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Theorem perfectlem1 22590
Description: Lemma for perfect 22592. (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 10500 . . 3  |-  2  e.  NN
2 perfectlem.1 . . . . 5  |-  ( ph  ->  A  e.  NN )
32nnnn0d 10657 . . . 4  |-  ( ph  ->  A  e.  NN0 )
4 peano2nn0 10641 . . . 4  |-  ( A  e.  NN0  ->  ( A  +  1 )  e. 
NN0 )
53, 4syl 16 . . 3  |-  ( ph  ->  ( A  +  1 )  e.  NN0 )
6 nnexpcl 11899 . . 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 10412 . . . . 5  |-  2  e.  RR
98a1i 11 . . . 4  |-  ( ph  ->  2  e.  RR )
102peano2nnd 10360 . . . 4  |-  ( ph  ->  ( A  +  1 )  e.  NN )
11 1lt2 10509 . . . . 5  |-  1  <  2
1211a1i 11 . . . 4  |-  ( ph  ->  1  <  2 )
13 expgt1 11923 . . . 4  |-  ( ( 2  e.  RR  /\  ( A  +  1
)  e.  NN  /\  1  <  2 )  -> 
1  <  ( 2 ^ ( A  + 
1 ) ) )
149, 10, 12, 13syl3anc 1218 . . 3  |-  ( ph  ->  1  <  ( 2 ^ ( A  + 
1 ) ) )
15 1nn 10354 . . . 4  |-  1  e.  NN
16 nnsub 10381 . . . 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 10767 . . . . . . 7  |-  ( ph  ->  ( 2 ^ ( A  +  1 ) )  e.  ZZ )
20 peano2zm 10709 . . . . . . 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 10616 . . . . . . . 8  |-  1  e.  NN0
23 perfectlem.2 . . . . . . . 8  |-  ( ph  ->  B  e.  NN )
24 sgmnncl 22507 . . . . . . . 8  |-  ( ( 1  e.  NN0  /\  B  e.  NN )  ->  ( 1  sigma  B )  e.  NN )
2522, 23, 24sylancr 663 . . . . . . 7  |-  ( ph  ->  ( 1  sigma  B )  e.  NN )
2625nnzd 10767 . . . . . 6  |-  ( ph  ->  ( 1  sigma  B )  e.  ZZ )
27 dvdsmul1 13575 . . . . . 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 10413 . . . . . . . . 9  |-  2  e.  CC
30 expp1 11893 . . . . . . . . 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 11899 . . . . . . . . . . 11  |-  ( ( 2  e.  NN  /\  A  e.  NN0 )  -> 
( 2 ^ A
)  e.  NN )
331, 3, 32sylancr 663 . . . . . . . . . 10  |-  ( ph  ->  ( 2 ^ A
)  e.  NN )
3433nncnd 10359 . . . . . . . . 9  |-  ( ph  ->  ( 2 ^ A
)  e.  CC )
35 mulcom 9389 . . . . . . . . 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 2475 . . . . . . 7  |-  ( ph  ->  ( 2 ^ ( A  +  1 ) )  =  ( 2  x.  ( 2 ^ A ) ) )
3837oveq1d 6127 . . . . . 6  |-  ( ph  ->  ( ( 2 ^ ( A  +  1 ) )  x.  B
)  =  ( ( 2  x.  ( 2 ^ A ) )  x.  B ) )
3929a1i 11 . . . . . . 7  |-  ( ph  ->  2  e.  CC )
4023nncnd 10359 . . . . . . 7  |-  ( ph  ->  B  e.  CC )
4139, 34, 40mulassd 9430 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  ( 2 ^ A
) )  x.  B
)  =  ( 2  x.  ( ( 2 ^ A )  x.  B ) ) )
42 ax-1cn 9361 . . . . . . . . 9  |-  1  e.  CC
4342a1i 11 . . . . . . . 8  |-  ( ph  ->  1  e.  CC )
44 perfectlem.3 . . . . . . . . . 10  |-  ( ph  ->  -.  2  ||  B
)
45 2prm 13800 . . . . . . . . . . 11  |-  2  e.  Prime
4623nnzd 10767 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  ZZ )
47 coprm 13807 . . . . . . . . . . 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 10699 . . . . . . . . . . 11  |-  2  e.  ZZ
5150a1i 11 . . . . . . . . . 10  |-  ( ph  ->  2  e.  ZZ )
52 rpexp1i 13828 . . . . . . . . . 10  |-  ( ( 2  e.  ZZ  /\  B  e.  ZZ  /\  A  e.  NN0 )  ->  (
( 2  gcd  B
)  =  1  -> 
( ( 2 ^ A )  gcd  B
)  =  1 ) )
5351, 46, 3, 52syl3anc 1218 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  gcd 
B )  =  1  ->  ( ( 2 ^ A )  gcd 
B )  =  1 ) )
5449, 53mpd 15 . . . . . . . 8  |-  ( ph  ->  ( ( 2 ^ A )  gcd  B
)  =  1 )
55 sgmmul 22562 . . . . . . . 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 1220 . . . . . . 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 10359 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  CC )
59 pncan 9637 . . . . . . . . . . . 12  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  + 
1 )  -  1 )  =  A )
6058, 42, 59sylancl 662 . . . . . . . . . . 11  |-  ( ph  ->  ( ( A  + 
1 )  -  1 )  =  A )
6160oveq2d 6128 . . . . . . . . . 10  |-  ( ph  ->  ( 2 ^ (
( A  +  1 )  -  1 ) )  =  ( 2 ^ A ) )
6261oveq2d 6128 . . . . . . . . 9  |-  ( ph  ->  ( 1  sigma  ( 2 ^ ( ( A  +  1 )  - 
1 ) ) )  =  ( 1  sigma 
( 2 ^ A
) ) )
63 1sgm2ppw 22561 . . . . . . . . . 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 2477 . . . . . . . 8  |-  ( ph  ->  ( 1  sigma  ( 2 ^ A ) )  =  ( ( 2 ^ ( A  + 
1 ) )  - 
1 ) )
6665oveq1d 6127 . . . . . . 7  |-  ( ph  ->  ( ( 1  sigma 
( 2 ^ A
) )  x.  (
1  sigma  B ) )  =  ( ( ( 2 ^ ( A  +  1 ) )  -  1 )  x.  ( 1  sigma  B ) ) )
6756, 57, 663eqtr3d 2483 . . . . . 6  |-  ( ph  ->  ( 2  x.  (
( 2 ^ A
)  x.  B ) )  =  ( ( ( 2 ^ ( A  +  1 ) )  -  1 )  x.  ( 1  sigma  B ) ) )
6838, 41, 673eqtrd 2479 . . . . 5  |-  ( ph  ->  ( ( 2 ^ ( A  +  1 ) )  x.  B
)  =  ( ( ( 2 ^ ( A  +  1 ) )  -  1 )  x.  ( 1  sigma  B ) ) )
6928, 68breqtrrd 4339 . . . 4  |-  ( ph  ->  ( ( 2 ^ ( A  +  1 ) )  -  1 )  ||  ( ( 2 ^ ( A  +  1 ) )  x.  B ) )
70 gcdcom 13725 . . . . . 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 13577 . . . . . . . . 9  |-  ( ( 2  e.  ZZ  /\  ( A  +  1
)  e.  NN )  ->  2  ||  (
2 ^ ( A  +  1 ) ) )
7350, 10, 72sylancr 663 . . . . . . . 8  |-  ( ph  ->  2  ||  ( 2 ^ ( A  + 
1 ) ) )
74 n2dvds1 13603 . . . . . . . . . 10  |-  -.  2  ||  1
75 1zzd 10698 . . . . . . . . . . . 12  |-  ( ph  ->  1  e.  ZZ )
7651, 19, 753jca 1168 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  e.  ZZ  /\  ( 2 ^ ( A  +  1 ) )  e.  ZZ  /\  1  e.  ZZ )
)
77 dvdssub2 13591 . . . . . . . . . . 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 13807 . . . . . . . 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 13828 . . . . . . 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 1218 . . . . . 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 2475 . . . 4  |-  ( ph  ->  ( ( ( 2 ^ ( A  + 
1 ) )  - 
1 )  gcd  (
2 ^ ( A  +  1 ) ) )  =  1 )
89 coprmdvds 13809 . . . . 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 1218 . . . 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 13562 . . . 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 1168 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 965    = wceq 1369    e. wcel 1756   class class class wbr 4313  (class class class)co 6112   CCcc 9301   RRcr 9302   1c1 9304    + caddc 9306    x. cmul 9308    < clt 9439    - cmin 9616    / cdiv 10014   NNcn 10343   2c2 10392   NN0cn0 10600   ZZcz 10667   ^cexp 11886    || cdivides 13556    gcd cgcd 13711   Primecprime 13784    sigma csgm 22455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381  ax-addf 9382  ax-mulf 9383
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-iin 4195  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-of 6341  df-om 6498  df-1st 6598  df-2nd 6599  df-supp 6712  df-recs 6853  df-rdg 6887  df-1o 6941  df-2o 6942  df-oadd 6945  df-er 7122  df-map 7237  df-pm 7238  df-ixp 7285  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-fsupp 7642  df-fi 7682  df-sup 7712  df-oi 7745  df-card 8130  df-cda 8358  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-7 10406  df-8 10407  df-9 10408  df-10 10409  df-n0 10601  df-z 10668  df-dec 10777  df-uz 10883  df-q 10975  df-rp 11013  df-xneg 11110  df-xadd 11111  df-xmul 11112  df-ioo 11325  df-ioc 11326  df-ico 11327  df-icc 11328  df-fz 11459  df-fzo 11570  df-fl 11663  df-mod 11730  df-seq 11828  df-exp 11887  df-fac 12073  df-bc 12100  df-hash 12125  df-shft 12577  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-limsup 12970  df-clim 12987  df-rlim 12988  df-sum 13185  df-ef 13374  df-sin 13376  df-cos 13377  df-pi 13379  df-dvds 13557  df-gcd 13712  df-prm 13785  df-pc 13925  df-struct 14197  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-ress 14202  df-plusg 14272  df-mulr 14273  df-starv 14274  df-sca 14275  df-vsca 14276  df-ip 14277  df-tset 14278  df-ple 14279  df-ds 14281  df-unif 14282  df-hom 14283  df-cco 14284  df-rest 14382  df-topn 14383  df-0g 14401  df-gsum 14402  df-topgen 14403  df-pt 14404  df-prds 14407  df-xrs 14461  df-qtop 14466  df-imas 14467  df-xps 14469  df-mre 14545  df-mrc 14546  df-acs 14548  df-mnd 15436  df-submnd 15486  df-mulg 15569  df-cntz 15856  df-cmn 16300  df-psmet 17831  df-xmet 17832  df-met 17833  df-bl 17834  df-mopn 17835  df-fbas 17836  df-fg 17837  df-cnfld 17841  df-top 18525  df-bases 18527  df-topon 18528  df-topsp 18529  df-cld 18645  df-ntr 18646  df-cls 18647  df-nei 18724  df-lp 18762  df-perf 18763  df-cn 18853  df-cnp 18854  df-haus 18941  df-tx 19157  df-hmeo 19350  df-fil 19441  df-fm 19533  df-flim 19534  df-flf 19535  df-xms 19917  df-ms 19918  df-tms 19919  df-cncf 20476  df-limc 21363  df-dv 21364  df-log 22030  df-cxp 22031  df-sgm 22461
This theorem is referenced by:  perfectlem2  22591
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