MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dirith2 Structured version   Unicode version

Theorem dirith2 22903
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 10432 . . . 4  |-  NN  e.  _V
2 inss1 3671 . . . . 5  |-  ( Prime  i^i  T )  C_  Prime
3 prmnn 13877 . . . . . 6  |-  ( p  e.  Prime  ->  p  e.  NN )
43ssriv 3461 . . . . 5  |-  Prime  C_  NN
52, 4sstri 3466 . . . 4  |-  ( Prime  i^i  T )  C_  NN
6 ssdomg 7458 . . . 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 22207 . . . 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 13958 . . . . . . . . 9  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( phi `  N )  e.  NN )
1312nnred 10441 . . . . . . . 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 3672 . . . . . . . . . 10  |-  ( ( 1 ... ( |_
`  x ) )  i^i  ( Prime  i^i  T ) )  C_  ( Prime  i^i  T )
17 ssfi 7637 . . . . . . . . . 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 3453 . . . . . . . . . 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 3455 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  n  e.  NN )
2221nnrpd 11130 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  n  e.  RR+ )
23 relogcl 22153 . . . . . . . . . . . 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 10474 . . . . . . . . . 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 13322 . . . . . . . 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 11105 . . . . . . . 8  |-  RR+  C_  RR
3013recnd 9516 . . . . . . . 8  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( phi `  N )  e.  CC )
31 o1const 13208 . . . . . . . 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 9505 . . . . . . . . 9  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  1  e.  RR )
3515, 25fsumrecl 13322 . . . . . . . . 9  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  sum_ n  e.  ( Prime  i^i  T )
( ( log `  n
)  /  n )  e.  RR )
36 log1 22160 . . . . . . . . . . . . 13  |-  ( log `  1 )  =  0
3721nnge1d 10468 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  1  <_  n )
38 1rp 11099 . . . . . . . . . . . . . . 15  |-  1  e.  RR+
39 logleb 22178 . . . . . . . . . . . . . . 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 4427 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  0  <_  ( log `  n ) )
4324, 22, 42divge0d 11167 . . . . . . . . . . 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 13370 . . . . . . . . . 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 13112 . . . . . . . 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 9491 . . . . . . . . 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 13369 . . . . . . . . . 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 13127 . . . . . . . 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 13213 . . . . . 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 9518 . . . . . . . . 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 9516 . . . . . . . 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 22153 . . . . . . . . 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 9516 . . . . . . 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 22902 . . . . . . . 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 13218 . . . . . 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 11912 . . . . 5  |-  NN  ~~  om
73 sdomentr 7548 . . . . 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 7674 . . . 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 7443 . 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 1370    e. wcel 1758   _Vcvv 3071    i^i cin 3428    C_ wss 3429   {csn 3978   class class class wbr 4393    |-> cmpt 4451   `'ccnv 4940   "cima 4944   ` cfv 5519  (class class class)co 6193   omcom 6579    ~~ cen 7410    ~<_ cdom 7411    ~< csdm 7412   Fincfn 7413   CCcc 9384   RRcr 9385   0cc0 9386   1c1 9387    x. cmul 9391    <_ cle 9523    - cmin 9699    / cdiv 10097   NNcn 10426   RR+crp 11095   ...cfz 11547   |_cfl 11750   O(1)co1 13075   <_O(1)clo1 13076   sum_csu 13274   Primecprime 13874   phicphi 13950  Unitcui 16846   ZRHomczrh 18049  ℤ/nczn 18052   logclog 22132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464  ax-addf 9465  ax-mulf 9466
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-iin 4275  df-disj 4364  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-of 6423  df-rpss 6463  df-om 6580  df-1st 6680  df-2nd 6681  df-supp 6794  df-tpos 6848  df-recs 6935  df-rdg 6969  df-1o 7023  df-2o 7024  df-oadd 7027  df-omul 7028  df-er 7204  df-ec 7206  df-qs 7210  df-map 7319  df-pm 7320  df-ixp 7367  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-fsupp 7725  df-fi 7765  df-sup 7795  df-oi 7828  df-card 8213  df-acn 8216  df-cda 8441  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-7 10489  df-8 10490  df-9 10491  df-10 10492  df-n0 10684  df-z 10751  df-dec 10860  df-uz 10966  df-q 11058  df-rp 11096  df-xneg 11193  df-xadd 11194  df-xmul 11195  df-ioo 11408  df-ioc 11409  df-ico 11410  df-icc 11411  df-fz 11548  df-fzo 11659  df-fl 11752  df-mod 11819  df-seq 11917  df-exp 11976  df-fac 12162  df-bc 12189  df-hash 12214  df-word 12340  df-concat 12342  df-s1 12343  df-shft 12667  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-limsup 13060  df-clim 13077  df-rlim 13078  df-o1 13079  df-lo1 13080  df-sum 13275  df-ef 13464  df-e 13465  df-sin 13466  df-cos 13467  df-tan 13468  df-pi 13469  df-dvds 13647  df-gcd 13802  df-prm 13875  df-numer 13924  df-denom 13925  df-phi 13952  df-pc 14015  df-struct 14287  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-mulr 14363  df-starv 14364  df-sca 14365  df-vsca 14366  df-ip 14367  df-tset 14368  df-ple 14369  df-ds 14371  df-unif 14372  df-hom 14373  df-cco 14374  df-rest 14472  df-topn 14473  df-0g 14491  df-gsum 14492  df-topgen 14493  df-pt 14494  df-prds 14497  df-xrs 14551  df-qtop 14556  df-imas 14557  df-divs 14558  df-xps 14559  df-mre 14635  df-mrc 14636  df-acs 14638  df-mnd 15526  df-mhm 15575  df-submnd 15576  df-grp 15656  df-minusg 15657  df-sbg 15658  df-mulg 15659  df-subg 15789  df-nsg 15790  df-eqg 15791  df-ghm 15856  df-gim 15898  df-ga 15919  df-cntz 15946  df-oppg 15972  df-od 16145  df-gex 16146  df-pgp 16147  df-lsm 16248  df-pj1 16249  df-cmn 16392  df-abl 16393  df-cyg 16468  df-dprd 16591  df-dpj 16592  df-mgp 16706  df-ur 16718  df-rng 16762  df-cring 16763  df-oppr 16830  df-dvdsr 16848  df-unit 16849  df-invr 16879  df-dvr 16890  df-rnghom 16921  df-drng 16949  df-subrg 16978  df-lmod 17065  df-lss 17129  df-lsp 17168  df-sra 17368  df-rgmod 17369  df-lidl 17370  df-rsp 17371  df-2idl 17429  df-psmet 17927  df-xmet 17928  df-met 17929  df-bl 17930  df-mopn 17931  df-fbas 17932  df-fg 17933  df-cnfld 17937  df-zring 18002  df-zrh 18053  df-zn 18056  df-top 18628  df-bases 18630  df-topon 18631  df-topsp 18632  df-cld 18748  df-ntr 18749  df-cls 18750  df-nei 18827  df-lp 18865  df-perf 18866  df-cn 18956  df-cnp 18957  df-haus 19044  df-cmp 19115  df-tx 19260  df-hmeo 19453  df-fil 19544  df-fm 19636  df-flim 19637  df-flf 19638  df-xms 20020  df-ms 20021  df-tms 20022  df-cncf 20579  df-0p 21274  df-limc 21467  df-dv 21468  df-ply 21782  df-idp 21783  df-coe 21784  df-dgr 21785  df-quot 21883  df-log 22134  df-cxp 22135  df-em 22512  df-cht 22560  df-vma 22561  df-chp 22562  df-ppi 22563  df-mu 22564  df-dchr 22698
This theorem is referenced by:  dirith  22904
  Copyright terms: Public domain W3C validator