Theorem prmorcht 16201

Description: Relate the primorial (product of the primes up to A) to the Chebyshev function. (Contributed by Mario Carneiro, 22-Sep-2014.)

Hypothesis

Ref	Expression
prmorcht.1	|- F = ( n e. NN |-> if ( n e. Prime , n , 1 ) )

Assertion

Ref	Expression
prmorcht	|- ( A e. NN -> ( exp ` ( theta ` A ) ) = ( seq 1 ( x. , F ) ` A ) )

Proof of Theorem prmorcht

Dummy variables k p j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnq 10042	. . . . . . 7 |- ( A e. NN -> A e. QQ )
2		chtqval 16164	. . . . . . 7 |- ( A e. QQ -> ( theta ` A ) = sum_ k e. ( ( 0 [,] A ) i^i Prime ) ( log ` k ) )
3	1, 2	syl 14	. . . . . 6 |- ( A e. NN -> ( theta ` A ) = sum_ k e. ( ( 0 [,] A ) i^i Prime ) ( log ` k ) )
4		2eluzge1 9985	. . . . . . . . . 10 |- 2 e. ( ZZ>= ` 1 )
5		ppiqsval2 16160	. . . . . . . . . 10 |- ( ( A e. QQ /\ 2 e. ( ZZ>= ` 1 ) ) -> ( ( 0 [,] A ) i^i Prime ) = ( ( 1 ... ( |_ ` A ) ) i^i Prime ) )
6	1, 4, 5	sylancl 417	. . . . . . . . 9 |- ( A e. NN -> ( ( 0 [,] A ) i^i Prime ) = ( ( 1 ... ( |_ ` A ) ) i^i Prime ) )
7		nnz 9667	. . . . . . . . . . . 12 |- ( A e. NN -> A e. ZZ )
8		flid 10732	. . . . . . . . . . . 12 |- ( A e. ZZ -> ( |_ ` A ) = A )
9	7, 8	syl 14	. . . . . . . . . . 11 |- ( A e. NN -> ( |_ ` A ) = A )
10	9	oveq2d 6101	. . . . . . . . . 10 |- ( A e. NN -> ( 1 ... ( |_ ` A ) ) = ( 1 ... A ) )
11	10	ineq1d 3431	. . . . . . . . 9 |- ( A e. NN -> ( ( 1 ... ( |_ ` A ) ) i^i Prime ) = ( ( 1 ... A ) i^i Prime ) )
12	6, 11	eqtrd 2271	. . . . . . . 8 |- ( A e. NN -> ( ( 0 [,] A ) i^i Prime ) = ( ( 1 ... A ) i^i Prime ) )
13	12	sumeq1d 12148	. . . . . . 7 |- ( A e. NN -> sum_ k e. ( ( 0 [,] A ) i^i Prime ) ( log ` k ) = sum_ k e. ( ( 1 ... A ) i^i Prime ) ( log ` k ) )
14		inss1 3451	. . . . . . . . 9 |- ( ( 1 ... A ) i^i Prime ) C_ ( 1 ... A )
15	14	a1i 9	. . . . . . . 8 |- ( A e. NN -> ( ( 1 ... A ) i^i Prime ) C_ ( 1 ... A ) )
16		animorrl 838	. . . . . . . . . . . 12 |- ( ( A e. NN /\ j e. ( 1 ... A ) ) -> ( j e. ( 1 ... A ) \/ -. j e. ( 1 ... A ) ) )
17		df-dc 847	. . . . . . . . . . . 12 |- (DECID j e. ( 1 ... A ) <-> ( j e. ( 1 ... A ) \/ -. j e. ( 1 ... A ) ) )
18	16, 17	sylibr 134	. . . . . . . . . . 11 |- ( ( A e. NN /\ j e. ( 1 ... A ) ) -> DECID j e. ( 1 ... A ) )
19		elfzelz 10438	. . . . . . . . . . . . 13 |- ( j e. ( 1 ... A ) -> j e. ZZ )
20	19	adantl 277	. . . . . . . . . . . 12 |- ( ( A e. NN /\ j e. ( 1 ... A ) ) -> j e. ZZ )
21		prmdcz 12925	. . . . . . . . . . . 12 |- ( j e. ZZ -> DECID j e. Prime )
22	20, 21	syl 14	. . . . . . . . . . 11 |- ( ( A e. NN /\ j e. ( 1 ... A ) ) -> DECID j e. Prime )
23	18, 22	dcand 945	. . . . . . . . . 10 |- ( ( A e. NN /\ j e. ( 1 ... A ) ) -> DECID ( j e. ( 1 ... A ) /\ j e. Prime ) )
24		elin 3412	. . . . . . . . . . 11 |- ( j e. ( ( 1 ... A ) i^i Prime ) <-> ( j e. ( 1 ... A ) /\ j e. Prime ) )
25	24	dcbii 852	. . . . . . . . . 10 |- (DECID j e. ( ( 1 ... A ) i^i Prime ) <-> DECID ( j e. ( 1 ... A ) /\ j e. Prime ) )
26	23, 25	sylibr 134	. . . . . . . . 9 |- ( ( A e. NN /\ j e. ( 1 ... A ) ) -> DECID j e. ( ( 1 ... A ) i^i Prime ) )
27	26	ralrimiva 2623	. . . . . . . 8 |- ( A e. NN -> A. j e. ( 1 ... A )DECID j e. ( ( 1 ... A ) i^i Prime ) )
28		elinel1 3415	. . . . . . . . . 10 |- ( k e. ( ( 1 ... A ) i^i Prime ) -> k e. ( 1 ... A ) )
29		elfznn 10470	. . . . . . . . . . . . . 14 |- ( k e. ( 1 ... A ) -> k e. NN )
30	29	adantl 277	. . . . . . . . . . . . 13 |- ( ( A e. NN /\ k e. ( 1 ... A ) ) -> k e. NN )
31	30	nnrpd 10105	. . . . . . . . . . . 12 |- ( ( A e. NN /\ k e. ( 1 ... A ) ) -> k e. RR+ )
32	31	relogcld 16034	. . . . . . . . . . 11 |- ( ( A e. NN /\ k e. ( 1 ... A ) ) -> ( log ` k ) e. RR )
33	32	recnd 8354	. . . . . . . . . 10 |- ( ( A e. NN /\ k e. ( 1 ... A ) ) -> ( log ` k ) e. CC )
34	28, 33	sylan2 286	. . . . . . . . 9 |- ( ( A e. NN /\ k e. ( ( 1 ... A ) i^i Prime ) ) -> ( log ` k ) e. CC )
35	34	ralrimiva 2623	. . . . . . . 8 |- ( A e. NN -> A. k e. ( ( 1 ... A ) i^i Prime ) ( log ` k ) e. CC )
36		1zzd 9675	. . . . . . . . . 10 |- ( A e. NN -> 1 e. ZZ )
37	36, 7	fzfigd 10881	. . . . . . . . 9 |- ( A e. NN -> ( 1 ... A ) e. Fin )
38	37	olcd 746	. . . . . . . 8 |- ( A e. NN -> ( ( 1 e. ZZ /\ ( 1 ... A ) C_ ( ZZ>= ` 1 ) /\ A. j e. ( ZZ>= ` 1 )DECID j e. ( 1 ... A ) ) \/ ( 1 ... A ) e. Fin ) )
39	15, 27, 35, 38	isumss2 12176	. . . . . . 7 |- ( A e. NN -> sum_ k e. ( ( 1 ... A ) i^i Prime ) ( log ` k ) = sum_ k e. ( 1 ... A ) if ( k e. ( ( 1 ... A ) i^i Prime ) , ( log ` k ) , 0 ) )
40	13, 39	eqtrd 2271	. . . . . 6 |- ( A e. NN -> sum_ k e. ( ( 0 [,] A ) i^i Prime ) ( log ` k ) = sum_ k e. ( 1 ... A ) if ( k e. ( ( 1 ... A ) i^i Prime ) , ( log ` k ) , 0 ) )
41	3, 40	eqtrd 2271	. . . . 5 |- ( A e. NN -> ( theta ` A ) = sum_ k e. ( 1 ... A ) if ( k e. ( ( 1 ... A ) i^i Prime ) , ( log ` k ) , 0 ) )
42		elin 3412	. . . . . . . 8 |- ( k e. ( ( 1 ... A ) i^i Prime ) <-> ( k e. ( 1 ... A ) /\ k e. Prime ) )
43	42	baibr 932	. . . . . . 7 |- ( k e. ( 1 ... A ) -> ( k e. Prime <-> k e. ( ( 1 ... A ) i^i Prime ) ) )
44	43	ifbid 3662	. . . . . 6 |- ( k e. ( 1 ... A ) -> if ( k e. Prime , ( log ` k ) , 0 ) = if ( k e. ( ( 1 ... A ) i^i Prime ) , ( log ` k ) , 0 ) )
45	44	sumeq2i 12146	. . . . 5 |- sum_ k e. ( 1 ... A ) if ( k e. Prime , ( log ` k ) , 0 ) = sum_ k e. ( 1 ... A ) if ( k e. ( ( 1 ... A ) i^i Prime ) , ( log ` k ) , 0 )
46	41, 45	eqtr4di 2289	. . . 4 |- ( A e. NN -> ( theta ` A ) = sum_ k e. ( 1 ... A ) if ( k e. Prime , ( log ` k ) , 0 ) )
47		eqid 2238	. . . . . 6 |- ( n e. NN |-> if ( n e. Prime , ( log ` n ) , 0 ) ) = ( n e. NN |-> if ( n e. Prime , ( log ` n ) , 0 ) )
48		eleq1w 2299	. . . . . . 7 |- ( n = k -> ( n e. Prime <-> k e. Prime ) )
49		fveq2 5695	. . . . . . 7 |- ( n = k -> ( log ` n ) = ( log ` k ) )
50	48, 49	ifbieq1d 3663	. . . . . 6 |- ( n = k -> if ( n e. Prime , ( log ` n ) , 0 ) = if ( k e. Prime , ( log ` k ) , 0 ) )
51		elnnuz 9968	. . . . . . 7 |- ( k e. NN <-> k e. ( ZZ>= ` 1 ) )
52	51	bilanri 389	. . . . . 6 |- ( ( A e. NN /\ k e. ( ZZ>= ` 1 ) ) -> k e. NN )
53	52	nnrpd 10105	. . . . . . . . 9 |- ( ( A e. NN /\ k e. ( ZZ>= ` 1 ) ) -> k e. RR+ )
54	53	relogcld 16034	. . . . . . . 8 |- ( ( A e. NN /\ k e. ( ZZ>= ` 1 ) ) -> ( log ` k ) e. RR )
55	54	recnd 8354	. . . . . . 7 |- ( ( A e. NN /\ k e. ( ZZ>= ` 1 ) ) -> ( log ` k ) e. CC )
56		0cnd 8319	. . . . . . 7 |- ( ( A e. NN /\ k e. ( ZZ>= ` 1 ) ) -> 0 e. CC )
57		eluzelz 9940	. . . . . . . . 9 |- ( k e. ( ZZ>= ` 1 ) -> k e. ZZ )
58	57	adantl 277	. . . . . . . 8 |- ( ( A e. NN /\ k e. ( ZZ>= ` 1 ) ) -> k e. ZZ )
59		prmdcz 12925	. . . . . . . 8 |- ( k e. ZZ -> DECID k e. Prime )
60	58, 59	syl 14	. . . . . . 7 |- ( ( A e. NN /\ k e. ( ZZ>= ` 1 ) ) -> DECID k e. Prime )
61	55, 56, 60	ifcldcd 3678	. . . . . 6 |- ( ( A e. NN /\ k e. ( ZZ>= ` 1 ) ) -> if ( k e. Prime , ( log ` k ) , 0 ) e. CC )
62	47, 50, 52, 61	fvmptd3 5799	. . . . 5 |- ( ( A e. NN /\ k e. ( ZZ>= ` 1 ) ) -> ( ( n e. NN |-> if ( n e. Prime , ( log ` n ) , 0 ) ) ` k ) = if ( k e. Prime , ( log ` k ) , 0 ) )
63		elnnuz 9968	. . . . . 6 |- ( A e. NN <-> A e. ( ZZ>= ` 1 ) )
64	63	biimpi 120	. . . . 5 |- ( A e. NN -> A e. ( ZZ>= ` 1 ) )
65	62, 64, 61	fsum3ser 12180	. . . 4 |- ( A e. NN -> sum_ k e. ( 1 ... A ) if ( k e. Prime , ( log ` k ) , 0 ) = ( seq 1 ( + , ( n e. NN |-> if ( n e. Prime , ( log ` n ) , 0 ) ) ) ` A ) )
66	46, 65	eqtrd 2271	. . 3 |- ( A e. NN -> ( theta ` A ) = ( seq 1 ( + , ( n e. NN |-> if ( n e. Prime , ( log ` n ) , 0 ) ) ) ` A ) )
67	66	fveq2d 5699	. 2 |- ( A e. NN -> ( exp ` ( theta ` A ) ) = ( exp ` ( seq 1 ( + , ( n e. NN |-> if ( n e. Prime , ( log ` n ) , 0 ) ) ) ` A ) ) )
68		addcl 8304	. . . 4 |- ( ( k e. CC /\ p e. CC ) -> ( k + p ) e. CC )
69	68	adantl 277	. . 3 |- ( ( A e. NN /\ ( k e. CC /\ p e. CC ) ) -> ( k + p ) e. CC )
70	62, 61	eqeltrd 2315	. . 3 |- ( ( A e. NN /\ k e. ( ZZ>= ` 1 ) ) -> ( ( n e. NN |-> if ( n e. Prime , ( log ` n ) , 0 ) ) ` k ) e. CC )
71		efadd 12458	. . . 4 |- ( ( k e. CC /\ p e. CC ) -> ( exp ` ( k + p ) ) = ( ( exp ` k ) x. ( exp ` p ) ) )
72	71	adantl 277	. . 3 |- ( ( A e. NN /\ ( k e. CC /\ p e. CC ) ) -> ( exp ` ( k + p ) ) = ( ( exp ` k ) x. ( exp ` p ) ) )
73		simpr 110	. . . . . . . 8 |- ( ( A e. NN /\ k e. NN ) -> k e. NN )
74		1nn 9317	. . . . . . . . 9 |- 1 e. NN
75	74	a1i 9	. . . . . . . 8 |- ( ( A e. NN /\ k e. NN ) -> 1 e. NN )
76	51, 60	sylan2b 287	. . . . . . . 8 |- ( ( A e. NN /\ k e. NN ) -> DECID k e. Prime )
77	73, 75, 76	ifcldcd 3678	. . . . . . 7 |- ( ( A e. NN /\ k e. NN ) -> if ( k e. Prime , k , 1 ) e. NN )
78	77	nnrpd 10105	. . . . . 6 |- ( ( A e. NN /\ k e. NN ) -> if ( k e. Prime , k , 1 ) e. RR+ )
79	78	reeflogd 16035	. . . . 5 |- ( ( A e. NN /\ k e. NN ) -> ( exp ` ( log ` if ( k e. Prime , k , 1 ) ) ) = if ( k e. Prime , k , 1 ) )
80	51, 62	sylan2b 287	. . . . . . 7 |- ( ( A e. NN /\ k e. NN ) -> ( ( n e. NN |-> if ( n e. Prime , ( log ` n ) , 0 ) ) ` k ) = if ( k e. Prime , ( log ` k ) , 0 ) )
81		prmdc 12924	. . . . . . . . . 10 |- ( k e. NN -> DECID k e. Prime )
82		fvifdc 5717	. . . . . . . . . 10 |- (DECID k e. Prime -> ( log ` if ( k e. Prime , k , 1 ) ) = if ( k e. Prime , ( log ` k ) , ( log ` 1 ) ) )
83	81, 82	syl 14	. . . . . . . . 9 |- ( k e. NN -> ( log ` if ( k e. Prime , k , 1 ) ) = if ( k e. Prime , ( log ` k ) , ( log ` 1 ) ) )
84	83	adantl 277	. . . . . . . 8 |- ( ( A e. NN /\ k e. NN ) -> ( log ` if ( k e. Prime , k , 1 ) ) = if ( k e. Prime , ( log ` k ) , ( log ` 1 ) ) )
85		log1 16017	. . . . . . . . 9 |- ( log ` 1 ) = 0
86		ifeq2 3644	. . . . . . . . 9 |- ( ( log ` 1 ) = 0 -> if ( k e. Prime , ( log ` k ) , ( log ` 1 ) ) = if ( k e. Prime , ( log ` k ) , 0 ) )
87	85, 86	ax-mp 5	. . . . . . . 8 |- if ( k e. Prime , ( log ` k ) , ( log ` 1 ) ) = if ( k e. Prime , ( log ` k ) , 0 )
88	84, 87	eqtrdi 2287	. . . . . . 7 |- ( ( A e. NN /\ k e. NN ) -> ( log ` if ( k e. Prime , k , 1 ) ) = if ( k e. Prime , ( log ` k ) , 0 ) )
89	80, 88	eqtr4d 2274	. . . . . 6 |- ( ( A e. NN /\ k e. NN ) -> ( ( n e. NN |-> if ( n e. Prime , ( log ` n ) , 0 ) ) ` k ) = ( log ` if ( k e. Prime , k , 1 ) ) )
90	89	fveq2d 5699	. . . . 5 |- ( ( A e. NN /\ k e. NN ) -> ( exp ` ( ( n e. NN |-> if ( n e. Prime , ( log ` n ) , 0 ) ) ` k ) ) = ( exp ` ( log ` if ( k e. Prime , k , 1 ) ) ) )
91		id 19	. . . . . . . 8 |- ( n = k -> n = k )
92	48, 91	ifbieq1d 3663	. . . . . . 7 |- ( n = k -> if ( n e. Prime , n , 1 ) = if ( k e. Prime , k , 1 ) )
93		prmorcht.1	. . . . . . 7 |- F = ( n e. NN |-> if ( n e. Prime , n , 1 ) )
94		vex 2824	. . . . . . . 8 |- k e. _V
95		1ex 8321	. . . . . . . 8 |- 1 e. _V
96	94, 95	ifex 4632	. . . . . . 7 |- if ( k e. Prime , k , 1 ) e. _V
97	92, 93, 96	fvmpt 5782	. . . . . 6 |- ( k e. NN -> ( F ` k ) = if ( k e. Prime , k , 1 ) )
98	97	adantl 277	. . . . 5 |- ( ( A e. NN /\ k e. NN ) -> ( F ` k ) = if ( k e. Prime , k , 1 ) )
99	79, 90, 98	3eqtr4d 2281	. . . 4 |- ( ( A e. NN /\ k e. NN ) -> ( exp ` ( ( n e. NN |-> if ( n e. Prime , ( log ` n ) , 0 ) ) ` k ) ) = ( F ` k ) )
100	52, 99	syldan 282	. . 3 |- ( ( A e. NN /\ k e. ( ZZ>= ` 1 ) ) -> ( exp ` ( ( n e. NN |-> if ( n e. Prime , ( log ` n ) , 0 ) ) ` k ) ) = ( F ` k ) )
101		efcl 12447	. . . . 5 |- ( ( ( n e. NN |-> if ( n e. Prime , ( log ` n ) , 0 ) ) ` k ) e. CC -> ( exp ` ( ( n e. NN |-> if ( n e. Prime , ( log ` n ) , 0 ) ) ` k ) ) e. CC )
102	70, 101	syl 14	. . . 4 |- ( ( A e. NN /\ k e. ( ZZ>= ` 1 ) ) -> ( exp ` ( ( n e. NN |-> if ( n e. Prime , ( log ` n ) , 0 ) ) ` k ) ) e. CC )
103	100, 102	eqeltrrd 2316	. . 3 |- ( ( A e. NN /\ k e. ( ZZ>= ` 1 ) ) -> ( F ` k ) e. CC )
104		mulcl 8306	. . . 4 |- ( ( k e. CC /\ p e. CC ) -> ( k x. p ) e. CC )
105	104	adantl 277	. . 3 |- ( ( A e. NN /\ ( k e. CC /\ p e. CC ) ) -> ( k x. p ) e. CC )
106	69, 70, 64, 72, 100, 103, 105	seq3homo 10977	. 2 |- ( A e. NN -> ( exp ` ( seq 1 ( + , ( n e. NN |-> if ( n e. Prime , ( log ` n ) , 0 ) ) ) ` A ) ) = ( seq 1 ( x. , F ) ` A ) )
107	67, 106	eqtrd 2271	1 |- ( A e. NN -> ( exp ` ( theta ` A ) ) = ( seq 1 ( x. , F ) ` A ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 720  DECID wdc 846   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   i^i cin 3219   C_ wss 3220   ifcif 3638   |-> cmpt 4192   ` cfv 5377  (class class class)co 6085   Fincfn 7022   CCcc 8177   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   NNcn 9306   2c2 9357   ZZcz 9648   ZZ>=cuz 9930   QQcq 10028   [,]cicc 10303   ...cfz 10421   |_cfl 10713   seqcseq 10897   sum_csu 12135   expce 12425   Primecprime 12901   logclog 16007   thetaccht 16152

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299  ax-pre-suploc 8300  ax-addf 8301  ax-mulf 8302

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-disj 4107  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-of 6302  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-2o 6688  df-oadd 6691  df-er 6807  df-map 6924  df-pm 6925  df-en 7023  df-dom 7024  df-fin 7025  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8500  df-neg 8501  df-reap 8905  df-ap 8912  df-div 9005  df-inn 9307  df-2 9365  df-3 9366  df-4 9367  df-n0 9568  df-z 9649  df-uz 9931  df-q 10029  df-rp 10065  df-xneg 10184  df-xadd 10185  df-ioo 10304  df-ico 10306  df-icc 10307  df-fz 10422  df-fzo 10560  df-fl 10715  df-mod 10773  df-seqfrec 10898  df-exp 10989  df-fac 11178  df-bc 11200  df-ihash 11229  df-shft 11594  df-cj 11621  df-re 11622  df-im 11623  df-rsqrt 11778  df-abs 11779  df-clim 12061  df-sumdc 12136  df-ef 12431  df-e 12432  df-dvds 12571  df-prm 12902  df-rest 13644  df-topgen 13663  df-psmet 14929  df-xmet 14930  df-met 14931  df-bl 14932  df-mopn 14933  df-top 15148  df-topon 15161  df-bases 15193  df-ntr 15246  df-cn 15338  df-cnp 15339  df-tx 15403  df-cncf 15721  df-limced 15806  df-dvap 15807  df-relog 16009  df-cht 16155

This theorem is used by:  chtublem  16214