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Theorem perfectlem1 20468
Description: Lemma for perfect 20470. (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 9877 . . 3  |-  2  e.  NN
2 perfectlem.1 . . . . 5  |-  ( ph  ->  A  e.  NN )
32nnnn0d 10018 . . . 4  |-  ( ph  ->  A  e.  NN0 )
4 peano2nn0 10004 . . . 4  |-  ( A  e.  NN0  ->  ( A  +  1 )  e. 
NN0 )
53, 4syl 15 . . 3  |-  ( ph  ->  ( A  +  1 )  e.  NN0 )
6 nnexpcl 11116 . . 3  |-  ( ( 2  e.  NN  /\  ( A  +  1
)  e.  NN0 )  ->  ( 2 ^ ( A  +  1 ) )  e.  NN )
71, 5, 6sylancr 644 . 2  |-  ( ph  ->  ( 2 ^ ( A  +  1 ) )  e.  NN )
8 2re 9815 . . . . 5  |-  2  e.  RR
98a1i 10 . . . 4  |-  ( ph  ->  2  e.  RR )
102peano2nnd 9763 . . . 4  |-  ( ph  ->  ( A  +  1 )  e.  NN )
11 1lt2 9886 . . . . 5  |-  1  <  2
1211a1i 10 . . . 4  |-  ( ph  ->  1  <  2 )
13 expgt1 11140 . . . 4  |-  ( ( 2  e.  RR  /\  ( A  +  1
)  e.  NN  /\  1  <  2 )  -> 
1  <  ( 2 ^ ( A  + 
1 ) ) )
149, 10, 12, 13syl3anc 1182 . . 3  |-  ( ph  ->  1  <  ( 2 ^ ( A  + 
1 ) ) )
15 1nn 9757 . . . 4  |-  1  e.  NN
16 nnsub 9784 . . . 4  |-  ( ( 1  e.  NN  /\  ( 2 ^ ( A  +  1 ) )  e.  NN )  ->  ( 1  < 
( 2 ^ ( A  +  1 ) )  <->  ( ( 2 ^ ( A  + 
1 ) )  - 
1 )  e.  NN ) )
1715, 7, 16sylancr 644 . . 3  |-  ( ph  ->  ( 1  <  (
2 ^ ( A  +  1 ) )  <-> 
( ( 2 ^ ( A  +  1 ) )  -  1 )  e.  NN ) )
1814, 17mpbid 201 . 2  |-  ( ph  ->  ( ( 2 ^ ( A  +  1 ) )  -  1 )  e.  NN )
197nnzd 10116 . . . . . . 7  |-  ( ph  ->  ( 2 ^ ( A  +  1 ) )  e.  ZZ )
20 peano2zm 10062 . . . . . . 7  |-  ( ( 2 ^ ( A  +  1 ) )  e.  ZZ  ->  (
( 2 ^ ( A  +  1 ) )  -  1 )  e.  ZZ )
2119, 20syl 15 . . . . . 6  |-  ( ph  ->  ( ( 2 ^ ( A  +  1 ) )  -  1 )  e.  ZZ )
22 1nn0 9981 . . . . . . . 8  |-  1  e.  NN0
23 perfectlem.2 . . . . . . . 8  |-  ( ph  ->  B  e.  NN )
24 sgmnncl 20385 . . . . . . . 8  |-  ( ( 1  e.  NN0  /\  B  e.  NN )  ->  ( 1  sigma  B )  e.  NN )
2522, 23, 24sylancr 644 . . . . . . 7  |-  ( ph  ->  ( 1  sigma  B )  e.  NN )
2625nnzd 10116 . . . . . 6  |-  ( ph  ->  ( 1  sigma  B )  e.  ZZ )
27 dvdsmul1 12550 . . . . . 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 642 . . . . 5  |-  ( ph  ->  ( ( 2 ^ ( A  +  1 ) )  -  1 )  ||  ( ( ( 2 ^ ( A  +  1 ) )  -  1 )  x.  ( 1  sigma  B ) ) )
29 2cn 9816 . . . . . . . . 9  |-  2  e.  CC
30 expp1 11110 . . . . . . . . 9  |-  ( ( 2  e.  CC  /\  A  e.  NN0 )  -> 
( 2 ^ ( A  +  1 ) )  =  ( ( 2 ^ A )  x.  2 ) )
3129, 3, 30sylancr 644 . . . . . . . 8  |-  ( ph  ->  ( 2 ^ ( A  +  1 ) )  =  ( ( 2 ^ A )  x.  2 ) )
32 nnexpcl 11116 . . . . . . . . . . 11  |-  ( ( 2  e.  NN  /\  A  e.  NN0 )  -> 
( 2 ^ A
)  e.  NN )
331, 3, 32sylancr 644 . . . . . . . . . 10  |-  ( ph  ->  ( 2 ^ A
)  e.  NN )
3433nncnd 9762 . . . . . . . . 9  |-  ( ph  ->  ( 2 ^ A
)  e.  CC )
35 mulcom 8823 . . . . . . . . 9  |-  ( ( ( 2 ^ A
)  e.  CC  /\  2  e.  CC )  ->  ( ( 2 ^ A )  x.  2 )  =  ( 2  x.  ( 2 ^ A ) ) )
3634, 29, 35sylancl 643 . . . . . . . 8  |-  ( ph  ->  ( ( 2 ^ A )  x.  2 )  =  ( 2  x.  ( 2 ^ A ) ) )
3731, 36eqtrd 2315 . . . . . . 7  |-  ( ph  ->  ( 2 ^ ( A  +  1 ) )  =  ( 2  x.  ( 2 ^ A ) ) )
3837oveq1d 5873 . . . . . 6  |-  ( ph  ->  ( ( 2 ^ ( A  +  1 ) )  x.  B
)  =  ( ( 2  x.  ( 2 ^ A ) )  x.  B ) )
3929a1i 10 . . . . . . . 8  |-  ( ph  ->  2  e.  CC )
4023nncnd 9762 . . . . . . . 8  |-  ( ph  ->  B  e.  CC )
4139, 34, 40mulassd 8858 . . . . . . 7  |-  ( ph  ->  ( ( 2  x.  ( 2 ^ A
) )  x.  B
)  =  ( 2  x.  ( ( 2 ^ A )  x.  B ) ) )
42 ax-1cn 8795 . . . . . . . . . 10  |-  1  e.  CC
4342a1i 10 . . . . . . . . 9  |-  ( ph  ->  1  e.  CC )
44 perfectlem.3 . . . . . . . . . . 11  |-  ( ph  ->  -.  2  ||  B
)
45 2prm 12774 . . . . . . . . . . . 12  |-  2  e.  Prime
4623nnzd 10116 . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  ZZ )
47 coprm 12779 . . . . . . . . . . . 12  |-  ( ( 2  e.  Prime  /\  B  e.  ZZ )  ->  ( -.  2  ||  B  <->  ( 2  gcd  B )  =  1 ) )
4845, 46, 47sylancr 644 . . . . . . . . . . 11  |-  ( ph  ->  ( -.  2  ||  B 
<->  ( 2  gcd  B
)  =  1 ) )
4944, 48mpbid 201 . . . . . . . . . 10  |-  ( ph  ->  ( 2  gcd  B
)  =  1 )
50 2z 10054 . . . . . . . . . . . 12  |-  2  e.  ZZ
5150a1i 10 . . . . . . . . . . 11  |-  ( ph  ->  2  e.  ZZ )
52 rpexp1i 12800 . . . . . . . . . . 11  |-  ( ( 2  e.  ZZ  /\  B  e.  ZZ  /\  A  e.  NN0 )  ->  (
( 2  gcd  B
)  =  1  -> 
( ( 2 ^ A )  gcd  B
)  =  1 ) )
5351, 46, 3, 52syl3anc 1182 . . . . . . . . . 10  |-  ( ph  ->  ( ( 2  gcd 
B )  =  1  ->  ( ( 2 ^ A )  gcd 
B )  =  1 ) )
5449, 53mpd 14 . . . . . . . . 9  |-  ( ph  ->  ( ( 2 ^ A )  gcd  B
)  =  1 )
55 sgmmul 20440 . . . . . . . . 9  |-  ( ( 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 1184 . . . . . . . 8  |-  ( ph  ->  ( 1  sigma  ( ( 2 ^ A )  x.  B ) )  =  ( ( 1 
sigma  ( 2 ^ A
) )  x.  (
1  sigma  B ) ) )
57 perfectlem.4 . . . . . . . 8  |-  ( ph  ->  ( 1  sigma  ( ( 2 ^ A )  x.  B ) )  =  ( 2  x.  ( ( 2 ^ A )  x.  B
) ) )
582nncnd 9762 . . . . . . . . . . . . 13  |-  ( ph  ->  A  e.  CC )
59 pncan 9057 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  + 
1 )  -  1 )  =  A )
6058, 42, 59sylancl 643 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( A  + 
1 )  -  1 )  =  A )
6160oveq2d 5874 . . . . . . . . . . 11  |-  ( ph  ->  ( 2 ^ (
( A  +  1 )  -  1 ) )  =  ( 2 ^ A ) )
6261oveq2d 5874 . . . . . . . . . 10  |-  ( ph  ->  ( 1  sigma  ( 2 ^ ( ( A  +  1 )  - 
1 ) ) )  =  ( 1  sigma 
( 2 ^ A
) ) )
63 1sgm2ppw 20439 . . . . . . . . . . 11  |-  ( ( A  +  1 )  e.  NN  ->  (
1  sigma  ( 2 ^ ( ( A  + 
1 )  -  1 ) ) )  =  ( ( 2 ^ ( A  +  1 ) )  -  1 ) )
6410, 63syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( 1  sigma  ( 2 ^ ( ( A  +  1 )  - 
1 ) ) )  =  ( ( 2 ^ ( A  + 
1 ) )  - 
1 ) )
6562, 64eqtr3d 2317 . . . . . . . . 9  |-  ( ph  ->  ( 1  sigma  ( 2 ^ A ) )  =  ( ( 2 ^ ( A  + 
1 ) )  - 
1 ) )
6665oveq1d 5873 . . . . . . . 8  |-  ( ph  ->  ( ( 1  sigma 
( 2 ^ A
) )  x.  (
1  sigma  B ) )  =  ( ( ( 2 ^ ( A  +  1 ) )  -  1 )  x.  ( 1  sigma  B ) ) )
6756, 57, 663eqtr3d 2323 . . . . . . 7  |-  ( ph  ->  ( 2  x.  (
( 2 ^ A
)  x.  B ) )  =  ( ( ( 2 ^ ( A  +  1 ) )  -  1 )  x.  ( 1  sigma  B ) ) )
6841, 67eqtrd 2315 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  ( 2 ^ A
) )  x.  B
)  =  ( ( ( 2 ^ ( A  +  1 ) )  -  1 )  x.  ( 1  sigma  B ) ) )
6938, 68eqtrd 2315 . . . . 5  |-  ( ph  ->  ( ( 2 ^ ( A  +  1 ) )  x.  B
)  =  ( ( ( 2 ^ ( A  +  1 ) )  -  1 )  x.  ( 1  sigma  B ) ) )
7028, 69breqtrrd 4049 . . . 4  |-  ( ph  ->  ( ( 2 ^ ( A  +  1 ) )  -  1 )  ||  ( ( 2 ^ ( A  +  1 ) )  x.  B ) )
71 gcdcom 12699 . . . . . 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 ) ) )
7221, 19, 71syl2anc 642 . . . . 5  |-  ( ph  ->  ( ( ( 2 ^ ( A  + 
1 ) )  - 
1 )  gcd  (
2 ^ ( A  +  1 ) ) )  =  ( ( 2 ^ ( A  +  1 ) )  gcd  ( ( 2 ^ ( A  + 
1 ) )  - 
1 ) ) )
73 iddvdsexp 12552 . . . . . . . . 9  |-  ( ( 2  e.  ZZ  /\  ( A  +  1
)  e.  NN )  ->  2  ||  (
2 ^ ( A  +  1 ) ) )
7450, 10, 73sylancr 644 . . . . . . . 8  |-  ( ph  ->  2  ||  ( 2 ^ ( A  + 
1 ) ) )
75 nprmdvds1 12790 . . . . . . . . . . 11  |-  ( 2  e.  Prime  ->  -.  2  ||  1 )
7645, 75ax-mp 8 . . . . . . . . . 10  |-  -.  2  ||  1
77 1z 10053 . . . . . . . . . . . . 13  |-  1  e.  ZZ
7877a1i 10 . . . . . . . . . . . 12  |-  ( ph  ->  1  e.  ZZ )
7951, 19, 783jca 1132 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  e.  ZZ  /\  ( 2 ^ ( A  +  1 ) )  e.  ZZ  /\  1  e.  ZZ )
)
80 dvdssub2 12566 . . . . . . . . . . 11  |-  ( ( ( 2  e.  ZZ  /\  ( 2 ^ ( A  +  1 ) )  e.  ZZ  /\  1  e.  ZZ )  /\  2  ||  ( ( 2 ^ ( A  +  1 ) )  -  1 ) )  ->  ( 2  ||  ( 2 ^ ( A  +  1 ) )  <->  2  ||  1
) )
8179, 80sylan 457 . . . . . . . . . 10  |-  ( (
ph  /\  2  ||  ( ( 2 ^ ( A  +  1 ) )  -  1 ) )  ->  (
2  ||  ( 2 ^ ( A  + 
1 ) )  <->  2  ||  1 ) )
8276, 81mtbiri 294 . . . . . . . . 9  |-  ( (
ph  /\  2  ||  ( ( 2 ^ ( A  +  1 ) )  -  1 ) )  ->  -.  2  ||  ( 2 ^ ( A  +  1 ) ) )
8382ex 423 . . . . . . . 8  |-  ( ph  ->  ( 2  ||  (
( 2 ^ ( A  +  1 ) )  -  1 )  ->  -.  2  ||  ( 2 ^ ( A  +  1 ) ) ) )
8474, 83mt2d 109 . . . . . . 7  |-  ( ph  ->  -.  2  ||  (
( 2 ^ ( A  +  1 ) )  -  1 ) )
85 coprm 12779 . . . . . . . 8  |-  ( ( 2  e.  Prime  /\  (
( 2 ^ ( A  +  1 ) )  -  1 )  e.  ZZ )  -> 
( -.  2  ||  ( ( 2 ^ ( A  +  1 ) )  -  1 )  <->  ( 2  gcd  ( ( 2 ^ ( A  +  1 ) )  -  1 ) )  =  1 ) )
8645, 21, 85sylancr 644 . . . . . . 7  |-  ( ph  ->  ( -.  2  ||  ( ( 2 ^ ( A  +  1 ) )  -  1 )  <->  ( 2  gcd  ( ( 2 ^ ( A  +  1 ) )  -  1 ) )  =  1 ) )
8784, 86mpbid 201 . . . . . 6  |-  ( ph  ->  ( 2  gcd  (
( 2 ^ ( A  +  1 ) )  -  1 ) )  =  1 )
88 rpexp1i 12800 . . . . . . 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 ) )
8951, 21, 5, 88syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( ( 2  gcd  ( ( 2 ^ ( A  +  1 ) )  -  1 ) )  =  1  ->  ( ( 2 ^ ( A  + 
1 ) )  gcd  ( ( 2 ^ ( A  +  1 ) )  -  1 ) )  =  1 ) )
9087, 89mpd 14 . . . . 5  |-  ( ph  ->  ( ( 2 ^ ( A  +  1 ) )  gcd  (
( 2 ^ ( A  +  1 ) )  -  1 ) )  =  1 )
9172, 90eqtrd 2315 . . . 4  |-  ( ph  ->  ( ( ( 2 ^ ( A  + 
1 ) )  - 
1 )  gcd  (
2 ^ ( A  +  1 ) ) )  =  1 )
92 coprmdvds 12781 . . . . 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 ) )
9321, 19, 46, 92syl3anc 1182 . . . 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 ) )
9470, 91, 93mp2and 660 . . 3  |-  ( ph  ->  ( ( 2 ^ ( A  +  1 ) )  -  1 )  ||  B )
95 nndivdvds 12537 . . . 4  |-  ( ( B  e.  NN  /\  ( ( 2 ^ ( A  +  1 ) )  -  1 )  e.  NN )  ->  ( ( ( 2 ^ ( A  +  1 ) )  -  1 )  ||  B 
<->  ( B  /  (
( 2 ^ ( A  +  1 ) )  -  1 ) )  e.  NN ) )
9623, 18, 95syl2anc 642 . . 3  |-  ( ph  ->  ( ( ( 2 ^ ( A  + 
1 ) )  - 
1 )  ||  B  <->  ( B  /  ( ( 2 ^ ( A  +  1 ) )  -  1 ) )  e.  NN ) )
9794, 96mpbid 201 . 2  |-  ( ph  ->  ( B  /  (
( 2 ^ ( A  +  1 ) )  -  1 ) )  e.  NN )
987, 18, 973jca 1132 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023  (class class class)co 5858   CCcc 8735   RRcr 8736   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    - cmin 9037    / cdiv 9423   NNcn 9746   2c2 9795   NN0cn0 9965   ZZcz 10024   ^cexp 11104    || cdivides 12531    gcd cgcd 12685   Primecprime 12758    sigma csgm 20333
This theorem is referenced by:  perfectlem2  20469
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  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  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-iin 3908  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-of 6078  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-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  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-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354  df-dvds 12532  df-gcd 12686  df-prm 12759  df-pc 12890  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914  df-cxp 19915  df-sgm 20339
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