Theorem dirith2 25217

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/nℤ ` 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.

Step	Hyp	Ref	Expression
1		nnex 11026	. . . 4 |- NN e. _V
2		inss1 3833	. . . . 5 |- ( Prime i^i T ) C_ Prime
3		prmnn 15388	. . . . . 6 |- ( p e. Prime -> p e. NN )
4	3	ssriv 3607	. . . . 5 |- Prime C_ NN
5	2, 4	sstri 3612	. . . 4 |- ( Prime i^i T ) C_ NN
6		ssdomg 8001	. . . 4 |- ( NN e. _V -> ( ( Prime i^i T ) C_ NN -> ( Prime i^i T ) ~<_ NN ) )
7	1, 5, 6	mp2 9	. . 3 |- ( Prime i^i T ) ~<_ NN
8	7	a1i 11	. 2 |- ( ph -> ( Prime i^i T ) ~<_ NN )
9		logno1 24382	. . . 4 |- -. ( x e. RR+ |-> ( log ` x ) ) e. O(1)
10		rpvmasum.a	. . . . . . . . . . 11 |- ( ph -> N e. NN )
11	10	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( Prime i^i T ) e. Fin ) -> N e. NN )
12	11	phicld 15477	. . . . . . . . 9 |- ( ( ph /\ ( Prime i^i T ) e. Fin ) -> ( phi ` N ) e. NN )
13	12	nnred 11035	. . . . . . . 8 |- ( ( ph /\ ( Prime i^i T ) e. Fin ) -> ( phi ` N ) e. RR )
14	13	adantr 481	. . . . . . 7 |- ( ( ( ph /\ ( Prime i^i T ) e. Fin ) /\ x e. RR+ ) -> ( phi ` N ) e. RR )
15		simpr 477	. . . . . . . . . 10 |- ( ( ph /\ ( Prime i^i T ) e. Fin ) -> ( Prime i^i T ) e. Fin )
16		inss2 3834	. . . . . . . . . 10 |- ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) C_ ( Prime i^i T )
17		ssfi 8180	. . . . . . . . . 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 )
18	15, 16, 17	sylancl 694	. . . . . . . . 9 |- ( ( ph /\ ( Prime i^i T ) e. Fin ) -> ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) e. Fin )
19	16	sseli 3599	. . . . . . . . . 10 |- ( n e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) -> n e. ( Prime i^i T ) )
20		simpr 477	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( Prime i^i T ) e. Fin ) /\ n e. ( Prime i^i T ) ) -> n e. ( Prime i^i T ) )
21	5, 20	sseldi 3601	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( Prime i^i T ) e. Fin ) /\ n e. ( Prime i^i T ) ) -> n e. NN )
22	21	nnrpd 11870	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( Prime i^i T ) e. Fin ) /\ n e. ( Prime i^i T ) ) -> n e. RR+ )
23		relogcl 24322	. . . . . . . . . . . 12 |- ( n e. RR+ -> ( log ` n ) e. RR )
24	22, 23	syl 17	. . . . . . . . . . 11 |- ( ( ( ph /\ ( Prime i^i T ) e. Fin ) /\ n e. ( Prime i^i T ) ) -> ( log ` n ) e. RR )
25	24, 21	nndivred 11069	. . . . . . . . . 10 |- ( ( ( ph /\ ( Prime i^i T ) e. Fin ) /\ n e. ( Prime i^i T ) ) -> ( ( log ` n ) / n ) e. RR )
26	19, 25	sylan2 491	. . . . . . . . 9 |- ( ( ( ph /\ ( Prime i^i T ) e. Fin ) /\ n e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ) -> ( ( log ` n ) / n ) e. RR )
27	18, 26	fsumrecl 14465	. . . . . . . 8 |- ( ( ph /\ ( Prime i^i T ) e. Fin ) -> sum_ n e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ( ( log ` n ) / n ) e. RR )
28	27	adantr 481	. . . . . . 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 11843	. . . . . . . 8 |- RR+ C_ RR
30	13	recnd 10068	. . . . . . . 8 |- ( ( ph /\ ( Prime i^i T ) e. Fin ) -> ( phi ` N ) e. CC )
31		o1const 14350	. . . . . . . 8 |- ( ( RR+ C_ RR /\ ( phi ` N ) e. CC ) -> ( x e. RR+ |-> ( phi ` N ) ) e. O(1) )
32	29, 30, 31	sylancr 695	. . . . . . 7 |- ( ( ph /\ ( Prime i^i T ) e. Fin ) -> ( x e. RR+ |-> ( phi ` N ) ) e. O(1) )
33	29	a1i 11	. . . . . . . . 9 |- ( ( ph /\ ( Prime i^i T ) e. Fin ) -> RR+ C_ RR )
34		1red 10055	. . . . . . . . 9 |- ( ( ph /\ ( Prime i^i T ) e. Fin ) -> 1 e. RR )
35	15, 25	fsumrecl 14465	. . . . . . . . 9 |- ( ( ph /\ ( Prime i^i T ) e. Fin ) -> sum_ n e. ( Prime i^i T ) ( ( log ` n ) / n ) e. RR )
36		log1 24332	. . . . . . . . . . . . 13 |- ( log ` 1 ) = 0
37	21	nnge1d 11063	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( Prime i^i T ) e. Fin ) /\ n e. ( Prime i^i T ) ) -> 1 <_ n )
38		1rp 11836	. . . . . . . . . . . . . . 15 |- 1 e. RR+
39		logleb 24349	. . . . . . . . . . . . . . 15 |- ( ( 1 e. RR+ /\ n e. RR+ ) -> ( 1 <_ n <-> ( log ` 1 ) <_ ( log ` n ) ) )
40	38, 22, 39	sylancr 695	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( Prime i^i T ) e. Fin ) /\ n e. ( Prime i^i T ) ) -> ( 1 <_ n <-> ( log ` 1 ) <_ ( log ` n ) ) )
41	37, 40	mpbid 222	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( Prime i^i T ) e. Fin ) /\ n e. ( Prime i^i T ) ) -> ( log ` 1 ) <_ ( log ` n ) )
42	36, 41	syl5eqbrr 4689	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( Prime i^i T ) e. Fin ) /\ n e. ( Prime i^i T ) ) -> 0 <_ ( log ` n ) )
43	24, 22, 42	divge0d 11912	. . . . . . . . . . 11 |- ( ( ( ph /\ ( Prime i^i T ) e. Fin ) /\ n e. ( Prime i^i T ) ) -> 0 <_ ( ( log ` n ) / n ) )
44	16	a1i 11	. . . . . . . . . . 11 |- ( ( ph /\ ( Prime i^i T ) e. Fin ) -> ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) C_ ( Prime i^i T ) )
45	15, 25, 43, 44	fsumless 14528	. . . . . . . . . 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 ) )
46	45	adantr 481	. . . . . . . . 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 ) )
47	33, 28, 34, 35, 46	ello1d 14254	. . . . . . . 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 10041	. . . . . . . . 9 |- ( ( ph /\ ( Prime i^i T ) e. Fin ) -> 0 e. RR )
49	19, 43	sylan2 491	. . . . . . . . . . 11 |- ( ( ( ph /\ ( Prime i^i T ) e. Fin ) /\ n e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ) -> 0 <_ ( ( log ` n ) / n ) )
50	18, 26, 49	fsumge0 14527	. . . . . . . . . 10 |- ( ( ph /\ ( Prime i^i T ) e. Fin ) -> 0 <_ sum_ n e. ( ( 1 ... ( |_ ` x ) ) i^i ( Prime i^i T ) ) ( ( log ` n ) / n ) )
51	50	adantr 481	. . . . . . . . 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 ) )
52	28, 48, 51	o1lo12 14269	. . . . . . . 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) ) )
53	47, 52	mpbird 247	. . . . . . 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) )
54	14, 28, 32, 53	o1mul2 14355	. . . . . 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) )
55	13, 27	remulcld 10070	. . . . . . . . 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 )
56	55	recnd 10068	. . . . . . . 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 )
57	56	adantr 481	. . . . . . 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 24322	. . . . . . . . 9 |- ( x e. RR+ -> ( log ` x ) e. RR )
59	58	adantl 482	. . . . . . . 8 |- ( ( ( ph /\ ( Prime i^i T ) e. Fin ) /\ x e. RR+ ) -> ( log ` x ) e. RR )
60	59	recnd 10068	. . . . . . 7 |- ( ( ( ph /\ ( Prime i^i T ) e. Fin ) /\ x e. RR+ ) -> ( log ` x ) e. CC )
61		rpvmasum.z	. . . . . . . . 9 |- Z = (ℤ/nℤ ` 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 } )
66	61, 62, 10, 63, 64, 65	rplogsum 25216	. . . . . . . 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) )
67	66	adantr 481	. . . . . . 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) )
68	57, 60, 67	o1dif 14360	. . . . . 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) ) )
69	54, 68	mpbid 222	. . . . 5 |- ( ( ph /\ ( Prime i^i T ) e. Fin ) -> ( x e. RR+ |-> ( log ` x ) ) e. O(1) )
70	69	ex 450	. . . 4 |- ( ph -> ( ( Prime i^i T ) e. Fin -> ( x e. RR+ |-> ( log ` x ) ) e. O(1) ) )
71	9, 70	mtoi 190	. . 3 |- ( ph -> -. ( Prime i^i T ) e. Fin )
72		nnenom 12779	. . . . 5 |- NN ~~ om
73		sdomentr 8094	. . . . 5 |- ( ( ( Prime i^i T ) ~< NN /\ NN ~~ om ) -> ( Prime i^i T ) ~< om )
74	72, 73	mpan2 707	. . . 4 |- ( ( Prime i^i T ) ~< NN -> ( Prime i^i T ) ~< om )
75		isfinite2 8218	. . . 4 |- ( ( Prime i^i T ) ~< om -> ( Prime i^i T ) e. Fin )
76	74, 75	syl 17	. . 3 |- ( ( Prime i^i T ) ~< NN -> ( Prime i^i T ) e. Fin )
77	71, 76	nsyl 135	. 2 |- ( ph -> -. ( Prime i^i T ) ~< NN )
78		bren2 7986	. 2 |- ( ( Prime i^i T ) ~~ NN <-> ( ( Prime i^i T ) ~<_ NN /\ -. ( Prime i^i T ) ~< NN ) )
79	8, 77, 78	sylanbrc 698	1 |- ( ph -> ( Prime i^i T ) ~~ NN )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   {csn 4177   class class class wbr 4653   |-> cmpt 4729   `'ccnv 5113   "cima 5117   ` cfv 5888  (class class class)co 6650   omcom 7065   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   RR+crp 11832   ...cfz 12326   |_cfl 12591   O(1)co1 14217   <_O(1)clo1 14218   sum_csu 14416   Primecprime 15385   phicphi 15469  Unitcui 18639   ZRHomczrh 19848  ℤ/nℤczn 19851   logclog 24301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-rpss 6937  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-o1 14221  df-lo1 14222  df-sum 14417  df-ef 14798  df-e 14799  df-sin 14800  df-cos 14801  df-tan 14802  df-pi 14803  df-dvds 14984  df-gcd 15217  df-prm 15386  df-numer 15443  df-denom 15444  df-phi 15471  df-pc 15542  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-qus 16169  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-nsg 17592  df-eqg 17593  df-ghm 17658  df-gim 17701  df-ga 17723  df-cntz 17750  df-oppg 17776  df-od 17948  df-gex 17949  df-pgp 17950  df-lsm 18051  df-pj1 18052  df-cmn 18195  df-abl 18196  df-cyg 18280  df-dprd 18394  df-dpj 18395  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-rnghom 18715  df-drng 18749  df-subrg 18778  df-lmod 18865  df-lss 18933  df-lsp 18972  df-sra 19172  df-rgmod 19173  df-lidl 19174  df-rsp 19175  df-2idl 19232  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-zring 19819  df-zrh 19852  df-zn 19855  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-0p 23437  df-limc 23630  df-dv 23631  df-ply 23944  df-idp 23945  df-coe 23946  df-dgr 23947  df-quot 24046  df-log 24303  df-cxp 24304  df-em 24719  df-cht 24823  df-vma 24824  df-chp 24825  df-ppi 24826  df-mu 24827  df-dchr 24958

This theorem is referenced by:  dirith  25218