  (
( 0..^ N )  u.  ( ZZ>= `  N
) ) ) )
46 nn0uz 10476 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
474, 46syl6eleq 2494 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  N  e.  ( ZZ>= ` 
0 ) )
48 fzouzsplit 11123 . . . . . . . . . 10  |-  ( N  e.  ( ZZ>= `  0
)  ->  ( ZZ>= ` 
0 )  =  ( ( 0..^ N )  u.  ( ZZ>= `  N
) ) )
4947, 48syl 16 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ZZ>= `  0 )  =  ( ( 0..^ N )  u.  ( ZZ>=
`  N ) ) )
5046, 49syl5eq 2448 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  NN0  =  ( ( 0..^ N )  u.  ( ZZ>=
`  N ) ) )
5128, 50syl5sseq 3356 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
(bits `  A )  C_  ( ( 0..^ N )  u.  ( ZZ>= `  N ) ) )
52 df-ss 3294 . . . . . . 7  |-  ( (bits `  A )  C_  (
( 0..^ N )  u.  ( ZZ>= `  N
) )  <->  ( (bits `  A )  i^i  (
( 0..^ N )  u.  ( ZZ>= `  N
) ) )  =  (bits `  A )
)
5351, 52sylib 189 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (bits `  A
)  i^i  ( (
0..^ N )  u.  ( ZZ>= `  N )
) )  =  (bits `  A ) )
5434, 45, 533eqtrd 2440 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (bits `  ( A  mod  ( 2 ^ N ) ) ) sadd  ( (bits `  A
)  i^i  ( ZZ>= `  N ) ) )  =  (bits `  A
) )
5554oveq2d 6056 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (bits `  -u ( A  mod  ( 2 ^ N ) ) ) sadd  ( (bits `  ( A  mod  ( 2 ^ N ) ) ) sadd  ( (bits `  A
)  i^i  ( ZZ>= `  N ) ) ) )  =  ( (bits `  -u ( A  mod  ( 2 ^ N
) ) ) sadd  (bits `  A ) ) )
5632, 55eqtrd 2436 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( (bits `  -u ( A  mod  (
2 ^ N ) ) ) sadd  (bits `  ( A  mod  ( 2 ^ N ) ) ) ) sadd  ( (bits `  A )  i^i  ( ZZ>=
`  N ) ) )  =  ( (bits `  -u ( A  mod  ( 2 ^ N
) ) ) sadd  (bits `  A ) ) )
57 sadid2 12936 . . . 4  |-  ( ( (bits `  A )  i^i  ( ZZ>= `  N )
)  C_  NN0  ->  ( (/) sadd  ( (bits `  A )  i^i  ( ZZ>= `  N )
) )  =  ( (bits `  A )  i^i  ( ZZ>= `  N )
) )
5830, 57syl 16 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (/) sadd  ( (bits `  A )  i^i  ( ZZ>=
`  N ) ) )  =  ( (bits `  A )  i^i  ( ZZ>=
`  N ) ) )
5922, 56, 583eqtr3d 2444 . 2  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (bits `  -u ( A  mod  ( 2 ^ N ) ) ) sadd  (bits `  A )
)  =  ( (bits `  A )  i^i  ( ZZ>=
`  N ) ) )
601zcnd 10332 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  A  e.  CC )
6113, 60addcomd 9224 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( -u ( A  mod  ( 2 ^ N
) )  +  A
)  =  ( A  +  -u ( A  mod  ( 2 ^ N
) ) ) )
6260, 14negsubd 9373 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( A  +  -u ( A  mod  ( 2 ^ N ) ) )  =  ( A  -  ( A  mod  ( 2 ^ N
) ) ) )
6360, 14subcld 9367 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( A  -  ( A  mod  ( 2 ^ N ) ) )  e.  CC )
645nncnd 9972 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( 2 ^ N
)  e.  CC )
655nnne0d 10000 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( 2 ^ N
)  =/=  0 )
6663, 64, 65divcan1d 9747 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( ( A  -  ( A  mod  ( 2 ^ N
) ) )  / 
( 2 ^ N
) )  x.  (
2 ^ N ) )  =  ( A  -  ( A  mod  ( 2 ^ N
) ) ) )
671zred 10331 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  A  e.  RR )
685nnrpd 10603 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( 2 ^ N
)  e.  RR+ )
69 moddiffl 11214 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( 2 ^ N
)  e.  RR+ )  ->  ( ( A  -  ( A  mod  ( 2 ^ N ) ) )  /  ( 2 ^ N ) )  =  ( |_ `  ( A  /  (
2 ^ N ) ) ) )
7067, 68, 69syl2anc 643 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( A  -  ( A  mod  ( 2 ^ N ) ) )  /  ( 2 ^ N ) )  =  ( |_ `  ( A  /  (
2 ^ N ) ) ) )
7170oveq1d 6055 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( ( A  -  ( A  mod  ( 2 ^ N
) ) )  / 
( 2 ^ N
) )  x.  (
2 ^ N ) )  =  ( ( |_ `  ( A  /  ( 2 ^ N ) ) )  x.  ( 2 ^ N ) ) )
7266, 71eqtr3d 2438 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( A  -  ( A  mod  ( 2 ^ N ) ) )  =  ( ( |_
`  ( A  / 
( 2 ^ N
) ) )  x.  ( 2 ^ N
) ) )
7361, 62, 723eqtrd 2440 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( -u ( A  mod  ( 2 ^ N
) )  +  A
)  =  ( ( |_ `  ( A  /  ( 2 ^ N ) ) )  x.  ( 2 ^ N ) ) )
7473fveq2d 5691 . 2  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
(bits `  ( -u ( A  mod  ( 2 ^ N ) )  +  A ) )  =  (bits `  ( ( |_ `  ( A  / 
( 2 ^ N
) ) )  x.  ( 2 ^ N
) ) ) )
7510, 59, 743eqtr3d 2444 1  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (bits `  A
)  i^i  ( ZZ>= `  N ) )  =  (bits `  ( ( |_ `  ( A  / 
( 2 ^ N
) ) )  x.  ( 2 ^ N
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    u. cun 3278    i^i cin 3279    C_ wss 3280   (/)c0 3588   ` cfv 5413  (class class class)co 6040   RRcr 8945   0cc0 8946    + caddc 8949    x. cmul 8951    - cmin 9247   -ucneg 9248    / cdiv 9633   NNcn 9956   2c2 10005   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   RR+crp 10568  ..^cfzo 11090   |_cfl 11156    mod cmo 11205   ^cexp 11337  bitscbits 12886   sadd csad 12887
This theorem is referenced by:  bitsuz  12941
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-xor 1311  df-tru 1325  df-had 1386  df-cad 1387  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-disj 4143  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435  df-dvds 12808  df-bits 12889  df-sad 12918
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