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Theorem perfectlem1 23329
Description: Lemma for perfect 23331. (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 10694 . . 3  |-  2  e.  NN
2 perfectlem.1 . . . . 5  |-  ( ph  ->  A  e.  NN )
32nnnn0d 10853 . . . 4  |-  ( ph  ->  A  e.  NN0 )
4 peano2nn0 10837 . . . 4  |-  ( A  e.  NN0  ->  ( A  +  1 )  e. 
NN0 )
53, 4syl 16 . . 3  |-  ( ph  ->  ( A  +  1 )  e.  NN0 )
6 nnexpcl 12148 . . 3  |-  ( ( 2  e.  NN  /\  ( A  +  1
)  e.  NN0 )  ->  ( 2 ^ ( A  +  1 ) )  e.  NN )
71, 5, 6sylancr 663 . 2  |-  ( ph  ->  ( 2 ^ ( A  +  1 ) )  e.  NN )
8 2re 10606 . . . . 5  |-  2  e.  RR
98a1i 11 . . . 4  |-  ( ph  ->  2  e.  RR )
102peano2nnd 10554 . . . 4  |-  ( ph  ->  ( A  +  1 )  e.  NN )
11 1lt2 10703 . . . . 5  |-  1  <  2
1211a1i 11 . . . 4  |-  ( ph  ->  1  <  2 )
13 expgt1 12173 . . . 4  |-  ( ( 2  e.  RR  /\  ( A  +  1
)  e.  NN  /\  1  <  2 )  -> 
1  <  ( 2 ^ ( A  + 
1 ) ) )
149, 10, 12, 13syl3anc 1228 . . 3  |-  ( ph  ->  1  <  ( 2 ^ ( A  + 
1 ) ) )
15 1nn 10548 . . . 4  |-  1  e.  NN
16 nnsub 10575 . . . 4  |-  ( ( 1  e.  NN  /\  ( 2 ^ ( A  +  1 ) )  e.  NN )  ->  ( 1  < 
( 2 ^ ( A  +  1 ) )  <->  ( ( 2 ^ ( A  + 
1 ) )  - 
1 )  e.  NN ) )
1715, 7, 16sylancr 663 . . 3  |-  ( ph  ->  ( 1  <  (
2 ^ ( A  +  1 ) )  <-> 
( ( 2 ^ ( A  +  1 ) )  -  1 )  e.  NN ) )
1814, 17mpbid 210 . 2  |-  ( ph  ->  ( ( 2 ^ ( A  +  1 ) )  -  1 )  e.  NN )
197nnzd 10966 . . . . . . 7  |-  ( ph  ->  ( 2 ^ ( A  +  1 ) )  e.  ZZ )
20 peano2zm 10907 . . . . . . 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 10812 . . . . . . . 8  |-  1  e.  NN0
23 perfectlem.2 . . . . . . . 8  |-  ( ph  ->  B  e.  NN )
24 sgmnncl 23246 . . . . . . . 8  |-  ( ( 1  e.  NN0  /\  B  e.  NN )  ->  ( 1  sigma  B )  e.  NN )
2522, 23, 24sylancr 663 . . . . . . 7  |-  ( ph  ->  ( 1  sigma  B )  e.  NN )
2625nnzd 10966 . . . . . 6  |-  ( ph  ->  ( 1  sigma  B )  e.  ZZ )
27 dvdsmul1 13869 . . . . . 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 661 . . . . 5  |-  ( ph  ->  ( ( 2 ^ ( A  +  1 ) )  -  1 )  ||  ( ( ( 2 ^ ( A  +  1 ) )  -  1 )  x.  ( 1  sigma  B ) ) )
29 2cn 10607 . . . . . . . . 9  |-  2  e.  CC
30 expp1 12142 . . . . . . . . 9  |-  ( ( 2  e.  CC  /\  A  e.  NN0 )  -> 
( 2 ^ ( A  +  1 ) )  =  ( ( 2 ^ A )  x.  2 ) )
3129, 3, 30sylancr 663 . . . . . . . 8  |-  ( ph  ->  ( 2 ^ ( A  +  1 ) )  =  ( ( 2 ^ A )  x.  2 ) )
32 nnexpcl 12148 . . . . . . . . . . 11  |-  ( ( 2  e.  NN  /\  A  e.  NN0 )  -> 
( 2 ^ A
)  e.  NN )
331, 3, 32sylancr 663 . . . . . . . . . 10  |-  ( ph  ->  ( 2 ^ A
)  e.  NN )
3433nncnd 10553 . . . . . . . . 9  |-  ( ph  ->  ( 2 ^ A
)  e.  CC )
35 mulcom 9579 . . . . . . . . 9  |-  ( ( ( 2 ^ A
)  e.  CC  /\  2  e.  CC )  ->  ( ( 2 ^ A )  x.  2 )  =  ( 2  x.  ( 2 ^ A ) ) )
3634, 29, 35sylancl 662 . . . . . . . 8  |-  ( ph  ->  ( ( 2 ^ A )  x.  2 )  =  ( 2  x.  ( 2 ^ A ) ) )
3731, 36eqtrd 2508 . . . . . . 7  |-  ( ph  ->  ( 2 ^ ( A  +  1 ) )  =  ( 2  x.  ( 2 ^ A ) ) )
3837oveq1d 6300 . . . . . 6  |-  ( ph  ->  ( ( 2 ^ ( A  +  1 ) )  x.  B
)  =  ( ( 2  x.  ( 2 ^ A ) )  x.  B ) )
3929a1i 11 . . . . . . 7  |-  ( ph  ->  2  e.  CC )
4023nncnd 10553 . . . . . . 7  |-  ( ph  ->  B  e.  CC )
4139, 34, 40mulassd 9620 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  ( 2 ^ A
) )  x.  B
)  =  ( 2  x.  ( ( 2 ^ A )  x.  B ) ) )
42 ax-1cn 9551 . . . . . . . . 9  |-  1  e.  CC
4342a1i 11 . . . . . . . 8  |-  ( ph  ->  1  e.  CC )
44 perfectlem.3 . . . . . . . . . 10  |-  ( ph  ->  -.  2  ||  B
)
45 2prm 14095 . . . . . . . . . . 11  |-  2  e.  Prime
4623nnzd 10966 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  ZZ )
47 coprm 14103 . . . . . . . . . . 11  |-  ( ( 2  e.  Prime  /\  B  e.  ZZ )  ->  ( -.  2  ||  B  <->  ( 2  gcd  B )  =  1 ) )
4845, 46, 47sylancr 663 . . . . . . . . . 10  |-  ( ph  ->  ( -.  2  ||  B 
<->  ( 2  gcd  B
)  =  1 ) )
4944, 48mpbid 210 . . . . . . . . 9  |-  ( ph  ->  ( 2  gcd  B
)  =  1 )
50 2z 10897 . . . . . . . . . . 11  |-  2  e.  ZZ
5150a1i 11 . . . . . . . . . 10  |-  ( ph  ->  2  e.  ZZ )
52 rpexp1i 14124 . . . . . . . . . 10  |-  ( ( 2  e.  ZZ  /\  B  e.  ZZ  /\  A  e.  NN0 )  ->  (
( 2  gcd  B
)  =  1  -> 
( ( 2 ^ A )  gcd  B
)  =  1 ) )
5351, 46, 3, 52syl3anc 1228 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  gcd 
B )  =  1  ->  ( ( 2 ^ A )  gcd 
B )  =  1 ) )
5449, 53mpd 15 . . . . . . . 8  |-  ( ph  ->  ( ( 2 ^ A )  gcd  B
)  =  1 )
55 sgmmul 23301 . . . . . . . 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 1230 . . . . . . 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 10553 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  CC )
59 pncan 9827 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  + 
1 )  -  1 )  =  A )
6058, 42, 59sylancl 662 . . . . . . . . . . 11  |-  ( ph  ->  ( ( A  + 
1 )  -  1 )  =  A )
6160oveq2d 6301 . . . . . . . . . 10  |-  ( ph  ->  ( 2 ^ (
( A  +  1 )  -  1 ) )  =  ( 2 ^ A ) )
6261oveq2d 6301 . . . . . . . . 9  |-  ( ph  ->  ( 1  sigma  ( 2 ^ ( ( A  +  1 )  - 
1 ) ) )  =  ( 1  sigma 
( 2 ^ A
) ) )
63 1sgm2ppw 23300 . . . . . . . . . 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 2510 . . . . . . . 8  |-  ( ph  ->  ( 1  sigma  ( 2 ^ A ) )  =  ( ( 2 ^ ( A  + 
1 ) )  - 
1 ) )
6665oveq1d 6300 . . . . . . 7  |-  ( ph  ->  ( ( 1  sigma 
( 2 ^ A
) )  x.  (
1  sigma  B ) )  =  ( ( ( 2 ^ ( A  +  1 ) )  -  1 )  x.  ( 1  sigma  B ) ) )
6756, 57, 663eqtr3d 2516 . . . . . 6  |-  ( ph  ->  ( 2  x.  (
( 2 ^ A
)  x.  B ) )  =  ( ( ( 2 ^ ( A  +  1 ) )  -  1 )  x.  ( 1  sigma  B ) ) )
6838, 41, 673eqtrd 2512 . . . . 5  |-  ( ph  ->  ( ( 2 ^ ( A  +  1 ) )  x.  B
)  =  ( ( ( 2 ^ ( A  +  1 ) )  -  1 )  x.  ( 1  sigma  B ) ) )
6928, 68breqtrrd 4473 . . . 4  |-  ( ph  ->  ( ( 2 ^ ( A  +  1 ) )  -  1 )  ||  ( ( 2 ^ ( A  +  1 ) )  x.  B ) )
70 gcdcom 14020 . . . . . 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 661 . . . . 5  |-  ( ph  ->  ( ( ( 2 ^ ( A  + 
1 ) )  - 
1 )  gcd  (
2 ^ ( A  +  1 ) ) )  =  ( ( 2 ^ ( A  +  1 ) )  gcd  ( ( 2 ^ ( A  + 
1 ) )  - 
1 ) ) )
72 iddvdsexp 13871 . . . . . . . . 9  |-  ( ( 2  e.  ZZ  /\  ( A  +  1
)  e.  NN )  ->  2  ||  (
2 ^ ( A  +  1 ) ) )
7350, 10, 72sylancr 663 . . . . . . . 8  |-  ( ph  ->  2  ||  ( 2 ^ ( A  + 
1 ) ) )
74 n2dvds1 13897 . . . . . . . . . 10  |-  -.  2  ||  1
75 1zzd 10896 . . . . . . . . . . . 12  |-  ( ph  ->  1  e.  ZZ )
7651, 19, 753jca 1176 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  e.  ZZ  /\  ( 2 ^ ( A  +  1 ) )  e.  ZZ  /\  1  e.  ZZ )
)
77 dvdssub2 13885 . . . . . . . . . . 11  |-  ( ( ( 2  e.  ZZ  /\  ( 2 ^ ( A  +  1 ) )  e.  ZZ  /\  1  e.  ZZ )  /\  2  ||  ( ( 2 ^ ( A  +  1 ) )  -  1 ) )  ->  ( 2  ||  ( 2 ^ ( A  +  1 ) )  <->  2  ||  1
) )
7876, 77sylan 471 . . . . . . . . . 10  |-  ( (
ph  /\  2  ||  ( ( 2 ^ ( A  +  1 ) )  -  1 ) )  ->  (
2  ||  ( 2 ^ ( A  + 
1 ) )  <->  2  ||  1 ) )
7974, 78mtbiri 303 . . . . . . . . 9  |-  ( (
ph  /\  2  ||  ( ( 2 ^ ( A  +  1 ) )  -  1 ) )  ->  -.  2  ||  ( 2 ^ ( A  +  1 ) ) )
8079ex 434 . . . . . . . 8  |-  ( ph  ->  ( 2  ||  (
( 2 ^ ( A  +  1 ) )  -  1 )  ->  -.  2  ||  ( 2 ^ ( A  +  1 ) ) ) )
8173, 80mt2d 117 . . . . . . 7  |-  ( ph  ->  -.  2  ||  (
( 2 ^ ( A  +  1 ) )  -  1 ) )
82 coprm 14103 . . . . . . . 8  |-  ( ( 2  e.  Prime  /\  (
( 2 ^ ( A  +  1 ) )  -  1 )  e.  ZZ )  -> 
( -.  2  ||  ( ( 2 ^ ( A  +  1 ) )  -  1 )  <->  ( 2  gcd  ( ( 2 ^ ( A  +  1 ) )  -  1 ) )  =  1 ) )
8345, 21, 82sylancr 663 . . . . . . 7  |-  ( ph  ->  ( -.  2  ||  ( ( 2 ^ ( A  +  1 ) )  -  1 )  <->  ( 2  gcd  ( ( 2 ^ ( A  +  1 ) )  -  1 ) )  =  1 ) )
8481, 83mpbid 210 . . . . . 6  |-  ( ph  ->  ( 2  gcd  (
( 2 ^ ( A  +  1 ) )  -  1 ) )  =  1 )
85 rpexp1i 14124 . . . . . . 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 ) )
8651, 21, 5, 85syl3anc 1228 . . . . . 6  |-  ( ph  ->  ( ( 2  gcd  ( ( 2 ^ ( A  +  1 ) )  -  1 ) )  =  1  ->  ( ( 2 ^ ( A  + 
1 ) )  gcd  ( ( 2 ^ ( A  +  1 ) )  -  1 ) )  =  1 ) )
8784, 86mpd 15 . . . . 5  |-  ( ph  ->  ( ( 2 ^ ( A  +  1 ) )  gcd  (
( 2 ^ ( A  +  1 ) )  -  1 ) )  =  1 )
8871, 87eqtrd 2508 . . . 4  |-  ( ph  ->  ( ( ( 2 ^ ( A  + 
1 ) )  - 
1 )  gcd  (
2 ^ ( A  +  1 ) ) )  =  1 )
89 coprmdvds 14105 . . . . 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 ) )
9021, 19, 46, 89syl3anc 1228 . . . 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 ) )
9169, 88, 90mp2and 679 . . 3  |-  ( ph  ->  ( ( 2 ^ ( A  +  1 ) )  -  1 )  ||  B )
92 nndivdvds 13856 . . . 4  |-  ( ( B  e.  NN  /\  ( ( 2 ^ ( A  +  1 ) )  -  1 )  e.  NN )  ->  ( ( ( 2 ^ ( A  +  1 ) )  -  1 )  ||  B 
<->  ( B  /  (
( 2 ^ ( A  +  1 ) )  -  1 ) )  e.  NN ) )
9323, 18, 92syl2anc 661 . . 3  |-  ( ph  ->  ( ( ( 2 ^ ( A  + 
1 ) )  - 
1 )  ||  B  <->  ( B  /  ( ( 2 ^ ( A  +  1 ) )  -  1 ) )  e.  NN ) )
9491, 93mpbid 210 . 2  |-  ( ph  ->  ( B  /  (
( 2 ^ ( A  +  1 ) )  -  1 ) )  e.  NN )
957, 18, 943jca 1176 1  |-  ( ph  ->  ( ( 2 ^ ( A  +  1 ) )  e.  NN  /\  ( ( 2 ^ ( A  +  1 ) )  -  1 )  e.  NN  /\  ( B  /  (
( 2 ^ ( A  +  1 ) )  -  1 ) )  e.  NN ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   class class class wbr 4447  (class class class)co 6285   CCcc 9491   RRcr 9492   1c1 9494    + caddc 9496    x. cmul 9498    < clt 9629    - cmin 9806    / cdiv 10207   NNcn 10537   2c2 10586   NN0cn0 10796   ZZcz 10865   ^cexp 12135    || cdivides 13850    gcd cgcd 14006   Primecprime 14079    sigma csgm 23194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-inf2 8059  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571  ax-addf 9572  ax-mulf 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6903  df-recs 7043  df-rdg 7077  df-1o 7131  df-2o 7132  df-oadd 7135  df-er 7312  df-map 7423  df-pm 7424  df-ixp 7471  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-fsupp 7831  df-fi 7872  df-sup 7902  df-oi 7936  df-card 8321  df-cda 8549  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10978  df-uz 11084  df-q 11184  df-rp 11222  df-xneg 11319  df-xadd 11320  df-xmul 11321  df-ioo 11534  df-ioc 11535  df-ico 11536  df-icc 11537  df-fz 11674  df-fzo 11794  df-fl 11898  df-mod 11966  df-seq 12077  df-exp 12136  df-fac 12323  df-bc 12350  df-hash 12375  df-shft 12866  df-cj 12898  df-re 12899  df-im 12900  df-sqrt 13034  df-abs 13035  df-limsup 13260  df-clim 13277  df-rlim 13278  df-sum 13475  df-ef 13668  df-sin 13670  df-cos 13671  df-pi 13673  df-dvds 13851  df-gcd 14007  df-prm 14080  df-pc 14223  df-struct 14495  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-ress 14500  df-plusg 14571  df-mulr 14572  df-starv 14573  df-sca 14574  df-vsca 14575  df-ip 14576  df-tset 14577  df-ple 14578  df-ds 14580  df-unif 14581  df-hom 14582  df-cco 14583  df-rest 14681  df-topn 14682  df-0g 14700  df-gsum 14701  df-topgen 14702  df-pt 14703  df-prds 14706  df-xrs 14760  df-qtop 14765  df-imas 14766  df-xps 14768  df-mre 14844  df-mrc 14845  df-acs 14847  df-mnd 15735  df-submnd 15790  df-mulg 15874  df-cntz 16169  df-cmn 16615  df-psmet 18222  df-xmet 18223  df-met 18224  df-bl 18225  df-mopn 18226  df-fbas 18227  df-fg 18228  df-cnfld 18232  df-top 19206  df-bases 19208  df-topon 19209  df-topsp 19210  df-cld 19326  df-ntr 19327  df-cls 19328  df-nei 19405  df-lp 19443  df-perf 19444  df-cn 19534  df-cnp 19535  df-haus 19622  df-tx 19890  df-hmeo 20083  df-fil 20174  df-fm 20266  df-flim 20267  df-flf 20268  df-xms 20650  df-ms 20651  df-tms 20652  df-cncf 21209  df-limc 22097  df-dv 22098  df-log 22769  df-cxp 22770  df-sgm 23200
This theorem is referenced by:  perfectlem2  23330
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