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Theorem dirith2 20625
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
StepHypRef Expression
1 nnex 9706 . . . 4  |-  NN  e.  _V
2 inss1 3350 . . . . 5  |-  ( Prime  i^i  T )  C_  Prime
3 prmnn 12710 . . . . . 6  |-  ( p  e.  Prime  ->  p  e.  NN )
43ssriv 3145 . . . . 5  |-  Prime  C_  NN
52, 4sstri 3149 . . . 4  |-  ( Prime  i^i  T )  C_  NN
6 ssdomg 6861 . . . 4  |-  ( NN  e.  _V  ->  (
( Prime  i^i  T ) 
C_  NN  ->  ( Prime  i^i  T )  ~<_  NN ) )
71, 5, 6mp2 19 . . 3  |-  ( Prime  i^i  T )  ~<_  NN
87a1i 12 . 2  |-  ( ph  ->  ( Prime  i^i  T )  ~<_  NN )
9 logno1 19931 . . . 4  |-  -.  (
x  e.  RR+  |->  ( log `  x ) )  e.  O ( 1 )
10 rpvmasum.a . . . . . . . . . . 11  |-  ( ph  ->  N  e.  NN )
1110adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  N  e.  NN )
1211phicld 12788 . . . . . . . . 9  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( phi `  N )  e.  NN )
1312nnred 9715 . . . . . . . 8  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( phi `  N )  e.  RR )
1413adantr 453 . . . . . . 7  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  x  e.  RR+ )  ->  ( phi `  N )  e.  RR )
15 simpr 449 . . . . . . . . . 10  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( Prime  i^i  T )  e.  Fin )
16 inss2 3351 . . . . . . . . . 10  |-  ( ( 1 ... ( |_
`  x ) )  i^i  ( Prime  i^i  T ) )  C_  ( Prime  i^i  T )
17 ssfi 7037 . . . . . . . . . 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 646 . . . . . . . . 9  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) )  e.  Fin )
1916sseli 3137 . . . . . . . . . 10  |-  ( n  e.  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) )  ->  n  e.  ( Prime  i^i  T )
)
20 simpr 449 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  n  e.  ( Prime  i^i  T )
)
215, 20sseldi 3139 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  n  e.  NN )
2221nnrpd 10342 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  n  e.  RR+ )
23 relogcl 19880 . . . . . . . . . . . 12  |-  ( n  e.  RR+  ->  ( log `  n )  e.  RR )
2422, 23syl 17 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  ( log `  n )  e.  RR )
2524, 21nndivred 9748 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  ( ( log `  n )  /  n )  e.  RR )
2619, 25sylan2 462 . . . . . . . . 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 12158 . . . . . . . 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 453 . . . . . . 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 10317 . . . . . . . 8  |-  RR+  C_  RR
3013recnd 8815 . . . . . . . 8  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( phi `  N )  e.  CC )
31 o1const 12044 . . . . . . . 8  |-  ( (
RR+  C_  RR  /\  ( phi `  N )  e.  CC )  ->  (
x  e.  RR+  |->  ( phi `  N ) )  e.  O ( 1 ) )
3229, 30, 31sylancr 647 . . . . . . 7  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( x  e.  RR+  |->  ( phi `  N ) )  e.  O ( 1 ) )
3329a1i 12 . . . . . . . . 9  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  RR+  C_  RR )
34 1re 8791 . . . . . . . . . 10  |-  1  e.  RR
3534a1i 12 . . . . . . . . 9  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  1  e.  RR )
3615, 25fsumrecl 12158 . . . . . . . . 9  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  sum_ n  e.  ( Prime  i^i  T )
( ( log `  n
)  /  n )  e.  RR )
37 log1 19887 . . . . . . . . . . . . 13  |-  ( log `  1 )  =  0
3821nnge1d 9742 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  1  <_  n )
39 1rp 10311 . . . . . . . . . . . . . . 15  |-  1  e.  RR+
40 logleb 19905 . . . . . . . . . . . . . . 15  |-  ( ( 1  e.  RR+  /\  n  e.  RR+ )  ->  (
1  <_  n  <->  ( log `  1 )  <_  ( log `  n ) ) )
4139, 22, 40sylancr 647 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  ( 1  <_  n  <->  ( log `  1 )  <_  ( log `  n ) ) )
4238, 41mpbid 203 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  ( log `  1 )  <_  ( log `  n ) )
4337, 42syl5eqbrr 4017 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  0  <_  ( log `  n ) )
4424, 22, 43divge0d 10379 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  0  <_  ( ( log `  n
)  /  n ) )
4516a1i 12 . . . . . . . . . . 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 12205 . . . . . . . . . 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 453 . . . . . . . . 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 11948 . . . . . . . 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 8792 . . . . . . . . . 10  |-  0  e.  RR
5049a1i 12 . . . . . . . . 9  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  0  e.  RR )
5119, 44sylan2 462 . . . . . . . . . . 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 12204 . . . . . . . . . 10  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  0  <_  sum_ n  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) ( ( log `  n
)  /  n ) )
5352adantr 453 . . . . . . . . 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 11963 . . . . . . . 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 225 . . . . . . 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 12049 . . . . . 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 8817 . . . . . . . . 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 8815 . . . . . . . 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 453 . . . . . . 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 19880 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
6160adantl 454 . . . . . . . 8  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  x  e.  RR+ )  ->  ( log `  x )  e.  RR )
6261recnd 8815 . . . . . . 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 20624 . . . . . . . 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 453 . . . . . . 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 12054 . . . . . 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 203 . . . . 5  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( x  e.  RR+  |->  ( log `  x
) )  e.  O
( 1 ) )
7271ex 425 . . . 4  |-  ( ph  ->  ( ( Prime  i^i  T )  e.  Fin  ->  ( x  e.  RR+  |->  ( log `  x ) )  e.  O ( 1 ) ) )
739, 72mtoi 171 . . 3  |-  ( ph  ->  -.  ( Prime  i^i  T )  e.  Fin )
74 nnenom 10994 . . . . 5  |-  NN  ~~  om
75 sdomentr 6949 . . . . 5  |-  ( ( ( Prime  i^i  T ) 
~<  NN  /\  NN  ~~  om )  ->  ( Prime  i^i 
T )  ~<  om )
7674, 75mpan2 655 . . . 4  |-  ( ( Prime  i^i  T )  ~<  NN  ->  ( Prime  i^i 
T )  ~<  om )
77 isfinite2 7069 . . . 4  |-  ( ( Prime  i^i  T )  ~<  om  ->  ( Prime  i^i 
T )  e.  Fin )
7876, 77syl 17 . . 3  |-  ( ( Prime  i^i  T )  ~<  NN  ->  ( Prime  i^i 
T )  e.  Fin )
7973, 78nsyl 115 . 2  |-  ( ph  ->  -.  ( Prime  i^i  T )  ~<  NN )
80 bren2 6846 . 2  |-  ( ( Prime  i^i  T )  ~~  NN  <->  ( ( Prime  i^i  T )  ~<_  NN  /\  -.  ( Prime  i^i  T ) 
~<  NN ) )
818, 79, 80sylanbrc 648 1  |-  ( ph  ->  ( Prime  i^i  T ) 
~~  NN )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   _Vcvv 2757    i^i cin 3112    C_ wss 3113   {csn 3600   class class class wbr 3983    e. cmpt 4037   omcom 4614   `'ccnv 4646   "cima 4650   ` cfv 4659  (class class class)co 5778    ~~ cen 6814    ~<_ cdom 6815    ~< csdm 6816   Fincfn 6817   CCcc 8689   RRcr 8690   0cc0 8691   1c1 8692    x. cmul 8696    <_ cle 8822    - cmin 8991    / cdiv 9377   NNcn 9700   RR+crp 10307   ...cfz 10734   |_cfl 10876   O ( 1 )co1 11911   <_ O ( 1 )clo1 11912   sum_csu 12109   Primecprime 12706   phicphi 12780  Unitcui 15369   ZRHomczrh 16399  ℤ/nczn 16402   logclog 19860
This theorem is referenced by:  dirith  20626
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-inf2 7296  ax-cnex 8747  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768  ax-pre-sup 8769  ax-addf 8770  ax-mulf 8771
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-fal 1316  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-iin 3868  df-disj 3954  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-se 4311  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-isom 4676  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-of 5998  df-1st 6042  df-2nd 6043  df-tpos 6154  df-rpss 6197  df-iota 6211  df-riota 6258  df-recs 6342  df-rdg 6377  df-1o 6433  df-2o 6434  df-oadd 6437  df-omul 6438  df-er 6614  df-ec 6616  df-qs 6620  df-map 6728  df-pm 6729  df-ixp 6772  df-en 6818  df-dom 6819  df-sdom 6820  df-fin 6821  df-fi 7119  df-sup 7148  df-oi 7179  df-card 7526  df-acn 7529  df-cda 7748  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-div 9378  df-n 9701  df-2 9758  df-3 9759  df-4 9760  df-5 9761  df-6 9762  df-7 9763  df-8 9764  df-9 9765  df-10 9766  df-n0 9919  df-z 9978  df-dec 10078  df-uz 10184  df-q 10270  df-rp 10308  df-xneg 10405  df-xadd 10406  df-xmul 10407  df-ioo 10612  df-ioc 10613  df-ico 10614  df-icc 10615  df-fz 10735  df-fzo 10823  df-fl 10877  df-mod 10926  df-seq 10999  df-exp 11057  df-fac 11241  df-bc 11268  df-hash 11290  df-word 11360  df-concat 11361  df-s1 11362  df-shft 11513  df-cj 11535  df-re 11536  df-im 11537  df-sqr 11671  df-abs 11672  df-limsup 11896  df-clim 11913  df-rlim 11914  df-o1 11915  df-lo1 11916  df-sum 12110  df-ef 12297  df-e 12298  df-sin 12299  df-cos 12300  df-tan 12301  df-pi 12302  df-divides 12480  df-gcd 12634  df-prime 12707  df-numer 12754  df-denom 12755  df-phi 12782  df-pc 12838  df-struct 13098  df-ndx 13099  df-slot 13100  df-base 13101  df-sets 13102  df-ress 13103  df-plusg 13169  df-mulr 13170  df-starv 13171  df-sca 13172  df-vsca 13173  df-tset 13175  df-ple 13176  df-ds 13178  df-hom 13180  df-cco 13181  df-rest 13275  df-topn 13276  df-topgen 13292  df-pt 13293  df-prds 13296  df-xrs 13351  df-0g 13352  df-gsum 13353  df-qtop 13358  df-imas 13359  df-divs 13360  df-xps 13361  df-mre 13436  df-mrc 13437  df-acs 13439  df-mnd 14315  df-mhm 14363  df-submnd 14364  df-grp 14437  df-minusg 14438  df-sbg 14439  df-mulg 14440  df-subg 14566  df-nsg 14567  df-eqg 14568  df-ghm 14629  df-gim 14671  df-ga 14692  df-cntz 14741  df-oppg 14767  df-od 14792  df-gex 14793  df-pgp 14794  df-lsm 14895  df-pj1 14896  df-cmn 15039  df-abl 15040  df-cyg 15113  df-dprd 15181  df-dpj 15182  df-mgp 15274  df-ring 15288  df-cring 15289  df-ur 15290  df-oppr 15353  df-dvdsr 15371  df-unit 15372  df-invr 15402  df-dvr 15413  df-rnghom 15444  df-drng 15462  df-subrg 15491  df-lmod 15577  df-lss 15638  df-lsp 15677  df-sra 15873  df-rgmod 15874  df-lidl 15875  df-rsp 15876  df-2idl 15932  df-xmet 16321  df-met 16322  df-bl 16323  df-mopn 16324  df-cnfld 16326  df-zrh 16403  df-zn 16406  df-top 16584  df-bases 16586  df-topon 16587  df-topsp 16588  df-cld 16704  df-ntr 16705  df-cls 16706  df-nei 16783  df-lp 16816  df-perf 16817  df-cn 16905  df-cnp 16906  df-haus 16991  df-cmp 17062  df-tx 17205  df-hmeo 17394  df-fbas 17468  df-fg 17469  df-fil 17489  df-fm 17581  df-flim 17582  df-flf 17583  df-xms 17833  df-ms 17834  df-tms 17835  df-cncf 18330  df-0p 18973  df-limc 19164  df-dv 19165  df-ply 19518  df-idp 19519  df-coe 19520  df-dgr 19521  df-quot 19619  df-log 19862  df-cxp 19863  df-em 20235  df-cht 20282  df-vma 20283  df-chp 20284  df-ppi 20285  df-mu 20286  df-dchr 20420
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