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Theorem perfectlem1 21013
Description: Lemma for perfect 21015. (Contributed by Mario Carneiro, 7-Jun-2016.)
Hypotheses
Ref Expression
perfectlem.1  |-  ( ph  ->  A  e.  NN )
perfectlem.2  |-  ( ph  ->  B  e.  NN )
perfectlem.3  |-  ( ph  ->  -.  2  ||  B
)
perfectlem.4  |-  ( ph  ->  ( 1  sigma  ( ( 2 ^ A )  x.  B ) )  =  ( 2  x.  ( ( 2 ^ A )  x.  B
) ) )
Assertion
Ref Expression
perfectlem1  |-  ( ph  ->  ( ( 2 ^ ( A  +  1 ) )  e.  NN  /\  ( ( 2 ^ ( A  +  1 ) )  -  1 )  e.  NN  /\  ( B  /  (
( 2 ^ ( A  +  1 ) )  -  1 ) )  e.  NN ) )

Proof of Theorem perfectlem1
StepHypRef Expression
1 2nn 10133 . . 3  |-  2  e.  NN
2 perfectlem.1 . . . . 5  |-  ( ph  ->  A  e.  NN )
32nnnn0d 10274 . . . 4  |-  ( ph  ->  A  e.  NN0 )
4 peano2nn0 10260 . . . 4  |-  ( A  e.  NN0  ->  ( A  +  1 )  e. 
NN0 )
53, 4syl 16 . . 3  |-  ( ph  ->  ( A  +  1 )  e.  NN0 )
6 nnexpcl 11394 . . 3  |-  ( ( 2  e.  NN  /\  ( A  +  1
)  e.  NN0 )  ->  ( 2 ^ ( A  +  1 ) )  e.  NN )
71, 5, 6sylancr 645 . 2  |-  ( ph  ->  ( 2 ^ ( A  +  1 ) )  e.  NN )
8 2re 10069 . . . . 5  |-  2  e.  RR
98a1i 11 . . . 4  |-  ( ph  ->  2  e.  RR )
102peano2nnd 10017 . . . 4  |-  ( ph  ->  ( A  +  1 )  e.  NN )
11 1lt2 10142 . . . . 5  |-  1  <  2
1211a1i 11 . . . 4  |-  ( ph  ->  1  <  2 )
13 expgt1 11418 . . . 4  |-  ( ( 2  e.  RR  /\  ( A  +  1
)  e.  NN  /\  1  <  2 )  -> 
1  <  ( 2 ^ ( A  + 
1 ) ) )
149, 10, 12, 13syl3anc 1184 . . 3  |-  ( ph  ->  1  <  ( 2 ^ ( A  + 
1 ) ) )
15 1nn 10011 . . . 4  |-  1  e.  NN
16 nnsub 10038 . . . 4  |-  ( ( 1  e.  NN  /\  ( 2 ^ ( A  +  1 ) )  e.  NN )  ->  ( 1  < 
( 2 ^ ( A  +  1 ) )  <->  ( ( 2 ^ ( A  + 
1 ) )  - 
1 )  e.  NN ) )
1715, 7, 16sylancr 645 . . 3  |-  ( ph  ->  ( 1  <  (
2 ^ ( A  +  1 ) )  <-> 
( ( 2 ^ ( A  +  1 ) )  -  1 )  e.  NN ) )
1814, 17mpbid 202 . 2  |-  ( ph  ->  ( ( 2 ^ ( A  +  1 ) )  -  1 )  e.  NN )
197nnzd 10374 . . . . . . 7  |-  ( ph  ->  ( 2 ^ ( A  +  1 ) )  e.  ZZ )
20 peano2zm 10320 . . . . . . 7  |-  ( ( 2 ^ ( A  +  1 ) )  e.  ZZ  ->  (
( 2 ^ ( A  +  1 ) )  -  1 )  e.  ZZ )
2119, 20syl 16 . . . . . 6  |-  ( ph  ->  ( ( 2 ^ ( A  +  1 ) )  -  1 )  e.  ZZ )
22 1nn0 10237 . . . . . . . 8  |-  1  e.  NN0
23 perfectlem.2 . . . . . . . 8  |-  ( ph  ->  B  e.  NN )
24 sgmnncl 20930 . . . . . . . 8  |-  ( ( 1  e.  NN0  /\  B  e.  NN )  ->  ( 1  sigma  B )  e.  NN )
2522, 23, 24sylancr 645 . . . . . . 7  |-  ( ph  ->  ( 1  sigma  B )  e.  NN )
2625nnzd 10374 . . . . . 6  |-  ( ph  ->  ( 1  sigma  B )  e.  ZZ )
27 dvdsmul1 12871 . . . . . 6  |-  ( ( ( ( 2 ^ ( A  +  1 ) )  -  1 )  e.  ZZ  /\  ( 1  sigma  B )  e.  ZZ )  -> 
( ( 2 ^ ( A  +  1 ) )  -  1 )  ||  ( ( ( 2 ^ ( A  +  1 ) )  -  1 )  x.  ( 1  sigma  B ) ) )
2821, 26, 27syl2anc 643 . . . . 5  |-  ( ph  ->  ( ( 2 ^ ( A  +  1 ) )  -  1 )  ||  ( ( ( 2 ^ ( A  +  1 ) )  -  1 )  x.  ( 1  sigma  B ) ) )
29 2cn 10070 . . . . . . . . 9  |-  2  e.  CC
30 expp1 11388 . . . . . . . . 9  |-  ( ( 2  e.  CC  /\  A  e.  NN0 )  -> 
( 2 ^ ( A  +  1 ) )  =  ( ( 2 ^ A )  x.  2 ) )
3129, 3, 30sylancr 645 . . . . . . . 8  |-  ( ph  ->  ( 2 ^ ( A  +  1 ) )  =  ( ( 2 ^ A )  x.  2 ) )
32 nnexpcl 11394 . . . . . . . . . . 11  |-  ( ( 2  e.  NN  /\  A  e.  NN0 )  -> 
( 2 ^ A
)  e.  NN )
331, 3, 32sylancr 645 . . . . . . . . . 10  |-  ( ph  ->  ( 2 ^ A
)  e.  NN )
3433nncnd 10016 . . . . . . . . 9  |-  ( ph  ->  ( 2 ^ A
)  e.  CC )
35 mulcom 9076 . . . . . . . . 9  |-  ( ( ( 2 ^ A
)  e.  CC  /\  2  e.  CC )  ->  ( ( 2 ^ A )  x.  2 )  =  ( 2  x.  ( 2 ^ A ) ) )
3634, 29, 35sylancl 644 . . . . . . . 8  |-  ( ph  ->  ( ( 2 ^ A )  x.  2 )  =  ( 2  x.  ( 2 ^ A ) ) )
3731, 36eqtrd 2468 . . . . . . 7  |-  ( ph  ->  ( 2 ^ ( A  +  1 ) )  =  ( 2  x.  ( 2 ^ A ) ) )
3837oveq1d 6096 . . . . . 6  |-  ( ph  ->  ( ( 2 ^ ( A  +  1 ) )  x.  B
)  =  ( ( 2  x.  ( 2 ^ A ) )  x.  B ) )
3929a1i 11 . . . . . . 7  |-  ( ph  ->  2  e.  CC )
4023nncnd 10016 . . . . . . 7  |-  ( ph  ->  B  e.  CC )
4139, 34, 40mulassd 9111 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  ( 2 ^ A
) )  x.  B
)  =  ( 2  x.  ( ( 2 ^ A )  x.  B ) ) )
42 ax-1cn 9048 . . . . . . . . 9  |-  1  e.  CC
4342a1i 11 . . . . . . . 8  |-  ( ph  ->  1  e.  CC )
44 perfectlem.3 . . . . . . . . . 10  |-  ( ph  ->  -.  2  ||  B
)
45 2prm 13095 . . . . . . . . . . 11  |-  2  e.  Prime
4623nnzd 10374 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  ZZ )
47 coprm 13100 . . . . . . . . . . 11  |-  ( ( 2  e.  Prime  /\  B  e.  ZZ )  ->  ( -.  2  ||  B  <->  ( 2  gcd  B )  =  1 ) )
4845, 46, 47sylancr 645 . . . . . . . . . 10  |-  ( ph  ->  ( -.  2  ||  B 
<->  ( 2  gcd  B
)  =  1 ) )
4944, 48mpbid 202 . . . . . . . . 9  |-  ( ph  ->  ( 2  gcd  B
)  =  1 )
50 2z 10312 . . . . . . . . . . 11  |-  2  e.  ZZ
5150a1i 11 . . . . . . . . . 10  |-  ( ph  ->  2  e.  ZZ )
52 rpexp1i 13121 . . . . . . . . . 10  |-  ( ( 2  e.  ZZ  /\  B  e.  ZZ  /\  A  e.  NN0 )  ->  (
( 2  gcd  B
)  =  1  -> 
( ( 2 ^ A )  gcd  B
)  =  1 ) )
5351, 46, 3, 52syl3anc 1184 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  gcd 
B )  =  1  ->  ( ( 2 ^ A )  gcd 
B )  =  1 ) )
5449, 53mpd 15 . . . . . . . 8  |-  ( ph  ->  ( ( 2 ^ A )  gcd  B
)  =  1 )
55 sgmmul 20985 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  ( ( 2 ^ A )  e.  NN  /\  B  e.  NN  /\  ( ( 2 ^ A )  gcd  B
)  =  1 ) )  ->  ( 1 
sigma  ( ( 2 ^ A )  x.  B
) )  =  ( ( 1  sigma  ( 2 ^ A ) )  x.  ( 1  sigma  B ) ) )
5643, 33, 23, 54, 55syl13anc 1186 . . . . . . 7  |-  ( ph  ->  ( 1  sigma  ( ( 2 ^ A )  x.  B ) )  =  ( ( 1 
sigma  ( 2 ^ A
) )  x.  (
1  sigma  B ) ) )
57 perfectlem.4 . . . . . . 7  |-  ( ph  ->  ( 1  sigma  ( ( 2 ^ A )  x.  B ) )  =  ( 2  x.  ( ( 2 ^ A )  x.  B
) ) )
582nncnd 10016 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  CC )
59 pncan 9311 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  + 
1 )  -  1 )  =  A )
6058, 42, 59sylancl 644 . . . . . . . . . . 11  |-  ( ph  ->  ( ( A  + 
1 )  -  1 )  =  A )
6160oveq2d 6097 . . . . . . . . . 10  |-  ( ph  ->  ( 2 ^ (
( A  +  1 )  -  1 ) )  =  ( 2 ^ A ) )
6261oveq2d 6097 . . . . . . . . 9  |-  ( ph  ->  ( 1  sigma  ( 2 ^ ( ( A  +  1 )  - 
1 ) ) )  =  ( 1  sigma 
( 2 ^ A
) ) )
63 1sgm2ppw 20984 . . . . . . . . . 10  |-  ( ( A  +  1 )  e.  NN  ->  (
1  sigma  ( 2 ^ ( ( A  + 
1 )  -  1 ) ) )  =  ( ( 2 ^ ( A  +  1 ) )  -  1 ) )
6410, 63syl 16 . . . . . . . . 9  |-  ( ph  ->  ( 1  sigma  ( 2 ^ ( ( A  +  1 )  - 
1 ) ) )  =  ( ( 2 ^ ( A  + 
1 ) )  - 
1 ) )
6562, 64eqtr3d 2470 . . . . . . . 8  |-  ( ph  ->  ( 1  sigma  ( 2 ^ A ) )  =  ( ( 2 ^ ( A  + 
1 ) )  - 
1 ) )
6665oveq1d 6096 . . . . . . 7  |-  ( ph  ->  ( ( 1  sigma 
( 2 ^ A
) )  x.  (
1  sigma  B ) )  =  ( ( ( 2 ^ ( A  +  1 ) )  -  1 )  x.  ( 1  sigma  B ) ) )
6756, 57, 663eqtr3d 2476 . . . . . 6  |-  ( ph  ->  ( 2  x.  (
( 2 ^ A
)  x.  B ) )  =  ( ( ( 2 ^ ( A  +  1 ) )  -  1 )  x.  ( 1  sigma  B ) ) )
6838, 41, 673eqtrd 2472 . . . . 5  |-  ( ph  ->  ( ( 2 ^ ( A  +  1 ) )  x.  B
)  =  ( ( ( 2 ^ ( A  +  1 ) )  -  1 )  x.  ( 1  sigma  B ) ) )
6928, 68breqtrrd 4238 . . . 4  |-  ( ph  ->  ( ( 2 ^ ( A  +  1 ) )  -  1 )  ||  ( ( 2 ^ ( A  +  1 ) )  x.  B ) )
70 gcdcom 13020 . . . . . 6  |-  ( ( ( ( 2 ^ ( A  +  1 ) )  -  1 )  e.  ZZ  /\  ( 2 ^ ( A  +  1 ) )  e.  ZZ )  ->  ( ( ( 2 ^ ( A  +  1 ) )  -  1 )  gcd  ( 2 ^ ( A  +  1 ) ) )  =  ( ( 2 ^ ( A  +  1 ) )  gcd  ( ( 2 ^ ( A  +  1 ) )  -  1 ) ) )
7121, 19, 70syl2anc 643 . . . . 5  |-  ( ph  ->  ( ( ( 2 ^ ( A  + 
1 ) )  - 
1 )  gcd  (
2 ^ ( A  +  1 ) ) )  =  ( ( 2 ^ ( A  +  1 ) )  gcd  ( ( 2 ^ ( A  + 
1 ) )  - 
1 ) ) )
72 iddvdsexp 12873 . . . . . . . . 9  |-  ( ( 2  e.  ZZ  /\  ( A  +  1
)  e.  NN )  ->  2  ||  (
2 ^ ( A  +  1 ) ) )
7350, 10, 72sylancr 645 . . . . . . . 8  |-  ( ph  ->  2  ||  ( 2 ^ ( A  + 
1 ) ) )
74 nprmdvds1 13111 . . . . . . . . . . 11  |-  ( 2  e.  Prime  ->  -.  2  ||  1 )
7545, 74ax-mp 8 . . . . . . . . . 10  |-  -.  2  ||  1
76 1z 10311 . . . . . . . . . . . . 13  |-  1  e.  ZZ
7776a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  1  e.  ZZ )
7851, 19, 773jca 1134 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  e.  ZZ  /\  ( 2 ^ ( A  +  1 ) )  e.  ZZ  /\  1  e.  ZZ )
)
79 dvdssub2 12887 . . . . . . . . . . 11  |-  ( ( ( 2  e.  ZZ  /\  ( 2 ^ ( A  +  1 ) )  e.  ZZ  /\  1  e.  ZZ )  /\  2  ||  ( ( 2 ^ ( A  +  1 ) )  -  1 ) )  ->  ( 2  ||  ( 2 ^ ( A  +  1 ) )  <->  2  ||  1
) )
8078, 79sylan 458 . . . . . . . . . 10  |-  ( (
ph  /\  2  ||  ( ( 2 ^ ( A  +  1 ) )  -  1 ) )  ->  (
2  ||  ( 2 ^ ( A  + 
1 ) )  <->  2  ||  1 ) )
8175, 80mtbiri 295 . . . . . . . . 9  |-  ( (
ph  /\  2  ||  ( ( 2 ^ ( A  +  1 ) )  -  1 ) )  ->  -.  2  ||  ( 2 ^ ( A  +  1 ) ) )
8281ex 424 . . . . . . . 8  |-  ( ph  ->  ( 2  ||  (
( 2 ^ ( A  +  1 ) )  -  1 )  ->  -.  2  ||  ( 2 ^ ( A  +  1 ) ) ) )
8373, 82mt2d 111 . . . . . . 7  |-  ( ph  ->  -.  2  ||  (
( 2 ^ ( A  +  1 ) )  -  1 ) )
84 coprm 13100 . . . . . . . 8  |-  ( ( 2  e.  Prime  /\  (
( 2 ^ ( A  +  1 ) )  -  1 )  e.  ZZ )  -> 
( -.  2  ||  ( ( 2 ^ ( A  +  1 ) )  -  1 )  <->  ( 2  gcd  ( ( 2 ^ ( A  +  1 ) )  -  1 ) )  =  1 ) )
8545, 21, 84sylancr 645 . . . . . . 7  |-  ( ph  ->  ( -.  2  ||  ( ( 2 ^ ( A  +  1 ) )  -  1 )  <->  ( 2  gcd  ( ( 2 ^ ( A  +  1 ) )  -  1 ) )  =  1 ) )
8683, 85mpbid 202 . . . . . 6  |-  ( ph  ->  ( 2  gcd  (
( 2 ^ ( A  +  1 ) )  -  1 ) )  =  1 )
87 rpexp1i 13121 . . . . . . 7  |-  ( ( 2  e.  ZZ  /\  ( ( 2 ^ ( A  +  1 ) )  -  1 )  e.  ZZ  /\  ( A  +  1
)  e.  NN0 )  ->  ( ( 2  gcd  ( ( 2 ^ ( A  +  1 ) )  -  1 ) )  =  1  ->  ( ( 2 ^ ( A  + 
1 ) )  gcd  ( ( 2 ^ ( A  +  1 ) )  -  1 ) )  =  1 ) )
8851, 21, 5, 87syl3anc 1184 . . . . . 6  |-  ( ph  ->  ( ( 2  gcd  ( ( 2 ^ ( A  +  1 ) )  -  1 ) )  =  1  ->  ( ( 2 ^ ( A  + 
1 ) )  gcd  ( ( 2 ^ ( A  +  1 ) )  -  1 ) )  =  1 ) )
8986, 88mpd 15 . . . . 5  |-  ( ph  ->  ( ( 2 ^ ( A  +  1 ) )  gcd  (
( 2 ^ ( A  +  1 ) )  -  1 ) )  =  1 )
9071, 89eqtrd 2468 . . . 4  |-  ( ph  ->  ( ( ( 2 ^ ( A  + 
1 ) )  - 
1 )  gcd  (
2 ^ ( A  +  1 ) ) )  =  1 )
91 coprmdvds 13102 . . . . 5  |-  ( ( ( ( 2 ^ ( A  +  1 ) )  -  1 )  e.  ZZ  /\  ( 2 ^ ( A  +  1 ) )  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( ( ( 2 ^ ( A  +  1 ) )  -  1 )  ||  ( ( 2 ^ ( A  +  1 ) )  x.  B
)  /\  ( (
( 2 ^ ( A  +  1 ) )  -  1 )  gcd  ( 2 ^ ( A  +  1 ) ) )  =  1 )  ->  (
( 2 ^ ( A  +  1 ) )  -  1 ) 
||  B ) )
9221, 19, 46, 91syl3anc 1184 . . . 4  |-  ( ph  ->  ( ( ( ( 2 ^ ( A  +  1 ) )  -  1 )  ||  ( ( 2 ^ ( A  +  1 ) )  x.  B
)  /\  ( (
( 2 ^ ( A  +  1 ) )  -  1 )  gcd  ( 2 ^ ( A  +  1 ) ) )  =  1 )  ->  (
( 2 ^ ( A  +  1 ) )  -  1 ) 
||  B ) )
9369, 90, 92mp2and 661 . . 3  |-  ( ph  ->  ( ( 2 ^ ( A  +  1 ) )  -  1 )  ||  B )
94 nndivdvds 12858 . . . 4  |-  ( ( B  e.  NN  /\  ( ( 2 ^ ( A  +  1 ) )  -  1 )  e.  NN )  ->  ( ( ( 2 ^ ( A  +  1 ) )  -  1 )  ||  B 
<->  ( B  /  (
( 2 ^ ( A  +  1 ) )  -  1 ) )  e.  NN ) )
9523, 18, 94syl2anc 643 . . 3  |-  ( ph  ->  ( ( ( 2 ^ ( A  + 
1 ) )  - 
1 )  ||  B  <->  ( B  /  ( ( 2 ^ ( A  +  1 ) )  -  1 ) )  e.  NN ) )
9693, 95mpbid 202 . 2  |-  ( ph  ->  ( B  /  (
( 2 ^ ( A  +  1 ) )  -  1 ) )  e.  NN )
977, 18, 963jca 1134 1  |-  ( ph  ->  ( ( 2 ^ ( A  +  1 ) )  e.  NN  /\  ( ( 2 ^ ( A  +  1 ) )  -  1 )  e.  NN  /\  ( B  /  (
( 2 ^ ( A  +  1 ) )  -  1 ) )  e.  NN ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4212  (class class class)co 6081   CCcc 8988   RRcr 8989   1c1 8991    + caddc 8993    x. cmul 8995    < clt 9120    - cmin 9291    / cdiv 9677   NNcn 10000   2c2 10049   NN0cn0 10221   ZZcz 10282   ^cexp 11382    || cdivides 12852    gcd cgcd 13006   Primecprime 13079    sigma csgm 20878
This theorem is referenced by:  perfectlem2  21014
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  ax-addf 9069  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  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-iin 4096  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-of 6305  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-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-fi 7416  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-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-q 10575  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-ioo 10920  df-ioc 10921  df-ico 10922  df-icc 10923  df-fz 11044  df-fzo 11136  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-fac 11567  df-bc 11594  df-hash 11619  df-shft 11882  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-limsup 12265  df-clim 12282  df-rlim 12283  df-sum 12480  df-ef 12670  df-sin 12672  df-cos 12673  df-pi 12675  df-dvds 12853  df-gcd 13007  df-prm 13080  df-pc 13211  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-starv 13544  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-hom 13553  df-cco 13554  df-rest 13650  df-topn 13651  df-topgen 13667  df-pt 13668  df-prds 13671  df-xrs 13726  df-0g 13727  df-gsum 13728  df-qtop 13733  df-imas 13734  df-xps 13736  df-mre 13811  df-mrc 13812  df-acs 13814  df-mnd 14690  df-submnd 14739  df-mulg 14815  df-cntz 15116  df-cmn 15414  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-mopn 16698  df-fbas 16699  df-fg 16700  df-cnfld 16704  df-top 16963  df-bases 16965  df-topon 16966  df-topsp 16967  df-cld 17083  df-ntr 17084  df-cls 17085  df-nei 17162  df-lp 17200  df-perf 17201  df-cn 17291  df-cnp 17292  df-haus 17379  df-tx 17594  df-hmeo 17787  df-fil 17878  df-fm 17970  df-flim 17971  df-flf 17972  df-xms 18350  df-ms 18351  df-tms 18352  df-cncf 18908  df-limc 19753  df-dv 19754  df-log 20454  df-cxp 20455  df-sgm 20884
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