Theorem pclogsum 24940

Description: The logarithmic analogue of pcprod 15599. The sum of the logarithms of the primes dividing A multiplied by their powers yields the logarithm of A. (Contributed by Mario Carneiro, 15-Apr-2016.)

Assertion

Ref	Expression
pclogsum	|- ( A e. NN -> sum_ p e. ( ( 1 ... A ) i^i Prime ) ( ( p pCnt A ) x. ( log ` p ) ) = ( log ` A ) )

Distinct variable group:   A, p

Proof of Theorem pclogsum

Dummy variables m n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3796	. . . . . 6 |- ( p e. ( ( 1 ... A ) i^i Prime ) <-> ( p e. ( 1 ... A ) /\ p e. Prime ) )
2	1	baib 944	. . . . 5 |- ( p e. ( 1 ... A ) -> ( p e. ( ( 1 ... A ) i^i Prime ) <-> p e. Prime ) )
3	2	ifbid 4108	. . . 4 |- ( p e. ( 1 ... A ) -> if ( p e. ( ( 1 ... A ) i^i Prime ) , ( log ` ( p ^ ( p pCnt A ) ) ) , 0 ) = if ( p e. Prime , ( log ` ( p ^ ( p pCnt A ) ) ) , 0 ) )
4		fvif 6204	. . . . 5 |- ( log ` if ( p e. Prime , ( p ^ ( p pCnt A ) ) , 1 ) ) = if ( p e. Prime , ( log ` ( p ^ ( p pCnt A ) ) ) , ( log ` 1 ) )
5		log1 24332	. . . . . 6 |- ( log ` 1 ) = 0
6		ifeq2 4091	. . . . . 6 |- ( ( log ` 1 ) = 0 -> if ( p e. Prime , ( log ` ( p ^ ( p pCnt A ) ) ) , ( log ` 1 ) ) = if ( p e. Prime , ( log ` ( p ^ ( p pCnt A ) ) ) , 0 ) )
7	5, 6	ax-mp 5	. . . . 5 |- if ( p e. Prime , ( log ` ( p ^ ( p pCnt A ) ) ) , ( log ` 1 ) ) = if ( p e. Prime , ( log ` ( p ^ ( p pCnt A ) ) ) , 0 )
8	4, 7	eqtri 2644	. . . 4 |- ( log ` if ( p e. Prime , ( p ^ ( p pCnt A ) ) , 1 ) ) = if ( p e. Prime , ( log ` ( p ^ ( p pCnt A ) ) ) , 0 )
9	3, 8	syl6eqr 2674	. . 3 |- ( p e. ( 1 ... A ) -> if ( p e. ( ( 1 ... A ) i^i Prime ) , ( log ` ( p ^ ( p pCnt A ) ) ) , 0 ) = ( log ` if ( p e. Prime , ( p ^ ( p pCnt A ) ) , 1 ) ) )
10	9	sumeq2i 14429	. 2 |- sum_ p e. ( 1 ... A ) if ( p e. ( ( 1 ... A ) i^i Prime ) , ( log ` ( p ^ ( p pCnt A ) ) ) , 0 ) = sum_ p e. ( 1 ... A ) ( log ` if ( p e. Prime , ( p ^ ( p pCnt A ) ) , 1 ) )
11		inss1 3833	. . . 4 |- ( ( 1 ... A ) i^i Prime ) C_ ( 1 ... A )
12		simpr 477	. . . . . . . . . . 11 |- ( ( A e. NN /\ p e. ( ( 1 ... A ) i^i Prime ) ) -> p e. ( ( 1 ... A ) i^i Prime ) )
13	11, 12	sseldi 3601	. . . . . . . . . 10 |- ( ( A e. NN /\ p e. ( ( 1 ... A ) i^i Prime ) ) -> p e. ( 1 ... A ) )
14		elfznn 12370	. . . . . . . . . 10 |- ( p e. ( 1 ... A ) -> p e. NN )
15	13, 14	syl 17	. . . . . . . . 9 |- ( ( A e. NN /\ p e. ( ( 1 ... A ) i^i Prime ) ) -> p e. NN )
16		inss2 3834	. . . . . . . . . . 11 |- ( ( 1 ... A ) i^i Prime ) C_ Prime
17	16, 12	sseldi 3601	. . . . . . . . . 10 |- ( ( A e. NN /\ p e. ( ( 1 ... A ) i^i Prime ) ) -> p e. Prime )
18		simpl 473	. . . . . . . . . 10 |- ( ( A e. NN /\ p e. ( ( 1 ... A ) i^i Prime ) ) -> A e. NN )
19	17, 18	pccld 15555	. . . . . . . . 9 |- ( ( A e. NN /\ p e. ( ( 1 ... A ) i^i Prime ) ) -> ( p pCnt A ) e. NN0 )
20	15, 19	nnexpcld 13030	. . . . . . . 8 |- ( ( A e. NN /\ p e. ( ( 1 ... A ) i^i Prime ) ) -> ( p ^ ( p pCnt A ) ) e. NN )
21	20	nnrpd 11870	. . . . . . 7 |- ( ( A e. NN /\ p e. ( ( 1 ... A ) i^i Prime ) ) -> ( p ^ ( p pCnt A ) ) e. RR+ )
22	21	relogcld 24369	. . . . . 6 |- ( ( A e. NN /\ p e. ( ( 1 ... A ) i^i Prime ) ) -> ( log ` ( p ^ ( p pCnt A ) ) ) e. RR )
23	22	recnd 10068	. . . . 5 |- ( ( A e. NN /\ p e. ( ( 1 ... A ) i^i Prime ) ) -> ( log ` ( p ^ ( p pCnt A ) ) ) e. CC )
24	23	ralrimiva 2966	. . . 4 |- ( A e. NN -> A. p e. ( ( 1 ... A ) i^i Prime ) ( log ` ( p ^ ( p pCnt A ) ) ) e. CC )
25		fzfi 12771	. . . . . 6 |- ( 1 ... A ) e. Fin
26	25	olci 406	. . . . 5 |- ( ( 1 ... A ) C_ ( ZZ>= ` 1 ) \/ ( 1 ... A ) e. Fin )
27		sumss2 14457	. . . . 5 |- ( ( ( ( ( 1 ... A ) i^i Prime ) C_ ( 1 ... A ) /\ A. p e. ( ( 1 ... A ) i^i Prime ) ( log ` ( p ^ ( p pCnt A ) ) ) e. CC ) /\ ( ( 1 ... A ) C_ ( ZZ>= ` 1 ) \/ ( 1 ... A ) e. Fin ) ) -> sum_ p e. ( ( 1 ... A ) i^i Prime ) ( log ` ( p ^ ( p pCnt A ) ) ) = sum_ p e. ( 1 ... A ) if ( p e. ( ( 1 ... A ) i^i Prime ) , ( log ` ( p ^ ( p pCnt A ) ) ) , 0 ) )
28	26, 27	mpan2 707	. . . 4 |- ( ( ( ( 1 ... A ) i^i Prime ) C_ ( 1 ... A ) /\ A. p e. ( ( 1 ... A ) i^i Prime ) ( log ` ( p ^ ( p pCnt A ) ) ) e. CC ) -> sum_ p e. ( ( 1 ... A ) i^i Prime ) ( log ` ( p ^ ( p pCnt A ) ) ) = sum_ p e. ( 1 ... A ) if ( p e. ( ( 1 ... A ) i^i Prime ) , ( log ` ( p ^ ( p pCnt A ) ) ) , 0 ) )
29	11, 24, 28	sylancr 695	. . 3 |- ( A e. NN -> sum_ p e. ( ( 1 ... A ) i^i Prime ) ( log ` ( p ^ ( p pCnt A ) ) ) = sum_ p e. ( 1 ... A ) if ( p e. ( ( 1 ... A ) i^i Prime ) , ( log ` ( p ^ ( p pCnt A ) ) ) , 0 ) )
30	15	nnrpd 11870	. . . . 5 |- ( ( A e. NN /\ p e. ( ( 1 ... A ) i^i Prime ) ) -> p e. RR+ )
31	19	nn0zd 11480	. . . . 5 |- ( ( A e. NN /\ p e. ( ( 1 ... A ) i^i Prime ) ) -> ( p pCnt A ) e. ZZ )
32		relogexp 24342	. . . . 5 |- ( ( p e. RR+ /\ ( p pCnt A ) e. ZZ ) -> ( log ` ( p ^ ( p pCnt A ) ) ) = ( ( p pCnt A ) x. ( log ` p ) ) )
33	30, 31, 32	syl2anc 693	. . . 4 |- ( ( A e. NN /\ p e. ( ( 1 ... A ) i^i Prime ) ) -> ( log ` ( p ^ ( p pCnt A ) ) ) = ( ( p pCnt A ) x. ( log ` p ) ) )
34	33	sumeq2dv 14433	. . 3 |- ( A e. NN -> sum_ p e. ( ( 1 ... A ) i^i Prime ) ( log ` ( p ^ ( p pCnt A ) ) ) = sum_ p e. ( ( 1 ... A ) i^i Prime ) ( ( p pCnt A ) x. ( log ` p ) ) )
35	29, 34	eqtr3d 2658	. 2 |- ( A e. NN -> sum_ p e. ( 1 ... A ) if ( p e. ( ( 1 ... A ) i^i Prime ) , ( log ` ( p ^ ( p pCnt A ) ) ) , 0 ) = sum_ p e. ( ( 1 ... A ) i^i Prime ) ( ( p pCnt A ) x. ( log ` p ) ) )
36	14	adantl 482	. . . . 5 |- ( ( A e. NN /\ p e. ( 1 ... A ) ) -> p e. NN )
37		eleq1 2689	. . . . . . . 8 |- ( n = p -> ( n e. Prime <-> p e. Prime ) )
38		id 22	. . . . . . . . 9 |- ( n = p -> n = p )
39		oveq1 6657	. . . . . . . . 9 |- ( n = p -> ( n pCnt A ) = ( p pCnt A ) )
40	38, 39	oveq12d 6668	. . . . . . . 8 |- ( n = p -> ( n ^ ( n pCnt A ) ) = ( p ^ ( p pCnt A ) ) )
41	37, 40	ifbieq1d 4109	. . . . . . 7 |- ( n = p -> if ( n e. Prime , ( n ^ ( n pCnt A ) ) , 1 ) = if ( p e. Prime , ( p ^ ( p pCnt A ) ) , 1 ) )
42	41	fveq2d 6195	. . . . . 6 |- ( n = p -> ( log ` if ( n e. Prime , ( n ^ ( n pCnt A ) ) , 1 ) ) = ( log ` if ( p e. Prime , ( p ^ ( p pCnt A ) ) , 1 ) ) )
43		eqid 2622	. . . . . 6 |- ( n e. NN |-> ( log ` if ( n e. Prime , ( n ^ ( n pCnt A ) ) , 1 ) ) ) = ( n e. NN |-> ( log ` if ( n e. Prime , ( n ^ ( n pCnt A ) ) , 1 ) ) )
44		fvex 6201	. . . . . 6 |- ( log ` if ( p e. Prime , ( p ^ ( p pCnt A ) ) , 1 ) ) e. _V
45	42, 43, 44	fvmpt 6282	. . . . 5 |- ( p e. NN -> ( ( n e. NN |-> ( log ` if ( n e. Prime , ( n ^ ( n pCnt A ) ) , 1 ) ) ) ` p ) = ( log ` if ( p e. Prime , ( p ^ ( p pCnt A ) ) , 1 ) ) )
46	36, 45	syl 17	. . . 4 |- ( ( A e. NN /\ p e. ( 1 ... A ) ) -> ( ( n e. NN |-> ( log ` if ( n e. Prime , ( n ^ ( n pCnt A ) ) , 1 ) ) ) ` p ) = ( log ` if ( p e. Prime , ( p ^ ( p pCnt A ) ) , 1 ) ) )
47		elnnuz 11724	. . . . 5 |- ( A e. NN <-> A e. ( ZZ>= ` 1 ) )
48	47	biimpi 206	. . . 4 |- ( A e. NN -> A e. ( ZZ>= ` 1 ) )
49	36	adantr 481	. . . . . . . . 9 |- ( ( ( A e. NN /\ p e. ( 1 ... A ) ) /\ p e. Prime ) -> p e. NN )
50		simpr 477	. . . . . . . . . 10 |- ( ( ( A e. NN /\ p e. ( 1 ... A ) ) /\ p e. Prime ) -> p e. Prime )
51		simpll 790	. . . . . . . . . 10 |- ( ( ( A e. NN /\ p e. ( 1 ... A ) ) /\ p e. Prime ) -> A e. NN )
52	50, 51	pccld 15555	. . . . . . . . 9 |- ( ( ( A e. NN /\ p e. ( 1 ... A ) ) /\ p e. Prime ) -> ( p pCnt A ) e. NN0 )
53	49, 52	nnexpcld 13030	. . . . . . . 8 |- ( ( ( A e. NN /\ p e. ( 1 ... A ) ) /\ p e. Prime ) -> ( p ^ ( p pCnt A ) ) e. NN )
54		1nn 11031	. . . . . . . . 9 |- 1 e. NN
55	54	a1i 11	. . . . . . . 8 |- ( ( ( A e. NN /\ p e. ( 1 ... A ) ) /\ -. p e. Prime ) -> 1 e. NN )
56	53, 55	ifclda 4120	. . . . . . 7 |- ( ( A e. NN /\ p e. ( 1 ... A ) ) -> if ( p e. Prime , ( p ^ ( p pCnt A ) ) , 1 ) e. NN )
57	56	nnrpd 11870	. . . . . 6 |- ( ( A e. NN /\ p e. ( 1 ... A ) ) -> if ( p e. Prime , ( p ^ ( p pCnt A ) ) , 1 ) e. RR+ )
58	57	relogcld 24369	. . . . 5 |- ( ( A e. NN /\ p e. ( 1 ... A ) ) -> ( log ` if ( p e. Prime , ( p ^ ( p pCnt A ) ) , 1 ) ) e. RR )
59	58	recnd 10068	. . . 4 |- ( ( A e. NN /\ p e. ( 1 ... A ) ) -> ( log ` if ( p e. Prime , ( p ^ ( p pCnt A ) ) , 1 ) ) e. CC )
60	46, 48, 59	fsumser 14461	. . 3 |- ( A e. NN -> sum_ p e. ( 1 ... A ) ( log ` if ( p e. Prime , ( p ^ ( p pCnt A ) ) , 1 ) ) = ( seq 1 ( + , ( n e. NN |-> ( log ` if ( n e. Prime , ( n ^ ( n pCnt A ) ) , 1 ) ) ) ) ` A ) )
61		rpmulcl 11855	. . . . 5 |- ( ( p e. RR+ /\ m e. RR+ ) -> ( p x. m ) e. RR+ )
62	61	adantl 482	. . . 4 |- ( ( A e. NN /\ ( p e. RR+ /\ m e. RR+ ) ) -> ( p x. m ) e. RR+ )
63		eqid 2622	. . . . . . 7 |- ( n e. NN |-> if ( n e. Prime , ( n ^ ( n pCnt A ) ) , 1 ) ) = ( n e. NN |-> if ( n e. Prime , ( n ^ ( n pCnt A ) ) , 1 ) )
64		ovex 6678	. . . . . . . 8 |- ( p ^ ( p pCnt A ) ) e. _V
65		1ex 10035	. . . . . . . 8 |- 1 e. _V
66	64, 65	ifex 4156	. . . . . . 7 |- if ( p e. Prime , ( p ^ ( p pCnt A ) ) , 1 ) e. _V
67	41, 63, 66	fvmpt 6282	. . . . . 6 |- ( p e. NN -> ( ( n e. NN |-> if ( n e. Prime , ( n ^ ( n pCnt A ) ) , 1 ) ) ` p ) = if ( p e. Prime , ( p ^ ( p pCnt A ) ) , 1 ) )
68	36, 67	syl 17	. . . . 5 |- ( ( A e. NN /\ p e. ( 1 ... A ) ) -> ( ( n e. NN |-> if ( n e. Prime , ( n ^ ( n pCnt A ) ) , 1 ) ) ` p ) = if ( p e. Prime , ( p ^ ( p pCnt A ) ) , 1 ) )
69	68, 57	eqeltrd 2701	. . . 4 |- ( ( A e. NN /\ p e. ( 1 ... A ) ) -> ( ( n e. NN |-> if ( n e. Prime , ( n ^ ( n pCnt A ) ) , 1 ) ) ` p ) e. RR+ )
70		relogmul 24338	. . . . 5 |- ( ( p e. RR+ /\ m e. RR+ ) -> ( log ` ( p x. m ) ) = ( ( log ` p ) + ( log ` m ) ) )
71	70	adantl 482	. . . 4 |- ( ( A e. NN /\ ( p e. RR+ /\ m e. RR+ ) ) -> ( log ` ( p x. m ) ) = ( ( log ` p ) + ( log ` m ) ) )
72	68	fveq2d 6195	. . . . 5 |- ( ( A e. NN /\ p e. ( 1 ... A ) ) -> ( log ` ( ( n e. NN |-> if ( n e. Prime , ( n ^ ( n pCnt A ) ) , 1 ) ) ` p ) ) = ( log ` if ( p e. Prime , ( p ^ ( p pCnt A ) ) , 1 ) ) )
73	72, 46	eqtr4d 2659	. . . 4 |- ( ( A e. NN /\ p e. ( 1 ... A ) ) -> ( log ` ( ( n e. NN |-> if ( n e. Prime , ( n ^ ( n pCnt A ) ) , 1 ) ) ` p ) ) = ( ( n e. NN |-> ( log ` if ( n e. Prime , ( n ^ ( n pCnt A ) ) , 1 ) ) ) ` p ) )
74	62, 69, 48, 71, 73	seqhomo 12848	. . 3 |- ( A e. NN -> ( log ` ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( n ^ ( n pCnt A ) ) , 1 ) ) ) ` A ) ) = ( seq 1 ( + , ( n e. NN |-> ( log ` if ( n e. Prime , ( n ^ ( n pCnt A ) ) , 1 ) ) ) ) ` A ) )
75	63	pcprod 15599	. . . 4 |- ( A e. NN -> ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( n ^ ( n pCnt A ) ) , 1 ) ) ) ` A ) = A )
76	75	fveq2d 6195	. . 3 |- ( A e. NN -> ( log ` ( seq 1 ( x. , ( n e. NN |-> if ( n e. Prime , ( n ^ ( n pCnt A ) ) , 1 ) ) ) ` A ) ) = ( log ` A ) )
77	60, 74, 76	3eqtr2d 2662	. 2 |- ( A e. NN -> sum_ p e. ( 1 ... A ) ( log ` if ( p e. Prime , ( p ^ ( p pCnt A ) ) , 1 ) ) = ( log ` A ) )
78	10, 35, 77	3eqtr3a 2680	1 |- ( A e. NN -> sum_ p e. ( ( 1 ... A ) i^i Prime ) ( ( p pCnt A ) x. ( log ` p ) ) = ( log ` A ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   i^i cin 3573   C_ wss 3574   ifcif 4086   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   NNcn 11020   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832   ...cfz 12326   seqcseq 12801   ^cexp 12860   sum_csu 14416   Primecprime 15385   pCnt cpc 15541   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-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-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  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-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-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-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-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-dvds 14984  df-gcd 15217  df-prm 15386  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-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  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-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-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-limc 23630  df-dv 23631  df-log 24303

This theorem is referenced by:  vmasum  24941  chebbnd1lem1  25158