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Theorem bitsres 12755
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 9966 . . . . . . . 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 11356 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( 2 ^ N
)  e.  NN )
61, 5zmodcld 11079 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( A  mod  (
2 ^ N ) )  e.  NN0 )
76nn0zd 10204 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( A  mod  (
2 ^ N ) )  e.  ZZ )
87znegcld 10208 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  -u ( A  mod  (
2 ^ N ) )  e.  ZZ )
9 sadadd 12749 . . 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 12749 . . . . . 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 10207 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  -u ( A  mod  (
2 ^ N ) )  e.  CC )
147zcnd 10207 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( A  mod  (
2 ^ N ) )  e.  CC )
1513, 14addcomd 9101 . . . . . . . 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 9234 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( A  mod  ( 2 ^ N
) )  +  -u ( A  mod  ( 2 ^ N ) ) )  =  0 )
1715, 16eqtrd 2390 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( -u ( A  mod  ( 2 ^ N
) )  +  ( A  mod  ( 2 ^ N ) ) )  =  0 )
1817fveq2d 5609 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
(bits `  ( -u ( A  mod  ( 2 ^ N ) )  +  ( A  mod  (
2 ^ N ) ) ) )  =  (bits `  0 )
)
19 0bits 12721 . . . . . 6  |-  (bits ` 
0 )  =  (/)
2018, 19syl6eq 2406 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
(bits `  ( -u ( A  mod  ( 2 ^ N ) )  +  ( A  mod  (
2 ^ N ) ) ) )  =  (/) )
2112, 20eqtrd 2390 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (bits `  -u ( A  mod  ( 2 ^ N ) ) ) sadd  (bits `  ( A  mod  ( 2 ^ N
) ) ) )  =  (/) )
2221oveq1d 5957 . . 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 12708 . . . . . 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 12708 . . . . . 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 3465 . . . . . 6  |-  ( (bits `  A )  i^i  ( ZZ>=
`  N ) ) 
C_  (bits `  A
)
28 bitsss 12708 . . . . . . 7  |-  (bits `  A )  C_  NN0
2928a1i 10 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
(bits `  A )  C_ 
NN0 )
3027, 29syl5ss 3266 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (bits `  A
)  i^i  ( ZZ>= `  N ) )  C_  NN0 )
31 sadass 12753 . . . . 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 12718 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
(bits `  ( A  mod  ( 2 ^ N
) ) )  =  ( (bits `  A
)  i^i  ( 0..^ N ) ) )
3433oveq1d 5957 . . . . . 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 3465 . . . . . . . . . 10  |-  ( (bits `  A )  i^i  (
0..^ N ) ) 
C_  (bits `  A
)
3635, 29syl5ss 3266 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (bits `  A
)  i^i  ( 0..^ N ) )  C_  NN0 )
37 fzouzdisj 10991 . . . . . . . . . . . 12  |-  ( ( 0..^ N )  i^i  ( ZZ>= `  N )
)  =  (/)
3837ineq2i 3443 . . . . . . . . . . 11  |-  ( (bits `  A )  i^i  (
( 0..^ N )  i^i  ( ZZ>= `  N
) ) )  =  ( (bits `  A
)  i^i  (/) )
39 inindi 3462 . . . . . . . . . . 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 3556 . . . . . . . . . . 11  |-  ( (bits `  A )  i^i  (/) )  =  (/)
4138, 39, 403eqtr3i 2386 . . . . . . . . . 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 12747 . . . . . . . 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 3491 . . . . . . . 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 2408 . . . . . . 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 10351 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
474, 46syl6eleq 2448 . . . . . . . . . . 11  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  N  e.  ( ZZ>= ` 
0 ) )
48 fzouzsplit 10990 . . . . . . . . . . 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 2402 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  NN0  =  ( ( 0..^ N )  u.  ( ZZ>=
`  N ) ) )
5128, 50syl5sseq 3302 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
(bits `  A )  C_  ( ( 0..^ N )  u.  ( ZZ>= `  N ) ) )
52 df-ss 3242 . . . . . . . 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 2390 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( (bits `  A )  i^i  (
0..^ N ) ) sadd  ( (bits `  A
)  i^i  ( ZZ>= `  N ) ) )  =  (bits `  A
) )
5534, 54eqtrd 2390 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (bits `  ( A  mod  ( 2 ^ N ) ) ) sadd  ( (bits `  A
)  i^i  ( ZZ>= `  N ) ) )  =  (bits `  A
) )
5655oveq2d 5958 . . . 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 2390 . . 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 12751 . . . 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 2398 . 2  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( (bits `  -u ( A  mod  ( 2 ^ N ) ) ) sadd  (bits `  A )
)  =  ( (bits `  A )  i^i  ( ZZ>=
`  N ) ) )
611zcnd 10207 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  A  e.  CC )
6213, 61addcomd 9101 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( -u ( A  mod  ( 2 ^ N
) )  +  A
)  =  ( A  +  -u ( A  mod  ( 2 ^ N
) ) ) )
6361, 14negsubd 9250 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( A  +  -u ( A  mod  ( 2 ^ N ) ) )  =  ( A  -  ( A  mod  ( 2 ^ N
) ) ) )
6461, 14subcld 9244 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( A  -  ( A  mod  ( 2 ^ N ) ) )  e.  CC )
655nncnd 9849 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( 2 ^ N
)  e.  CC )
665nnne0d 9877 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( 2 ^ N
)  =/=  0 )
6764, 65, 66divcan1d 9624 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( ( ( A  -  ( A  mod  ( 2 ^ N
) ) )  / 
( 2 ^ N
) )  x.  (
2 ^ N ) )  =  ( A  -  ( A  mod  ( 2 ^ N
) ) ) )
681zred 10206 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  ->  A  e.  RR )
695nnrpd 10478 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( 2 ^ N
)  e.  RR+ )
70 moddiffl 11071 . . . . . . 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 5957 . . . . 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 2392 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( A  -  ( A  mod  ( 2 ^ N ) ) )  =  ( ( |_
`  ( A  / 
( 2 ^ N
) ) )  x.  ( 2 ^ N
) ) )
7462, 63, 733eqtrd 2394 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
( -u ( A  mod  ( 2 ^ N
) )  +  A
)  =  ( ( |_ `  ( A  /  ( 2 ^ N ) ) )  x.  ( 2 ^ N ) ) )
7574fveq2d 5609 . 2  |-  ( ( A  e.  ZZ  /\  N  e.  NN0 )  -> 
(bits `  ( -u ( A  mod  ( 2 ^ N ) )  +  A ) )  =  (bits `  ( ( |_ `  ( A  / 
( 2 ^ N
) ) )  x.  ( 2 ^ N
) ) ) )
7610, 60, 753eqtr3d 2398 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 1642    e. wcel 1710    u. cun 3226    i^i cin 3227    C_ wss 3228   (/)c0 3531   ` cfv 5334  (class class class)co 5942   RRcr 8823   0cc0 8824    + caddc 8827    x. cmul 8829    - cmin 9124   -ucneg 9125    / cdiv 9510   NNcn 9833   2c2 9882   NN0cn0 10054   ZZcz 10113   ZZ>=cuz 10319   RR+crp 10443  ..^cfzo 10959   |_cfl 11013    mod cmo 11062   ^cexp 11194  bitscbits 12701   sadd csad 12702
This theorem is referenced by:  bitsuz  12756
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-inf2 7429  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-xor 1305  df-tru 1319  df-had 1380  df-cad 1381  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-disj 4073  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-se 4432  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-isom 5343  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-2o 6564  df-oadd 6567  df-er 6744  df-map 6859  df-pm 6860  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-sup 7281  df-oi 7312  df-card 7659  df-cda 7881  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-2 9891  df-3 9892  df-n0 10055  df-z 10114  df-uz 10320  df-rp 10444  df-fz 10872  df-fzo 10960  df-fl 11014  df-mod 11063  df-seq 11136  df-exp 11195  df-hash 11428  df-cj 11674  df-re 11675  df-im 11676  df-sqr 11810  df-abs 11811  df-clim 12052  df-sum 12250  df-dvds 12623  df-bits 12704  df-sad 12733
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