Theorem prmorcht 16294

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 10050	. . . . . . 7 |- ( A e. NN -> A e. QQ )
2		chtqval 16257	. . . . . . 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 9993	. . . . . . . . . 10 |- 2 e. ( ZZ>= ` 1 )
5		ppiqsval2 16253	. . . . . . . . . 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 9675	. . . . . . . . . . . 12 |- ( A e. NN -> A e. ZZ )
8		flid 10741	. . . . . . . . . . . 12 |- ( A e. ZZ -> ( |_ ` A ) = A )
9	7, 8	syl 14	. . . . . . . . . . 11 |- ( A e. NN -> ( |_ ` A ) = A )
10	9	oveq2d 6104	. . . . . . . . . 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 12158	. . . . . . 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 10446	. . . . . . . . . . . . 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 12935	. . . . . . . . . . . 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 10478	. . . . . . . . . . . . . 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 10113	. . . . . . . . . . . 12 |- ( ( A e. NN /\ k e. ( 1 ... A ) ) -> k e. RR+ )
32	31	relogcld 16124	. . . . . . . . . . 11 |- ( ( A e. NN /\ k e. ( 1 ... A ) ) -> ( log ` k ) e. RR )
33	32	recnd 8361	. . . . . . . . . 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 9683	. . . . . . . . . 10 |- ( A e. NN -> 1 e. ZZ )
37	36, 7	fzfigd 10890	. . . . . . . . 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 12186	. . . . . . 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 12156	. . . . 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 5698	. . . . . . 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 9976	. . . . . . 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 10113	. . . . . . . . 9 |- ( ( A e. NN /\ k e. ( ZZ>= ` 1 ) ) -> k e. RR+ )
54	53	relogcld 16124	. . . . . . . 8 |- ( ( A e. NN /\ k e. ( ZZ>= ` 1 ) ) -> ( log ` k ) e. RR )
55	54	recnd 8361	. . . . . . 7 |- ( ( A e. NN /\ k e. ( ZZ>= ` 1 ) ) -> ( log ` k ) e. CC )
56		0cnd 8326	. . . . . . 7 |- ( ( A e. NN /\ k e. ( ZZ>= ` 1 ) ) -> 0 e. CC )
57		eluzelz 9948	. . . . . . . . 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 12935	. . . . . . . 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 5802	. . . . 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 9976	. . . . . 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 12190	. . . 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 5702	. 2 |- ( A e. NN -> ( exp ` ( theta ` A ) ) = ( exp ` ( seq 1 ( + , ( n e. NN |-> if ( n e. Prime , ( log ` n ) , 0 ) ) ) ` A ) ) )
68		addcl 8311	. . . 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 12468	. . . 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 9325	. . . . . . . . 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 10113	. . . . . 6 |- ( ( A e. NN /\ k e. NN ) -> if ( k e. Prime , k , 1 ) e. RR+ )
79	78	reeflogd 16125	. . . . 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 12934	. . . . . . . . . 10 |- ( k e. NN -> DECID k e. Prime )
82		fvifdc 5720	. . . . . . . . . 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 16107	. . . . . . . . 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 5702	. . . . 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 8328	. . . . . . . 8 |- 1 e. _V
96	94, 95	ifex 4634	. . . . . . 7 |- if ( k e. Prime , k , 1 ) e. _V
97	92, 93, 96	fvmpt 5785	. . . . . 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 12457	. . . . 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 8313	. . . 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 10986	. 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 4193   ` cfv 5380  (class class class)co 6088   Fincfn 7027   CCcc 8184   0cc0 8186   1c1 8187   + caddc 8189   x. cmul 8191   NNcn 9314   2c2 9365   ZZcz 9656   ZZ>=cuz 9938   QQcq 10036   [,]cicc 10311   ...cfz 10429   |_cfl 10721   seqcseq 10906   sum_csu 12145   expce 12435   Primecprime 12911   logclog 16097   thetaccht 16245

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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-mulrcl 8285  ax-addcom 8286  ax-mulcom 8287  ax-addass 8288  ax-mulass 8289  ax-distr 8290  ax-i2m1 8291  ax-0lt1 8292  ax-1rid 8293  ax-0id 8294  ax-rnegex 8295  ax-precex 8296  ax-cnre 8297  ax-pre-ltirr 8298  ax-pre-ltwlin 8299  ax-pre-lttrn 8300  ax-pre-apti 8301  ax-pre-ltadd 8302  ax-pre-mulgt0 8303  ax-pre-mulext 8304  ax-arch 8305  ax-caucvg 8306  ax-pre-suploc 8307  ax-addf 8308  ax-mulf 8309

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 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-of 6305  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-frec 6666  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6811  df-map 6928  df-pm 6929  df-en 7028  df-dom 7029  df-fin 7030  df-sup 7331  df-inf 7332  df-pnf 8369  df-mnf 8370  df-xr 8371  df-ltxr 8372  df-le 8373  df-sub 8507  df-neg 8508  df-reap 8912  df-ap 8919  df-div 9012  df-inn 9315  df-2 9373  df-3 9374  df-4 9375  df-n0 9576  df-z 9657  df-uz 9939  df-q 10037  df-rp 10073  df-xneg 10192  df-xadd 10193  df-ioo 10312  df-ico 10314  df-icc 10315  df-fz 10430  df-fzo 10568  df-fl 10723  df-mod 10782  df-seqfrec 10907  df-exp 10998  df-fac 11187  df-bc 11209  df-ihash 11238  df-shft 11603  df-cj 11630  df-re 11631  df-im 11632  df-rsqrt 11787  df-abs 11788  df-clim 12071  df-sumdc 12146  df-ef 12441  df-e 12442  df-dvds 12581  df-prm 12912  df-rest 13655  df-topgen 13674  df-psmet 14990  df-xmet 14991  df-met 14992  df-bl 14993  df-mopn 14994  df-top 15236  df-topon 15249  df-bases 15281  df-ntr 15334  df-cn 15426  df-cnp 15427  df-tx 15491  df-cncf 15809  df-limced 15894  df-dvap 15895  df-relog 16099  df-cht 16248

This theorem is used by:  chtublem  16307  bposlem6  16328