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Theorem bitsres 12680
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 443 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  A  e.  ZZ )
2 2nn 9893 . . . . . . . 8  |-  2  e.  NN
32a1i 10 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
2  e.  NN )
4 simpr 447 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  N  e.  NN0 )
53, 4nnexpcld 11282 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( 2 ^ N
)  e.  NN )
61, 5zmodcld 11006 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( A  mod  (
2 ^ N ) )  e.  NN0 )
76nn0zd 10131 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( A  mod  (
2 ^ N ) )  e.  ZZ )
87znegcld 10135 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  -u ( A  mod  (
2 ^ N ) )  e.  ZZ )
9 sadadd 12674 . . 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 642 . 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 12674 . . . . . 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 642 . . . . 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 10134 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  -u ( A  mod  (
2 ^ N ) )  e.  CC )
147zcnd 10134 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( A  mod  (
2 ^ N ) )  e.  CC )
1513, 14addcomd 9030 . . . . . . . 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 9163 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( A  mod  ( 2 ^ N
) )  +  -u ( A  mod  ( 2 ^ N ) ) )  =  0 )
1715, 16eqtrd 2328 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( -u ( A  mod  ( 2 ^ N
) )  +  ( A  mod  ( 2 ^ N ) ) )  =  0 )
1817fveq2d 5545 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
(bits `  ( -u ( A  mod  ( 2 ^ N ) )  +  ( A  mod  (
2 ^ N ) ) ) )  =  (bits `  0 )
)
19 0bits 12646 . . . . . 6  |-  (bits ` 
0 )  =  (/)
2018, 19syl6eq 2344 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
(bits `  ( -u ( A  mod  ( 2 ^ N ) )  +  ( A  mod  (
2 ^ N ) ) ) )  =  (/) )
2112, 20eqtrd 2328 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (bits `  -u ( A  mod  ( 2 ^ N ) ) ) sadd  (bits `  ( A  mod  ( 2 ^ N
) ) ) )  =  (/) )
2221oveq1d 5889 . . 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 12633 . . . . . 6  |-  (bits `  -u ( A  mod  (
2 ^ N ) ) )  C_  NN0
2423a1i 10 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
(bits `  -u ( A  mod  ( 2 ^ N ) ) ) 
C_  NN0 )
25 bitsss 12633 . . . . . 6  |-  (bits `  ( A  mod  ( 2 ^ N ) ) )  C_  NN0
2625a1i 10 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
(bits `  ( A  mod  ( 2 ^ N
) ) )  C_  NN0 )
27 inss1 3402 . . . . . 6  |-  ( (bits `  A )  i^i  ( ZZ>=
`  N ) ) 
C_  (bits `  A
)
28 bitsss 12633 . . . . . . 7  |-  (bits `  A )  C_  NN0
2928a1i 10 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
(bits `  A )  C_ 
NN0 )
3027, 29syl5ss 3203 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (bits `  A
)  i^i  ( ZZ>= `  N ) )  C_  NN0 )
31 sadass 12678 . . . . 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 1182 . . . 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 12643 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
(bits `  ( A  mod  ( 2 ^ N
) ) )  =  ( (bits `  A
)  i^i  ( 0..^ N ) ) )
3433oveq1d 5889 . . . . . 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 3402 . . . . . . . . . 10  |-  ( (bits `  A )  i^i  (
0..^ N ) ) 
C_  (bits `  A
)
3635, 29syl5ss 3203 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (bits `  A
)  i^i  ( 0..^ N ) )  C_  NN0 )
37 fzouzdisj 10918 . . . . . . . . . . . 12  |-  ( ( 0..^ N )  i^i  ( ZZ>= `  N )
)  =  (/)
3837ineq2i 3380 . . . . . . . . . . 11  |-  ( (bits `  A )  i^i  (
( 0..^ N )  i^i  ( ZZ>= `  N
) ) )  =  ( (bits `  A
)  i^i  (/) )
39 inindi 3399 . . . . . . . . . . 11  |-  ( (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 3493 . . . . . . . . . . 11  |-  ( (bits `  A )  i^i  (/) )  =  (/)
4138, 39, 403eqtr3i 2324 . . . . . . . . . 10  |-  ( ( (bits `  A )  i^i  ( 0..^ N ) )  i^i  ( (bits `  A )  i^i  ( ZZ>=
`  N ) ) )  =  (/)
4241a1i 10 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( (bits `  A )  i^i  (
0..^ N ) )  i^i  ( (bits `  A )  i^i  ( ZZ>=
`  N ) ) )  =  (/) )
4336, 30, 42saddisj 12672 . . . . . . . 8  |-  ( ( 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 3428 . . . . . . . 8  |-  ( (bits `  A )  i^i  (
( 0..^ N )  u.  ( ZZ>= `  N
) ) )  =  ( ( (bits `  A )  i^i  (
0..^ N ) )  u.  ( (bits `  A )  i^i  ( ZZ>=
`  N ) ) )
4543, 44syl6eqr 2346 . . . . . . 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.  ( ZZ>= `  N
) ) ) )
46 nn0uz 10278 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
474, 46syl6eleq 2386 . . . . . . . . . . 11  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  N  e.  ( ZZ>= ` 
0 ) )
48 fzouzsplit 10917 . . . . . . . . . . 11  |-  ( N  e.  ( ZZ>= `  0
)  ->  ( ZZ>= ` 
0 )  =  ( ( 0..^ N )  u.  ( ZZ>= `  N
) ) )
4947, 48syl 15 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ZZ>= `  0 )  =  ( ( 0..^ N )  u.  ( ZZ>=
`  N ) ) )
5046, 49syl5eq 2340 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  NN0  =  ( ( 0..^ N )  u.  ( ZZ>=
`  N ) ) )
5128, 50syl5sseq 3239 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
(bits `  A )  C_  ( ( 0..^ N )  u.  ( ZZ>= `  N ) ) )
52 df-ss 3179 . . . . . . . 8  |-  ( (bits `  A )  C_  (
( 0..^ N )  u.  ( ZZ>= `  N
) )  <->  ( (bits `  A )  i^i  (
( 0..^ N )  u.  ( ZZ>= `  N
) ) )  =  (bits `  A )
)
5351, 52sylib 188 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (bits `  A
)  i^i  ( (
0..^ N )  u.  ( ZZ>= `  N )
) )  =  (bits `  A ) )
5445, 53eqtrd 2328 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( (bits `  A )  i^i  (
0..^ N ) ) sadd  ( (bits `  A
)  i^i  ( ZZ>= `  N ) ) )  =  (bits `  A
) )
5534, 54eqtrd 2328 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (bits `  ( A  mod  ( 2 ^ N ) ) ) sadd  ( (bits `  A
)  i^i  ( ZZ>= `  N ) ) )  =  (bits `  A
) )
5655oveq2d 5890 . . . 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 ) ) )
5732, 56eqtrd 2328 . . 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 ) ) )
58 sadid2 12676 . . . 4  |-  ( ( (bits `  A )  i^i  ( ZZ>= `  N )
)  C_  NN0  ->  ( (/) sadd  ( (bits `  A )  i^i  ( ZZ>= `  N )
) )  =  ( (bits `  A )  i^i  ( ZZ>= `  N )
) )
5930, 58syl 15 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (/) sadd  ( (bits `  A )  i^i  ( ZZ>=
`  N ) ) )  =  ( (bits `  A )  i^i  ( ZZ>=
`  N ) ) )
6022, 57, 593eqtr3d 2336 . 2  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (bits `  -u ( A  mod  ( 2 ^ N ) ) ) sadd  (bits `  A )
)  =  ( (bits `  A )  i^i  ( ZZ>=
`  N ) ) )
611zcnd 10134 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  A  e.  CC )
6213, 61addcomd 9030 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( -u ( A  mod  ( 2 ^ N
) )  +  A
)  =  ( A  +  -u ( A  mod  ( 2 ^ N
) ) ) )
6361, 14negsubd 9179 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( A  +  -u ( A  mod  ( 2 ^ N ) ) )  =  ( A  -  ( A  mod  ( 2 ^ N
) ) ) )
6461, 14subcld 9173 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( A  -  ( A  mod  ( 2 ^ N ) ) )  e.  CC )
655nncnd 9778 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( 2 ^ N
)  e.  CC )
665nnne0d 9806 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( 2 ^ N
)  =/=  0 )
6764, 65, 66divcan1d 9553 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( ( A  -  ( A  mod  ( 2 ^ N
) ) )  / 
( 2 ^ N
) )  x.  (
2 ^ N ) )  =  ( A  -  ( A  mod  ( 2 ^ N
) ) ) )
681zred 10133 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  A  e.  RR )
695nnrpd 10405 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( 2 ^ N
)  e.  RR+ )
70 moddiffl 10998 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( 2 ^ N
)  e.  RR+ )  ->  ( ( A  -  ( A  mod  ( 2 ^ N ) ) )  /  ( 2 ^ N ) )  =  ( |_ `  ( A  /  (
2 ^ N ) ) ) )
7168, 69, 70syl2anc 642 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( A  -  ( A  mod  ( 2 ^ N ) ) )  /  ( 2 ^ N ) )  =  ( |_ `  ( A  /  (
2 ^ N ) ) ) )
7271oveq1d 5889 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( ( A  -  ( A  mod  ( 2 ^ N
) ) )  / 
( 2 ^ N
) )  x.  (
2 ^ N ) )  =  ( ( |_ `  ( A  /  ( 2 ^ N ) ) )  x.  ( 2 ^ N ) ) )
7367, 72eqtr3d 2330 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( A  -  ( A  mod  ( 2 ^ N ) ) )  =  ( ( |_
`  ( A  / 
( 2 ^ N
) ) )  x.  ( 2 ^ N
) ) )
7462, 63, 733eqtrd 2332 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( -u ( A  mod  ( 2 ^ N
) )  +  A
)  =  ( ( |_ `  ( A  /  ( 2 ^ N ) ) )  x.  ( 2 ^ N ) ) )
7574fveq2d 5545 . 2  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
(bits `  ( -u ( A  mod  ( 2 ^ N ) )  +  A ) )  =  (bits `  ( ( |_ `  ( A  / 
( 2 ^ N
) ) )  x.  ( 2 ^ N
) ) ) )
7610, 60, 753eqtr3d 2336 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 358    = wceq 1632    e. wcel 1696    u. cun 3163    i^i cin 3164    C_ wss 3165   (/)c0 3468   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753    + caddc 8756    x. cmul 8758    - cmin 9053   -ucneg 9054    / cdiv 9439   NNcn 9762   2c2 9811   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   RR+crp 10370  ..^cfzo 10886   |_cfl 10940    mod cmo 10989   ^cexp 11120  bitscbits 12626   sadd csad 12627
This theorem is referenced by:  bitsuz  12681
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-xor 1296  df-tru 1310  df-had 1370  df-cad 1371  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-disj 4010  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-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  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-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  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  df-bits 12629  df-sad 12658
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