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Theorem bitsres 12664
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 9877 . . . . . . . 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 11266 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( 2 ^ N
)  e.  NN )
61, 5zmodcld 10990 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( A  mod  (
2 ^ N ) )  e.  NN0 )
76nn0zd 10115 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( A  mod  (
2 ^ N ) )  e.  ZZ )
87znegcld 10119 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  -u ( A  mod  (
2 ^ N ) )  e.  ZZ )
9 sadadd 12658 . . 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 12658 . . . . . 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 10118 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  -u ( A  mod  (
2 ^ N ) )  e.  CC )
147zcnd 10118 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( A  mod  (
2 ^ N ) )  e.  CC )
1513, 14addcomd 9014 . . . . . . . 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 9147 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( A  mod  ( 2 ^ N
) )  +  -u ( A  mod  ( 2 ^ N ) ) )  =  0 )
1715, 16eqtrd 2315 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( -u ( A  mod  ( 2 ^ N
) )  +  ( A  mod  ( 2 ^ N ) ) )  =  0 )
1817fveq2d 5529 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
(bits `  ( -u ( A  mod  ( 2 ^ N ) )  +  ( A  mod  (
2 ^ N ) ) ) )  =  (bits `  0 )
)
19 0bits 12630 . . . . . 6  |-  (bits ` 
0 )  =  (/)
2018, 19syl6eq 2331 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
(bits `  ( -u ( A  mod  ( 2 ^ N ) )  +  ( A  mod  (
2 ^ N ) ) ) )  =  (/) )
2112, 20eqtrd 2315 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (bits `  -u ( A  mod  ( 2 ^ N ) ) ) sadd  (bits `  ( A  mod  ( 2 ^ N
) ) ) )  =  (/) )
2221oveq1d 5873 . . 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 12617 . . . . . 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 12617 . . . . . 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 3389 . . . . . 6  |-  ( (bits `  A )  i^i  ( ZZ>=
`  N ) ) 
C_  (bits `  A
)
28 bitsss 12617 . . . . . . 7  |-  (bits `  A )  C_  NN0
2928a1i 10 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
(bits `  A )  C_ 
NN0 )
3027, 29syl5ss 3190 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (bits `  A
)  i^i  ( ZZ>= `  N ) )  C_  NN0 )
31 sadass 12662 . . . . 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 12627 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
(bits `  ( A  mod  ( 2 ^ N
) ) )  =  ( (bits `  A
)  i^i  ( 0..^ N ) ) )
3433oveq1d 5873 . . . . . 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 3389 . . . . . . . . . 10  |-  ( (bits `  A )  i^i  (
0..^ N ) ) 
C_  (bits `  A
)
3635, 29syl5ss 3190 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (bits `  A
)  i^i  ( 0..^ N ) )  C_  NN0 )
37 fzouzdisj 10902 . . . . . . . . . . . 12  |-  ( ( 0..^ N )  i^i  ( ZZ>= `  N )
)  =  (/)
3837ineq2i 3367 . . . . . . . . . . 11  |-  ( (bits `  A )  i^i  (
( 0..^ N )  i^i  ( ZZ>= `  N
) ) )  =  ( (bits `  A
)  i^i  (/) )
39 inindi 3386 . . . . . . . . . . 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 3480 . . . . . . . . . . 11  |-  ( (bits `  A )  i^i  (/) )  =  (/)
4138, 39, 403eqtr3i 2311 . . . . . . . . . 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 12656 . . . . . . . 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 3415 . . . . . . . 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 2333 . . . . . . 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 10262 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
474, 46syl6eleq 2373 . . . . . . . . . . 11  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  N  e.  ( ZZ>= ` 
0 ) )
48 fzouzsplit 10901 . . . . . . . . . . 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 2327 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  NN0  =  ( ( 0..^ N )  u.  ( ZZ>=
`  N ) ) )
5128, 50syl5sseq 3226 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
(bits `  A )  C_  ( ( 0..^ N )  u.  ( ZZ>= `  N ) ) )
52 df-ss 3166 . . . . . . . 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 2315 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( (bits `  A )  i^i  (
0..^ N ) ) sadd  ( (bits `  A
)  i^i  ( ZZ>= `  N ) ) )  =  (bits `  A
) )
5534, 54eqtrd 2315 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (bits `  ( A  mod  ( 2 ^ N ) ) ) sadd  ( (bits `  A
)  i^i  ( ZZ>= `  N ) ) )  =  (bits `  A
) )
5655oveq2d 5874 . . . 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 2315 . . 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 12660 . . . 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 2323 . 2  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (bits `  -u ( A  mod  ( 2 ^ N ) ) ) sadd  (bits `  A )
)  =  ( (bits `  A )  i^i  ( ZZ>=
`  N ) ) )
611zcnd 10118 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  A  e.  CC )
6213, 61addcomd 9014 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( -u ( A  mod  ( 2 ^ N
) )  +  A
)  =  ( A  +  -u ( A  mod  ( 2 ^ N
) ) ) )
6361, 14negsubd 9163 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( A  +  -u ( A  mod  ( 2 ^ N ) ) )  =  ( A  -  ( A  mod  ( 2 ^ N
) ) ) )
6461, 14subcld 9157 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( A  -  ( A  mod  ( 2 ^ N ) ) )  e.  CC )
655nncnd 9762 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( 2 ^ N
)  e.  CC )
665nnne0d 9790 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( 2 ^ N
)  =/=  0 )
6764, 65, 66divcan1d 9537 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( ( A  -  ( A  mod  ( 2 ^ N
) ) )  / 
( 2 ^ N
) )  x.  (
2 ^ N ) )  =  ( A  -  ( A  mod  ( 2 ^ N
) ) ) )
681zred 10117 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  A  e.  RR )
695nnrpd 10389 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( 2 ^ N
)  e.  RR+ )
70 moddiffl 10982 . . . . . . 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 5873 . . . . 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 2317 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( A  -  ( A  mod  ( 2 ^ N ) ) )  =  ( ( |_
`  ( A  / 
( 2 ^ N
) ) )  x.  ( 2 ^ N
) ) )
7462, 63, 733eqtrd 2319 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( -u ( A  mod  ( 2 ^ N
) )  +  A
)  =  ( ( |_ `  ( A  /  ( 2 ^ N ) ) )  x.  ( 2 ^ N ) ) )
7574fveq2d 5529 . 2  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
(bits `  ( -u ( A  mod  ( 2 ^ N ) )  +  A ) )  =  (bits `  ( ( |_ `  ( A  / 
( 2 ^ N
) ) )  x.  ( 2 ^ N
) ) ) )
7610, 60, 753eqtr3d 2323 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 1623    e. wcel 1684    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737    + caddc 8740    x. cmul 8742    - cmin 9037   -ucneg 9038    / cdiv 9423   NNcn 9746   2c2 9795   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   RR+crp 10354  ..^cfzo 10870   |_cfl 10924    mod cmo 10973   ^cexp 11104  bitscbits 12610   sadd csad 12611
This theorem is referenced by:  bitsuz  12665
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-disj 3994  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-dvds 12532  df-bits 12613  df-sad 12642
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