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Theorem bitsres 12985
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 10133 . . . . . . . 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 11544 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( 2 ^ N
)  e.  NN )
61, 5zmodcld 11267 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( A  mod  (
2 ^ N ) )  e.  NN0 )
76nn0zd 10373 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( A  mod  (
2 ^ N ) )  e.  ZZ )
87znegcld 10377 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  -u ( A  mod  (
2 ^ N ) )  e.  ZZ )
9 sadadd 12979 . . 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 12979 . . . . . 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 10376 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  -u ( A  mod  (
2 ^ N ) )  e.  CC )
147zcnd 10376 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( A  mod  (
2 ^ N ) )  e.  CC )
1513, 14addcomd 9268 . . . . . . . 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 9401 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( A  mod  ( 2 ^ N
) )  +  -u ( A  mod  ( 2 ^ N ) ) )  =  0 )
1715, 16eqtrd 2468 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( -u ( A  mod  ( 2 ^ N
) )  +  ( A  mod  ( 2 ^ N ) ) )  =  0 )
1817fveq2d 5732 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
(bits `  ( -u ( A  mod  ( 2 ^ N ) )  +  ( A  mod  (
2 ^ N ) ) ) )  =  (bits `  0 )
)
19 0bits 12951 . . . . . 6  |-  (bits ` 
0 )  =  (/)
2018, 19syl6eq 2484 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
(bits `  ( -u ( A  mod  ( 2 ^ N ) )  +  ( A  mod  (
2 ^ N ) ) ) )  =  (/) )
2112, 20eqtrd 2468 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (bits `  -u ( A  mod  ( 2 ^ N ) ) ) sadd  (bits `  ( A  mod  ( 2 ^ N
) ) ) )  =  (/) )
2221oveq1d 6096 . . 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 12938 . . . . . 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 12938 . . . . . 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 3561 . . . . . 6  |-  ( (bits `  A )  i^i  ( ZZ>=
`  N ) ) 
C_  (bits `  A
)
28 bitsss 12938 . . . . . . 7  |-  (bits `  A )  C_  NN0
2928a1i 11 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
(bits `  A )  C_ 
NN0 )
3027, 29syl5ss 3359 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (bits `  A
)  i^i  ( ZZ>= `  N ) )  C_  NN0 )
31 sadass 12983 . . . . 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 12948 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
(bits `  ( A  mod  ( 2 ^ N
) ) )  =  ( (bits `  A
)  i^i  ( 0..^ N ) ) )
3433oveq1d 6096 . . . . . 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 3561 . . . . . . . . 9  |-  ( (bits `  A )  i^i  (
0..^ N ) ) 
C_  (bits `  A
)
3635, 29syl5ss 3359 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (bits `  A
)  i^i  ( 0..^ N ) )  C_  NN0 )
37 fzouzdisj 11169 . . . . . . . . . . 11  |-  ( ( 0..^ N )  i^i  ( ZZ>= `  N )
)  =  (/)
3837ineq2i 3539 . . . . . . . . . 10  |-  ( (bits `  A )  i^i  (
( 0..^ N )  i^i  ( ZZ>= `  N
) ) )  =  ( (bits `  A
)  i^i  (/) )
39 inindi 3558 . . . . . . . . . 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 3653 . . . . . . . . . 10  |-  ( (bits `  A )  i^i  (/) )  =  (/)
4138, 39, 403eqtr3i 2464 . . . . . . . . 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 12977 . . . . . . 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 3587 . . . . . . 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 2486 . . . . . 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 10520 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
474, 46syl6eleq 2526 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  N  e.  ( ZZ>= ` 
0 ) )
48 fzouzsplit 11168 . . . . . . . . . 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 2480 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  NN0  =  ( ( 0..^ N )  u.  ( ZZ>=
`  N ) ) )
5128, 50syl5sseq 3396 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
(bits `  A )  C_  ( ( 0..^ N )  u.  ( ZZ>= `  N ) ) )
52 df-ss 3334 . . . . . . 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 2472 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (bits `  ( A  mod  ( 2 ^ N ) ) ) sadd  ( (bits `  A
)  i^i  ( ZZ>= `  N ) ) )  =  (bits `  A
) )
5554oveq2d 6097 . . . 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 2468 . . 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 12981 . . . 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 2476 . 2  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (bits `  -u ( A  mod  ( 2 ^ N ) ) ) sadd  (bits `  A )
)  =  ( (bits `  A )  i^i  ( ZZ>=
`  N ) ) )
601zcnd 10376 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  A  e.  CC )
6113, 60addcomd 9268 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( -u ( A  mod  ( 2 ^ N
) )  +  A
)  =  ( A  +  -u ( A  mod  ( 2 ^ N
) ) ) )
6260, 14negsubd 9417 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( A  +  -u ( A  mod  ( 2 ^ N ) ) )  =  ( A  -  ( A  mod  ( 2 ^ N
) ) ) )
6360, 14subcld 9411 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( A  -  ( A  mod  ( 2 ^ N ) ) )  e.  CC )
645nncnd 10016 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( 2 ^ N
)  e.  CC )
655nnne0d 10044 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( 2 ^ N
)  =/=  0 )
6663, 64, 65divcan1d 9791 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( ( A  -  ( A  mod  ( 2 ^ N
) ) )  / 
( 2 ^ N
) )  x.  (
2 ^ N ) )  =  ( A  -  ( A  mod  ( 2 ^ N
) ) ) )
671zred 10375 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  A  e.  RR )
685nnrpd 10647 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( 2 ^ N
)  e.  RR+ )
69 moddiffl 11259 . . . . . . 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 6096 . . . . 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 2470 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( A  -  ( A  mod  ( 2 ^ N ) ) )  =  ( ( |_
`  ( A  / 
( 2 ^ N
) ) )  x.  ( 2 ^ N
) ) )
7361, 62, 723eqtrd 2472 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( -u ( A  mod  ( 2 ^ N
) )  +  A
)  =  ( ( |_ `  ( A  /  ( 2 ^ N ) ) )  x.  ( 2 ^ N ) ) )
7473fveq2d 5732 . 2  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
(bits `  ( -u ( A  mod  ( 2 ^ N ) )  +  A ) )  =  (bits `  ( ( |_ `  ( A  / 
( 2 ^ N
) ) )  x.  ( 2 ^ N
) ) ) )
7510, 59, 743eqtr3d 2476 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 1652    e. wcel 1725    u. cun 3318    i^i cin 3319    C_ wss 3320   (/)c0 3628   ` cfv 5454  (class class class)co 6081   RRcr 8989   0cc0 8990    + caddc 8993    x. cmul 8995    - cmin 9291   -ucneg 9292    / cdiv 9677   NNcn 10000   2c2 10049   NN0cn0 10221   ZZcz 10282   ZZ>=cuz 10488   RR+crp 10612  ..^cfzo 11135   |_cfl 11201    mod cmo 11250   ^cexp 11382  bitscbits 12931   sadd csad 12932
This theorem is referenced by:  bitsuz  12986
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-xor 1314  df-tru 1328  df-had 1389  df-cad 1390  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-disj 4183  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fz 11044  df-fzo 11136  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-sum 12480  df-dvds 12853  df-bits 12934  df-sad 12963
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