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Theorem dirith2 22752
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 10320 . . . 4  |-  NN  e.  _V
2 inss1 3565 . . . . 5  |-  ( Prime  i^i  T )  C_  Prime
3 prmnn 13758 . . . . . 6  |-  ( p  e.  Prime  ->  p  e.  NN )
43ssriv 3355 . . . . 5  |-  Prime  C_  NN
52, 4sstri 3360 . . . 4  |-  ( Prime  i^i  T )  C_  NN
6 ssdomg 7347 . . . 4  |-  ( NN  e.  _V  ->  (
( Prime  i^i  T ) 
C_  NN  ->  ( Prime  i^i  T )  ~<_  NN ) )
71, 5, 6mp2 9 . . 3  |-  ( Prime  i^i  T )  ~<_  NN
87a1i 11 . 2  |-  ( ph  ->  ( Prime  i^i  T )  ~<_  NN )
9 logno1 22056 . . . 4  |-  -.  (
x  e.  RR+  |->  ( log `  x ) )  e.  O(1)
10 rpvmasum.a . . . . . . . . . . 11  |-  ( ph  ->  N  e.  NN )
1110adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  N  e.  NN )
1211phicld 13839 . . . . . . . . 9  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( phi `  N )  e.  NN )
1312nnred 10329 . . . . . . . 8  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( phi `  N )  e.  RR )
1413adantr 465 . . . . . . 7  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  x  e.  RR+ )  ->  ( phi `  N )  e.  RR )
15 simpr 461 . . . . . . . . . 10  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( Prime  i^i  T )  e.  Fin )
16 inss2 3566 . . . . . . . . . 10  |-  ( ( 1 ... ( |_
`  x ) )  i^i  ( Prime  i^i  T ) )  C_  ( Prime  i^i  T )
17 ssfi 7525 . . . . . . . . . 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 662 . . . . . . . . 9  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) )  e.  Fin )
1916sseli 3347 . . . . . . . . . 10  |-  ( n  e.  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) )  ->  n  e.  ( Prime  i^i  T )
)
20 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  n  e.  ( Prime  i^i  T )
)
215, 20sseldi 3349 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  n  e.  NN )
2221nnrpd 11018 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  n  e.  RR+ )
23 relogcl 22002 . . . . . . . . . . . 12  |-  ( n  e.  RR+  ->  ( log `  n )  e.  RR )
2422, 23syl 16 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  ( log `  n )  e.  RR )
2524, 21nndivred 10362 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  ( ( log `  n )  /  n )  e.  RR )
2619, 25sylan2 474 . . . . . . . . 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 13203 . . . . . . . 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 465 . . . . . . 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 10993 . . . . . . . 8  |-  RR+  C_  RR
3013recnd 9404 . . . . . . . 8  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( phi `  N )  e.  CC )
31 o1const 13089 . . . . . . . 8  |-  ( (
RR+  C_  RR  /\  ( phi `  N )  e.  CC )  ->  (
x  e.  RR+  |->  ( phi `  N ) )  e.  O(1) )
3229, 30, 31sylancr 663 . . . . . . 7  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( x  e.  RR+  |->  ( phi `  N ) )  e.  O(1) )
3329a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  RR+  C_  RR )
34 1red 9393 . . . . . . . . 9  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  1  e.  RR )
3515, 25fsumrecl 13203 . . . . . . . . 9  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  sum_ n  e.  ( Prime  i^i  T )
( ( log `  n
)  /  n )  e.  RR )
36 log1 22009 . . . . . . . . . . . . 13  |-  ( log `  1 )  =  0
3721nnge1d 10356 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  1  <_  n )
38 1rp 10987 . . . . . . . . . . . . . . 15  |-  1  e.  RR+
39 logleb 22027 . . . . . . . . . . . . . . 15  |-  ( ( 1  e.  RR+  /\  n  e.  RR+ )  ->  (
1  <_  n  <->  ( log `  1 )  <_  ( log `  n ) ) )
4038, 22, 39sylancr 663 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  ( 1  <_  n  <->  ( log `  1 )  <_  ( log `  n ) ) )
4137, 40mpbid 210 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  ( log `  1 )  <_  ( log `  n ) )
4236, 41syl5eqbrr 4321 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  0  <_  ( log `  n ) )
4324, 22, 42divge0d 11055 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  0  <_  ( ( log `  n
)  /  n ) )
4416a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) )  C_  ( Prime  i^i 
T ) )
4515, 25, 43, 44fsumless 13251 . . . . . . . . . 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 ) )
4645adantr 465 . . . . . . . . 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 ) )
4733, 28, 34, 35, 46ello1d 12993 . . . . . . . 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) )
48 0red 9379 . . . . . . . . 9  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  0  e.  RR )
4919, 43sylan2 474 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) )  ->  0  <_  ( ( log `  n
)  /  n ) )
5018, 26, 49fsumge0 13250 . . . . . . . . . 10  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  0  <_  sum_ n  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) ( ( log `  n
)  /  n ) )
5150adantr 465 . . . . . . . . 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 ) )
5228, 48, 51o1lo12 13008 . . . . . . . 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) ) )
5347, 52mpbird 232 . . . . . . 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) )
5414, 28, 32, 53o1mul2 13094 . . . . . 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) )
5513, 27remulcld 9406 . . . . . . . . 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 )
5655recnd 9404 . . . . . . . 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 )
5756adantr 465 . . . . . . 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 )
58 relogcl 22002 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
5958adantl 466 . . . . . . . 8  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  x  e.  RR+ )  ->  ( log `  x )  e.  RR )
6059recnd 9404 . . . . . . 7  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  x  e.  RR+ )  ->  ( log `  x )  e.  CC )
61 rpvmasum.z . . . . . . . . 9  |-  Z  =  (ℤ/n `  N )
62 rpvmasum.l . . . . . . . . 9  |-  L  =  ( ZRHom `  Z
)
63 rpvmasum.u . . . . . . . . 9  |-  U  =  (Unit `  Z )
64 rpvmasum.b . . . . . . . . 9  |-  ( ph  ->  A  e.  U )
65 rpvmasum.t . . . . . . . . 9  |-  T  =  ( `' L " { A } )
6661, 62, 10, 63, 64, 65rplogsum 22751 . . . . . . . 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) )
6766adantr 465 . . . . . . 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) )
6857, 60, 67o1dif 13099 . . . . . 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) ) )
6954, 68mpbid 210 . . . . 5  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( x  e.  RR+  |->  ( log `  x
) )  e.  O(1) )
7069ex 434 . . . 4  |-  ( ph  ->  ( ( Prime  i^i  T )  e.  Fin  ->  ( x  e.  RR+  |->  ( log `  x ) )  e.  O(1) ) )
719, 70mtoi 178 . . 3  |-  ( ph  ->  -.  ( Prime  i^i  T )  e.  Fin )
72 nnenom 11794 . . . . 5  |-  NN  ~~  om
73 sdomentr 7437 . . . . 5  |-  ( ( ( Prime  i^i  T ) 
~<  NN  /\  NN  ~~  om )  ->  ( Prime  i^i 
T )  ~<  om )
7472, 73mpan2 671 . . . 4  |-  ( ( Prime  i^i  T )  ~<  NN  ->  ( Prime  i^i 
T )  ~<  om )
75 isfinite2 7562 . . . 4  |-  ( ( Prime  i^i  T )  ~<  om  ->  ( Prime  i^i 
T )  e.  Fin )
7674, 75syl 16 . . 3  |-  ( ( Prime  i^i  T )  ~<  NN  ->  ( Prime  i^i 
T )  e.  Fin )
7771, 76nsyl 121 . 2  |-  ( ph  ->  -.  ( Prime  i^i  T )  ~<  NN )
78 bren2 7332 . 2  |-  ( ( Prime  i^i  T )  ~~  NN  <->  ( ( Prime  i^i  T )  ~<_  NN  /\  -.  ( Prime  i^i  T ) 
~<  NN ) )
798, 77, 78sylanbrc 664 1  |-  ( ph  ->  ( Prime  i^i  T ) 
~~  NN )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2967    i^i cin 3322    C_ wss 3323   {csn 3872   class class class wbr 4287    e. cmpt 4345   `'ccnv 4834   "cima 4838   ` cfv 5413  (class class class)co 6086   omcom 6471    ~~ cen 7299    ~<_ cdom 7300    ~< csdm 7301   Fincfn 7302   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275    x. cmul 9279    <_ cle 9411    - cmin 9587    / cdiv 9985   NNcn 10314   RR+crp 10983   ...cfz 11429   |_cfl 11632   O(1)co1 12956   <_O(1)clo1 12957   sum_csu 13155   Primecprime 13755   phicphi 13831  Unitcui 16719   ZRHomczrh 17906  ℤ/nczn 17909   logclog 21981
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-disj 4258  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-rpss 6355  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-tpos 6740  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-omul 6917  df-er 7093  df-ec 7095  df-qs 7099  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-acn 8104  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ioc 11297  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-mod 11701  df-seq 11799  df-exp 11858  df-fac 12044  df-bc 12071  df-hash 12096  df-word 12221  df-concat 12223  df-s1 12224  df-shft 12548  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-limsup 12941  df-clim 12958  df-rlim 12959  df-o1 12960  df-lo1 12961  df-sum 13156  df-ef 13345  df-e 13346  df-sin 13347  df-cos 13348  df-tan 13349  df-pi 13350  df-dvds 13528  df-gcd 13683  df-prm 13756  df-numer 13805  df-denom 13806  df-phi 13833  df-pc 13896  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-divs 14439  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-mhm 15456  df-submnd 15457  df-grp 15536  df-minusg 15537  df-sbg 15538  df-mulg 15539  df-subg 15669  df-nsg 15670  df-eqg 15671  df-ghm 15736  df-gim 15778  df-ga 15799  df-cntz 15826  df-oppg 15852  df-od 16023  df-gex 16024  df-pgp 16025  df-lsm 16126  df-pj1 16127  df-cmn 16270  df-abl 16271  df-cyg 16346  df-dprd 16465  df-dpj 16466  df-mgp 16580  df-ur 16592  df-rng 16635  df-cring 16636  df-oppr 16703  df-dvdsr 16721  df-unit 16722  df-invr 16752  df-dvr 16763  df-rnghom 16794  df-drng 16812  df-subrg 16841  df-lmod 16928  df-lss 16991  df-lsp 17030  df-sra 17230  df-rgmod 17231  df-lidl 17232  df-rsp 17233  df-2idl 17291  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787  df-mopn 17788  df-fbas 17789  df-fg 17790  df-cnfld 17794  df-zring 17859  df-zrh 17910  df-zn 17913  df-top 18478  df-bases 18480  df-topon 18481  df-topsp 18482  df-cld 18598  df-ntr 18599  df-cls 18600  df-nei 18677  df-lp 18715  df-perf 18716  df-cn 18806  df-cnp 18807  df-haus 18894  df-cmp 18965  df-tx 19110  df-hmeo 19303  df-fil 19394  df-fm 19486  df-flim 19487  df-flf 19488  df-xms 19870  df-ms 19871  df-tms 19872  df-cncf 20429  df-0p 21123  df-limc 21316  df-dv 21317  df-ply 21631  df-idp 21632  df-coe 21633  df-dgr 21634  df-quot 21732  df-log 21983  df-cxp 21984  df-em 22361  df-cht 22409  df-vma 22410  df-chp 22411  df-ppi 22412  df-mu 22413  df-dchr 22547
This theorem is referenced by:  dirith  22753
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