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Theorem dirith2 20677
Description: Dirichlet's theorem: there are infinitely many primes in any arithmetic progression coprime to  N. Theorem 9.4.1 of [Shapiro], p. 375. (Contributed by Mario Carneiro, 30-Apr-2016.) (Proof shortened by Mario Carneiro, 26-May-2016.)
Hypotheses
Ref Expression
rpvmasum.z  |-  Z  =  (ℤ/n `  N )
rpvmasum.l  |-  L  =  ( ZRHom `  Z
)
rpvmasum.a  |-  ( ph  ->  N  e.  NN )
rpvmasum.u  |-  U  =  (Unit `  Z )
rpvmasum.b  |-  ( ph  ->  A  e.  U )
rpvmasum.t  |-  T  =  ( `' L " { A } )
Assertion
Ref Expression
dirith2  |-  ( ph  ->  ( Prime  i^i  T ) 
~~  NN )

Proof of Theorem dirith2
Dummy variables  n  x  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nnex 9752 . . . 4  |-  NN  e.  _V
2 inss1 3389 . . . . 5  |-  ( Prime  i^i  T )  C_  Prime
3 prmnn 12761 . . . . . 6  |-  ( p  e.  Prime  ->  p  e.  NN )
43ssriv 3184 . . . . 5  |-  Prime  C_  NN
52, 4sstri 3188 . . . 4  |-  ( Prime  i^i  T )  C_  NN
6 ssdomg 6907 . . . 4  |-  ( NN  e.  _V  ->  (
( Prime  i^i  T ) 
C_  NN  ->  ( Prime  i^i  T )  ~<_  NN ) )
71, 5, 6mp2 17 . . 3  |-  ( Prime  i^i  T )  ~<_  NN
87a1i 10 . 2  |-  ( ph  ->  ( Prime  i^i  T )  ~<_  NN )
9 logno1 19983 . . . 4  |-  -.  (
x  e.  RR+  |->  ( log `  x ) )  e.  O ( 1 )
10 rpvmasum.a . . . . . . . . . . 11  |-  ( ph  ->  N  e.  NN )
1110adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  N  e.  NN )
1211phicld 12840 . . . . . . . . 9  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( phi `  N )  e.  NN )
1312nnred 9761 . . . . . . . 8  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( phi `  N )  e.  RR )
1413adantr 451 . . . . . . 7  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  x  e.  RR+ )  ->  ( phi `  N )  e.  RR )
15 simpr 447 . . . . . . . . . 10  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( Prime  i^i  T )  e.  Fin )
16 inss2 3390 . . . . . . . . . 10  |-  ( ( 1 ... ( |_
`  x ) )  i^i  ( Prime  i^i  T ) )  C_  ( Prime  i^i  T )
17 ssfi 7083 . . . . . . . . . 10  |-  ( ( ( Prime  i^i  T )  e.  Fin  /\  (
( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) )  C_  ( Prime  i^i  T ) )  ->  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) )  e.  Fin )
1815, 16, 17sylancl 643 . . . . . . . . 9  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) )  e.  Fin )
1916sseli 3176 . . . . . . . . . 10  |-  ( n  e.  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) )  ->  n  e.  ( Prime  i^i  T )
)
20 simpr 447 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  n  e.  ( Prime  i^i  T )
)
215, 20sseldi 3178 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  n  e.  NN )
2221nnrpd 10389 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  n  e.  RR+ )
23 relogcl 19932 . . . . . . . . . . . 12  |-  ( n  e.  RR+  ->  ( log `  n )  e.  RR )
2422, 23syl 15 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  ( log `  n )  e.  RR )
2524, 21nndivred 9794 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  ( ( log `  n )  /  n )  e.  RR )
2619, 25sylan2 460 . . . . . . . . 9  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) )  ->  ( ( log `  n )  /  n )  e.  RR )
2718, 26fsumrecl 12207 . . . . . . . 8  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  sum_ n  e.  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) ) ( ( log `  n )  /  n )  e.  RR )
2827adantr 451 . . . . . . 7  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  x  e.  RR+ )  ->  sum_ n  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) ( ( log `  n
)  /  n )  e.  RR )
29 rpssre 10364 . . . . . . . 8  |-  RR+  C_  RR
3013recnd 8861 . . . . . . . 8  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( phi `  N )  e.  CC )
31 o1const 12093 . . . . . . . 8  |-  ( (
RR+  C_  RR  /\  ( phi `  N )  e.  CC )  ->  (
x  e.  RR+  |->  ( phi `  N ) )  e.  O ( 1 ) )
3229, 30, 31sylancr 644 . . . . . . 7  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( x  e.  RR+  |->  ( phi `  N ) )  e.  O ( 1 ) )
3329a1i 10 . . . . . . . . 9  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  RR+  C_  RR )
34 1re 8837 . . . . . . . . . 10  |-  1  e.  RR
3534a1i 10 . . . . . . . . 9  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  1  e.  RR )
3615, 25fsumrecl 12207 . . . . . . . . 9  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  sum_ n  e.  ( Prime  i^i  T )
( ( log `  n
)  /  n )  e.  RR )
37 log1 19939 . . . . . . . . . . . . 13  |-  ( log `  1 )  =  0
3821nnge1d 9788 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  1  <_  n )
39 1rp 10358 . . . . . . . . . . . . . . 15  |-  1  e.  RR+
40 logleb 19957 . . . . . . . . . . . . . . 15  |-  ( ( 1  e.  RR+  /\  n  e.  RR+ )  ->  (
1  <_  n  <->  ( log `  1 )  <_  ( log `  n ) ) )
4139, 22, 40sylancr 644 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  ( 1  <_  n  <->  ( log `  1 )  <_  ( log `  n ) ) )
4238, 41mpbid 201 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  ( log `  1 )  <_  ( log `  n ) )
4337, 42syl5eqbrr 4057 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  0  <_  ( log `  n ) )
4424, 22, 43divge0d 10426 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  0  <_  ( ( log `  n
)  /  n ) )
4516a1i 10 . . . . . . . . . . 11  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) )  C_  ( Prime  i^i 
T ) )
4615, 25, 44, 45fsumless 12254 . . . . . . . . . 10  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  sum_ n  e.  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) ) ( ( log `  n )  /  n )  <_  sum_ n  e.  ( Prime  i^i  T ) ( ( log `  n )  /  n ) )
4746adantr 451 . . . . . . . . 9  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  (
x  e.  RR+  /\  1  <_  x ) )  ->  sum_ n  e.  ( ( 1 ... ( |_
`  x ) )  i^i  ( Prime  i^i  T ) ) ( ( log `  n )  /  n )  <_  sum_ n  e.  ( Prime  i^i  T ) ( ( log `  n )  /  n ) )
4833, 28, 35, 36, 47ello1d 11997 . . . . . . . 8  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( x  e.  RR+  |->  sum_ n  e.  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) ) ( ( log `  n )  /  n ) )  e.  <_ O ( 1 ) )
49 0re 8838 . . . . . . . . . 10  |-  0  e.  RR
5049a1i 10 . . . . . . . . 9  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  0  e.  RR )
5119, 44sylan2 460 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) )  ->  0  <_  ( ( log `  n
)  /  n ) )
5218, 26, 51fsumge0 12253 . . . . . . . . . 10  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  0  <_  sum_ n  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) ( ( log `  n
)  /  n ) )
5352adantr 451 . . . . . . . . 9  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  x  e.  RR+ )  ->  0  <_ 
sum_ n  e.  (
( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) ) ( ( log `  n )  /  n ) )
5428, 50, 53o1lo12 12012 . . . . . . . 8  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( ( x  e.  RR+  |->  sum_ n  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) ( ( log `  n
)  /  n ) )  e.  O ( 1 )  <->  ( x  e.  RR+  |->  sum_ n  e.  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) ) ( ( log `  n )  /  n ) )  e.  <_ O ( 1 ) ) )
5548, 54mpbird 223 . . . . . . 7  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( x  e.  RR+  |->  sum_ n  e.  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) ) ( ( log `  n )  /  n ) )  e.  O ( 1 ) )
5614, 28, 32, 55o1mul2 12098 . . . . . 6  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( x  e.  RR+  |->  ( ( phi `  N )  x.  sum_ n  e.  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) ) ( ( log `  n )  /  n
) ) )  e.  O ( 1 ) )
5713, 27remulcld 8863 . . . . . . . . 9  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( ( phi `  N )  x.  sum_ n  e.  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) ) ( ( log `  n )  /  n
) )  e.  RR )
5857recnd 8861 . . . . . . . 8  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( ( phi `  N )  x.  sum_ n  e.  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) ) ( ( log `  n )  /  n
) )  e.  CC )
5958adantr 451 . . . . . . 7  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  x  e.  RR+ )  ->  (
( phi `  N
)  x.  sum_ n  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) ( ( log `  n
)  /  n ) )  e.  CC )
60 relogcl 19932 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
6160adantl 452 . . . . . . . 8  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  x  e.  RR+ )  ->  ( log `  x )  e.  RR )
6261recnd 8861 . . . . . . 7  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  x  e.  RR+ )  ->  ( log `  x )  e.  CC )
63 rpvmasum.z . . . . . . . . 9  |-  Z  =  (ℤ/n `  N )
64 rpvmasum.l . . . . . . . . 9  |-  L  =  ( ZRHom `  Z
)
65 rpvmasum.u . . . . . . . . 9  |-  U  =  (Unit `  Z )
66 rpvmasum.b . . . . . . . . 9  |-  ( ph  ->  A  e.  U )
67 rpvmasum.t . . . . . . . . 9  |-  T  =  ( `' L " { A } )
6863, 64, 10, 65, 66, 67rplogsum 20676 . . . . . . . 8  |-  ( ph  ->  ( x  e.  RR+  |->  ( ( ( phi `  N )  x.  sum_ n  e.  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) ) ( ( log `  n )  /  n
) )  -  ( log `  x ) ) )  e.  O ( 1 ) )
6968adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( x  e.  RR+  |->  ( ( ( phi `  N )  x.  sum_ n  e.  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) ) ( ( log `  n )  /  n ) )  -  ( log `  x
) ) )  e.  O ( 1 ) )
7059, 62, 69o1dif 12103 . . . . . 6  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( ( x  e.  RR+  |->  ( ( phi `  N )  x.  sum_ n  e.  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) ) ( ( log `  n )  /  n ) ) )  e.  O ( 1 )  <->  ( x  e.  RR+  |->  ( log `  x
) )  e.  O
( 1 ) ) )
7156, 70mpbid 201 . . . . 5  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( x  e.  RR+  |->  ( log `  x
) )  e.  O
( 1 ) )
7271ex 423 . . . 4  |-  ( ph  ->  ( ( Prime  i^i  T )  e.  Fin  ->  ( x  e.  RR+  |->  ( log `  x ) )  e.  O ( 1 ) ) )
739, 72mtoi 169 . . 3  |-  ( ph  ->  -.  ( Prime  i^i  T )  e.  Fin )
74 nnenom 11042 . . . . 5  |-  NN  ~~  om
75 sdomentr 6995 . . . . 5  |-  ( ( ( Prime  i^i  T ) 
~<  NN  /\  NN  ~~  om )  ->  ( Prime  i^i 
T )  ~<  om )
7674, 75mpan2 652 . . . 4  |-  ( ( Prime  i^i  T )  ~<  NN  ->  ( Prime  i^i 
T )  ~<  om )
77 isfinite2 7115 . . . 4  |-  ( ( Prime  i^i  T )  ~<  om  ->  ( Prime  i^i 
T )  e.  Fin )
7876, 77syl 15 . . 3  |-  ( ( Prime  i^i  T )  ~<  NN  ->  ( Prime  i^i 
T )  e.  Fin )
7973, 78nsyl 113 . 2  |-  ( ph  ->  -.  ( Prime  i^i  T )  ~<  NN )
80 bren2 6892 . 2  |-  ( ( Prime  i^i  T )  ~~  NN  <->  ( ( Prime  i^i  T )  ~<_  NN  /\  -.  ( Prime  i^i  T ) 
~<  NN ) )
818, 79, 80sylanbrc 645 1  |-  ( ph  ->  ( Prime  i^i  T ) 
~~  NN )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151    C_ wss 3152   {csn 3640   class class class wbr 4023    e. cmpt 4077   omcom 4656   `'ccnv 4688   "cima 4692   ` cfv 5255  (class class class)co 5858    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862   Fincfn 6863   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742    <_ cle 8868    - cmin 9037    / cdiv 9423   NNcn 9746   RR+crp 10354   ...cfz 10782   |_cfl 10924   O ( 1 )co1 11960   <_ O ( 1 )clo1 11961   sum_csu 12158   Primecprime 12758   phicphi 12832  Unitcui 15421   ZRHomczrh 16451  ℤ/nczn 16454   logclog 19912
This theorem is referenced by:  dirith  20678
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-fal 1311  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-disj 3994  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-tpos 6234  df-rpss 6277  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-er 6660  df-ec 6662  df-qs 6666  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-acn 7575  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-word 11409  df-concat 11410  df-s1 11411  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-o1 11964  df-lo1 11965  df-sum 12159  df-ef 12349  df-e 12350  df-sin 12351  df-cos 12352  df-tan 12353  df-pi 12354  df-dvds 12532  df-gcd 12686  df-prm 12759  df-numer 12806  df-denom 12807  df-phi 12834  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-divs 13412  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-nsg 14619  df-eqg 14620  df-ghm 14681  df-gim 14723  df-ga 14744  df-cntz 14793  df-oppg 14819  df-od 14844  df-gex 14845  df-pgp 14846  df-lsm 14947  df-pj1 14948  df-cmn 15091  df-abl 15092  df-cyg 15165  df-dprd 15233  df-dpj 15234  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-rnghom 15496  df-drng 15514  df-subrg 15543  df-lmod 15629  df-lss 15690  df-lsp 15729  df-sra 15925  df-rgmod 15926  df-lidl 15927  df-rsp 15928  df-2idl 15984  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-zrh 16455  df-zn 16458  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-cmp 17114  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-0p 19025  df-limc 19216  df-dv 19217  df-ply 19570  df-idp 19571  df-coe 19572  df-dgr 19573  df-quot 19671  df-log 19914  df-cxp 19915  df-em 20287  df-cht 20334  df-vma 20335  df-chp 20336  df-ppi 20337  df-mu 20338  df-dchr 20472
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