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Theorem dirith2 24445
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 10637 . . . 4  |-  NN  e.  _V
2 inss1 3643 . . . . 5  |-  ( Prime  i^i  T )  C_  Prime
3 prmnn 14704 . . . . . 6  |-  ( p  e.  Prime  ->  p  e.  NN )
43ssriv 3422 . . . . 5  |-  Prime  C_  NN
52, 4sstri 3427 . . . 4  |-  ( Prime  i^i  T )  C_  NN
6 ssdomg 7633 . . . 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 23660 . . . 4  |-  -.  (
x  e.  RR+  |->  ( log `  x ) )  e.  O(1)
10 rpvmasum.a . . . . . . . . . . 11  |-  ( ph  ->  N  e.  NN )
1110adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  N  e.  NN )
1211phicld 14799 . . . . . . . . 9  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( phi `  N )  e.  NN )
1312nnred 10646 . . . . . . . 8  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( phi `  N )  e.  RR )
1413adantr 472 . . . . . . 7  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  x  e.  RR+ )  ->  ( phi `  N )  e.  RR )
15 simpr 468 . . . . . . . . . 10  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( Prime  i^i  T )  e.  Fin )
16 inss2 3644 . . . . . . . . . 10  |-  ( ( 1 ... ( |_
`  x ) )  i^i  ( Prime  i^i  T ) )  C_  ( Prime  i^i  T )
17 ssfi 7810 . . . . . . . . . 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 675 . . . . . . . . 9  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) )  e.  Fin )
1916sseli 3414 . . . . . . . . . 10  |-  ( n  e.  ( ( 1 ... ( |_ `  x ) )  i^i  ( Prime  i^i  T ) )  ->  n  e.  ( Prime  i^i  T )
)
20 simpr 468 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  n  e.  ( Prime  i^i  T )
)
215, 20sseldi 3416 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  n  e.  NN )
2221nnrpd 11362 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  n  e.  RR+ )
23 relogcl 23604 . . . . . . . . . . . 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 10680 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  ( ( log `  n )  /  n )  e.  RR )
2619, 25sylan2 482 . . . . . . . . 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 13877 . . . . . . . 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 472 . . . . . . 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 11335 . . . . . . . 8  |-  RR+  C_  RR
3013recnd 9687 . . . . . . . 8  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( phi `  N )  e.  CC )
31 o1const 13760 . . . . . . . 8  |-  ( (
RR+  C_  RR  /\  ( phi `  N )  e.  CC )  ->  (
x  e.  RR+  |->  ( phi `  N ) )  e.  O(1) )
3229, 30, 31sylancr 676 . . . . . . 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 9676 . . . . . . . . 9  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  1  e.  RR )
3515, 25fsumrecl 13877 . . . . . . . . 9  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  sum_ n  e.  ( Prime  i^i  T )
( ( log `  n
)  /  n )  e.  RR )
36 log1 23614 . . . . . . . . . . . . 13  |-  ( log `  1 )  =  0
3721nnge1d 10674 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  1  <_  n )
38 1rp 11329 . . . . . . . . . . . . . . 15  |-  1  e.  RR+
39 logleb 23631 . . . . . . . . . . . . . . 15  |-  ( ( 1  e.  RR+  /\  n  e.  RR+ )  ->  (
1  <_  n  <->  ( log `  1 )  <_  ( log `  n ) ) )
4038, 22, 39sylancr 676 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  ( 1  <_  n  <->  ( log `  1 )  <_  ( log `  n ) ) )
4137, 40mpbid 215 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  ( log `  1 )  <_  ( log `  n ) )
4236, 41syl5eqbrr 4430 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  n  e.  ( Prime  i^i  T ) )  ->  0  <_  ( log `  n ) )
4324, 22, 42divge0d 11401 . . . . . . . . . . 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 13933 . . . . . . . . . 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 472 . . . . . . . . 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 13664 . . . . . . . 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 9662 . . . . . . . . 9  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  0  e.  RR )
4919, 43sylan2 482 . . . . . . . . . . 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 13932 . . . . . . . . . 10  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  0  <_  sum_ n  e.  ( ( 1 ... ( |_ `  x
) )  i^i  ( Prime  i^i  T ) ) ( ( log `  n
)  /  n ) )
5150adantr 472 . . . . . . . . 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 13679 . . . . . . . 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 240 . . . . . . 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 13765 . . . . . 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 9689 . . . . . . . . 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 9687 . . . . . . . 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 472 . . . . . . 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 23604 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
5958adantl 473 . . . . . . . 8  |-  ( ( ( ph  /\  ( Prime  i^i  T )  e. 
Fin )  /\  x  e.  RR+ )  ->  ( log `  x )  e.  RR )
6059recnd 9687 . . . . . . 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 24444 . . . . . . . 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 472 . . . . . . 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 13770 . . . . . 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 215 . . . . 5  |-  ( (
ph  /\  ( Prime  i^i 
T )  e.  Fin )  ->  ( x  e.  RR+  |->  ( log `  x
) )  e.  O(1) )
7069ex 441 . . . 4  |-  ( ph  ->  ( ( Prime  i^i  T )  e.  Fin  ->  ( x  e.  RR+  |->  ( log `  x ) )  e.  O(1) ) )
719, 70mtoi 183 . . 3  |-  ( ph  ->  -.  ( Prime  i^i  T )  e.  Fin )
72 nnenom 12231 . . . . 5  |-  NN  ~~  om
73 sdomentr 7724 . . . . 5  |-  ( ( ( Prime  i^i  T ) 
~<  NN  /\  NN  ~~  om )  ->  ( Prime  i^i 
T )  ~<  om )
7472, 73mpan2 685 . . . 4  |-  ( ( Prime  i^i  T )  ~<  NN  ->  ( Prime  i^i 
T )  ~<  om )
75 isfinite2 7847 . . . 4  |-  ( ( Prime  i^i  T )  ~<  om  ->  ( Prime  i^i 
T )  e.  Fin )
7674, 75syl 17 . . 3  |-  ( ( Prime  i^i  T )  ~<  NN  ->  ( Prime  i^i 
T )  e.  Fin )
7771, 76nsyl 125 . 2  |-  ( ph  ->  -.  ( Prime  i^i  T )  ~<  NN )
78 bren2 7618 . 2  |-  ( ( Prime  i^i  T )  ~~  NN  <->  ( ( Prime  i^i  T )  ~<_  NN  /\  -.  ( Prime  i^i  T ) 
~<  NN ) )
798, 77, 78sylanbrc 677 1  |-  ( ph  ->  ( Prime  i^i  T ) 
~~  NN )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   _Vcvv 3031    i^i cin 3389    C_ wss 3390   {csn 3959   class class class wbr 4395    |-> cmpt 4454   `'ccnv 4838   "cima 4842   ` cfv 5589  (class class class)co 6308   omcom 6711    ~~ cen 7584    ~<_ cdom 7585    ~< csdm 7586   Fincfn 7587   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    x. cmul 9562    <_ cle 9694    - cmin 9880    / cdiv 10291   NNcn 10631   RR+crp 11325   ...cfz 11810   |_cfl 12059   O(1)co1 13627   <_O(1)clo1 13628   sum_csu 13829   Primecprime 14701   phicphi 14790  Unitcui 17945   ZRHomczrh 19148  ℤ/nczn 19151   logclog 23583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-disj 4367  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-rpss 6590  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-tpos 6991  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-omul 7205  df-er 7381  df-ec 7383  df-qs 7387  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-acn 8394  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-word 12711  df-concat 12713  df-s1 12714  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-o1 13631  df-lo1 13632  df-sum 13830  df-ef 14198  df-e 14199  df-sin 14200  df-cos 14201  df-tan 14202  df-pi 14203  df-dvds 14383  df-gcd 14548  df-prm 14702  df-numer 14763  df-denom 14764  df-phi 14793  df-pc 14866  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-qus 15487  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-mhm 16660  df-submnd 16661  df-grp 16751  df-minusg 16752  df-sbg 16753  df-mulg 16754  df-subg 16892  df-nsg 16893  df-eqg 16894  df-ghm 16959  df-gim 17001  df-ga 17022  df-cntz 17049  df-oppg 17075  df-od 17250  df-gex 17252  df-pgp 17254  df-lsm 17366  df-pj1 17367  df-cmn 17510  df-abl 17511  df-cyg 17591  df-dprd 17705  df-dpj 17706  df-mgp 17802  df-ur 17814  df-ring 17860  df-cring 17861  df-oppr 17929  df-dvdsr 17947  df-unit 17948  df-invr 17978  df-dvr 17989  df-rnghom 18021  df-drng 18055  df-subrg 18084  df-lmod 18171  df-lss 18234  df-lsp 18273  df-sra 18473  df-rgmod 18474  df-lidl 18475  df-rsp 18476  df-2idl 18533  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-zring 19117  df-zrh 19152  df-zn 19155  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-cmp 20479  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-0p 22707  df-limc 22900  df-dv 22901  df-ply 23221  df-idp 23222  df-coe 23223  df-dgr 23224  df-quot 23323  df-log 23585  df-cxp 23586  df-em 23997  df-cht 24102  df-vma 24103  df-chp 24104  df-ppi 24105  df-mu 24106  df-dchr 24240
This theorem is referenced by:  dirith  24446
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