MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  perfectlem1 Unicode version

Theorem perfectlem1 20691
Description: Lemma for perfect 20693. (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 10026 . . 3  |-  2  e.  NN
2 perfectlem.1 . . . . 5  |-  ( ph  ->  A  e.  NN )
32nnnn0d 10167 . . . 4  |-  ( ph  ->  A  e.  NN0 )
4 peano2nn0 10153 . . . 4  |-  ( A  e.  NN0  ->  ( A  +  1 )  e. 
NN0 )
53, 4syl 15 . . 3  |-  ( ph  ->  ( A  +  1 )  e.  NN0 )
6 nnexpcl 11281 . . 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 9962 . . . . 5  |-  2  e.  RR
98a1i 10 . . . 4  |-  ( ph  ->  2  e.  RR )
102peano2nnd 9910 . . . 4  |-  ( ph  ->  ( A  +  1 )  e.  NN )
11 1lt2 10035 . . . . 5  |-  1  <  2
1211a1i 10 . . . 4  |-  ( ph  ->  1  <  2 )
13 expgt1 11305 . . . 4  |-  ( ( 2  e.  RR  /\  ( A  +  1
)  e.  NN  /\  1  <  2 )  -> 
1  <  ( 2 ^ ( A  + 
1 ) ) )
149, 10, 12, 13syl3anc 1183 . . 3  |-  ( ph  ->  1  <  ( 2 ^ ( A  + 
1 ) ) )
15 1nn 9904 . . . 4  |-  1  e.  NN
16 nnsub 9931 . . . 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 10267 . . . . . . 7  |-  ( ph  ->  ( 2 ^ ( A  +  1 ) )  e.  ZZ )
20 peano2zm 10213 . . . . . . 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 10130 . . . . . . . 8  |-  1  e.  NN0
23 perfectlem.2 . . . . . . . 8  |-  ( ph  ->  B  e.  NN )
24 sgmnncl 20608 . . . . . . . 8  |-  ( ( 1  e.  NN0  /\  B  e.  NN )  ->  ( 1  sigma  B )  e.  NN )
2522, 23, 24sylancr 644 . . . . . . 7  |-  ( ph  ->  ( 1  sigma  B )  e.  NN )
2625nnzd 10267 . . . . . 6  |-  ( ph  ->  ( 1  sigma  B )  e.  ZZ )
27 dvdsmul1 12758 . . . . . 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 9963 . . . . . . . . 9  |-  2  e.  CC
30 expp1 11275 . . . . . . . . 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 11281 . . . . . . . . . . 11  |-  ( ( 2  e.  NN  /\  A  e.  NN0 )  -> 
( 2 ^ A
)  e.  NN )
331, 3, 32sylancr 644 . . . . . . . . . 10  |-  ( ph  ->  ( 2 ^ A
)  e.  NN )
3433nncnd 9909 . . . . . . . . 9  |-  ( ph  ->  ( 2 ^ A
)  e.  CC )
35 mulcom 8970 . . . . . . . . 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 2398 . . . . . . 7  |-  ( ph  ->  ( 2 ^ ( A  +  1 ) )  =  ( 2  x.  ( 2 ^ A ) ) )
3837oveq1d 5996 . . . . . 6  |-  ( ph  ->  ( ( 2 ^ ( A  +  1 ) )  x.  B
)  =  ( ( 2  x.  ( 2 ^ A ) )  x.  B ) )
3929a1i 10 . . . . . . . 8  |-  ( ph  ->  2  e.  CC )
4023nncnd 9909 . . . . . . . 8  |-  ( ph  ->  B  e.  CC )
4139, 34, 40mulassd 9005 . . . . . . 7  |-  ( ph  ->  ( ( 2  x.  ( 2 ^ A
) )  x.  B
)  =  ( 2  x.  ( ( 2 ^ A )  x.  B ) ) )
42 ax-1cn 8942 . . . . . . . . . 10  |-  1  e.  CC
4342a1i 10 . . . . . . . . 9  |-  ( ph  ->  1  e.  CC )
44 perfectlem.3 . . . . . . . . . . 11  |-  ( ph  ->  -.  2  ||  B
)
45 2prm 12982 . . . . . . . . . . . 12  |-  2  e.  Prime
4623nnzd 10267 . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  ZZ )
47 coprm 12987 . . . . . . . . . . . 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 10205 . . . . . . . . . . . 12  |-  2  e.  ZZ
5150a1i 10 . . . . . . . . . . 11  |-  ( ph  ->  2  e.  ZZ )
52 rpexp1i 13008 . . . . . . . . . . 11  |-  ( ( 2  e.  ZZ  /\  B  e.  ZZ  /\  A  e.  NN0 )  ->  (
( 2  gcd  B
)  =  1  -> 
( ( 2 ^ A )  gcd  B
)  =  1 ) )
5351, 46, 3, 52syl3anc 1183 . . . . . . . . . 10  |-  ( ph  ->  ( ( 2  gcd 
B )  =  1  ->  ( ( 2 ^ A )  gcd 
B )  =  1 ) )
5449, 53mpd 14 . . . . . . . . 9  |-  ( ph  ->  ( ( 2 ^ A )  gcd  B
)  =  1 )
55 sgmmul 20663 . . . . . . . . 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 1185 . . . . . . . 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 9909 . . . . . . . . . . . . 13  |-  ( ph  ->  A  e.  CC )
59 pncan 9204 . . . . . . . . . . . . 13  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  + 
1 )  -  1 )  =  A )
6058, 42, 59sylancl 643 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( A  + 
1 )  -  1 )  =  A )
6160oveq2d 5997 . . . . . . . . . . 11  |-  ( ph  ->  ( 2 ^ (
( A  +  1 )  -  1 ) )  =  ( 2 ^ A ) )
6261oveq2d 5997 . . . . . . . . . 10  |-  ( ph  ->  ( 1  sigma  ( 2 ^ ( ( A  +  1 )  - 
1 ) ) )  =  ( 1  sigma 
( 2 ^ A
) ) )
63 1sgm2ppw 20662 . . . . . . . . . . 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 2400 . . . . . . . . 9  |-  ( ph  ->  ( 1  sigma  ( 2 ^ A ) )  =  ( ( 2 ^ ( A  + 
1 ) )  - 
1 ) )
6665oveq1d 5996 . . . . . . . 8  |-  ( ph  ->  ( ( 1  sigma 
( 2 ^ A
) )  x.  (
1  sigma  B ) )  =  ( ( ( 2 ^ ( A  +  1 ) )  -  1 )  x.  ( 1  sigma  B ) ) )
6756, 57, 663eqtr3d 2406 . . . . . . 7  |-  ( ph  ->  ( 2  x.  (
( 2 ^ A
)  x.  B ) )  =  ( ( ( 2 ^ ( A  +  1 ) )  -  1 )  x.  ( 1  sigma  B ) ) )
6841, 67eqtrd 2398 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  ( 2 ^ A
) )  x.  B
)  =  ( ( ( 2 ^ ( A  +  1 ) )  -  1 )  x.  ( 1  sigma  B ) ) )
6938, 68eqtrd 2398 . . . . 5  |-  ( ph  ->  ( ( 2 ^ ( A  +  1 ) )  x.  B
)  =  ( ( ( 2 ^ ( A  +  1 ) )  -  1 )  x.  ( 1  sigma  B ) ) )
7028, 69breqtrrd 4151 . . . 4  |-  ( ph  ->  ( ( 2 ^ ( A  +  1 ) )  -  1 )  ||  ( ( 2 ^ ( A  +  1 ) )  x.  B ) )
71 gcdcom 12907 . . . . . 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 12760 . . . . . . . . 9  |-  ( ( 2  e.  ZZ  /\  ( A  +  1
)  e.  NN )  ->  2  ||  (
2 ^ ( A  +  1 ) ) )
7450, 10, 73sylancr 644 . . . . . . . 8  |-  ( ph  ->  2  ||  ( 2 ^ ( A  + 
1 ) ) )
75 nprmdvds1 12998 . . . . . . . . . . 11  |-  ( 2  e.  Prime  ->  -.  2  ||  1 )
7645, 75ax-mp 8 . . . . . . . . . 10  |-  -.  2  ||  1
77 1z 10204 . . . . . . . . . . . . 13  |-  1  e.  ZZ
7877a1i 10 . . . . . . . . . . . 12  |-  ( ph  ->  1  e.  ZZ )
7951, 19, 783jca 1133 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  e.  ZZ  /\  ( 2 ^ ( A  +  1 ) )  e.  ZZ  /\  1  e.  ZZ )
)
80 dvdssub2 12774 . . . . . . . . . . 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 12987 . . . . . . . 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 13008 . . . . . . 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 1183 . . . . . 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 2398 . . . 4  |-  ( ph  ->  ( ( ( 2 ^ ( A  + 
1 ) )  - 
1 )  gcd  (
2 ^ ( A  +  1 ) ) )  =  1 )
92 coprmdvds 12989 . . . . 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 1183 . . . 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 12745 . . . 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 1133 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 935    = wceq 1647    e. wcel 1715   class class class wbr 4125  (class class class)co 5981   CCcc 8882   RRcr 8883   1c1 8885    + caddc 8887    x. cmul 8889    < clt 9014    - cmin 9184    / cdiv 9570   NNcn 9893   2c2 9942   NN0cn0 10114   ZZcz 10175   ^cexp 11269    || cdivides 12739    gcd cgcd 12893   Primecprime 12966    sigma csgm 20556
This theorem is referenced by:  perfectlem2  20692
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-inf2 7489  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-pre-sup 8962  ax-addf 8963  ax-mulf 8964
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-iin 4010  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-se 4456  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-isom 5367  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-of 6205  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-2o 6622  df-oadd 6625  df-er 6802  df-map 6917  df-pm 6918  df-ixp 6961  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-fi 7312  df-sup 7341  df-oi 7372  df-card 7719  df-cda 7941  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-nn 9894  df-2 9951  df-3 9952  df-4 9953  df-5 9954  df-6 9955  df-7 9956  df-8 9957  df-9 9958  df-10 9959  df-n0 10115  df-z 10176  df-dec 10276  df-uz 10382  df-q 10468  df-rp 10506  df-xneg 10603  df-xadd 10604  df-xmul 10605  df-ioo 10813  df-ioc 10814  df-ico 10815  df-icc 10816  df-fz 10936  df-fzo 11026  df-fl 11089  df-mod 11138  df-seq 11211  df-exp 11270  df-fac 11454  df-bc 11481  df-hash 11506  df-shft 11769  df-cj 11791  df-re 11792  df-im 11793  df-sqr 11927  df-abs 11928  df-limsup 12152  df-clim 12169  df-rlim 12170  df-sum 12367  df-ef 12557  df-sin 12559  df-cos 12560  df-pi 12562  df-dvds 12740  df-gcd 12894  df-prm 12967  df-pc 13098  df-struct 13358  df-ndx 13359  df-slot 13360  df-base 13361  df-sets 13362  df-ress 13363  df-plusg 13429  df-mulr 13430  df-starv 13431  df-sca 13432  df-vsca 13433  df-tset 13435  df-ple 13436  df-ds 13438  df-unif 13439  df-hom 13440  df-cco 13441  df-rest 13537  df-topn 13538  df-topgen 13554  df-pt 13555  df-prds 13558  df-xrs 13613  df-0g 13614  df-gsum 13615  df-qtop 13620  df-imas 13621  df-xps 13623  df-mre 13698  df-mrc 13699  df-acs 13701  df-mnd 14577  df-submnd 14626  df-mulg 14702  df-cntz 15003  df-cmn 15301  df-xmet 16586  df-met 16587  df-bl 16588  df-mopn 16589  df-fbas 16590  df-fg 16591  df-cnfld 16594  df-top 16853  df-bases 16855  df-topon 16856  df-topsp 16857  df-cld 16973  df-ntr 16974  df-cls 16975  df-nei 17052  df-lp 17085  df-perf 17086  df-cn 17174  df-cnp 17175  df-haus 17260  df-tx 17474  df-hmeo 17663  df-fil 17754  df-fm 17846  df-flim 17847  df-flf 17848  df-xms 18098  df-ms 18099  df-tms 18100  df-cncf 18596  df-limc 19431  df-dv 19432  df-log 20132  df-cxp 20133  df-sgm 20562
  Copyright terms: Public domain W3C validator