Theorem chtppilimlem2 25163

Description: Lemma for chtppilim 25164. (Contributed by Mario Carneiro, 22-Sep-2014.)

Hypotheses

Ref	Expression
chtppilim.1	|- ( ph -> A e. RR+ )
chtppilim.2	|- ( ph -> A < 1 )

Assertion

Ref	Expression
chtppilimlem2	|- ( ph -> E. z e. RR A. x e. ( 2 [,) +oo ) ( z <_ x -> ( ( A ^ 2 ) x. ( (π ` x ) x. ( log ` x ) ) ) < ( theta ` x ) ) )

Distinct variable groups:   x, z, A   ph, x, z

Proof of Theorem chtppilimlem2

Step	Hyp	Ref	Expression
1		simpr 477	. . . . . . . . 9 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> x e. ( 2 [,) +oo ) )
2		2re 11090	. . . . . . . . . 10 |- 2 e. RR
3		elicopnf 12269	. . . . . . . . . 10 |- ( 2 e. RR -> ( x e. ( 2 [,) +oo ) <-> ( x e. RR /\ 2 <_ x ) ) )
4	2, 3	ax-mp 5	. . . . . . . . 9 |- ( x e. ( 2 [,) +oo ) <-> ( x e. RR /\ 2 <_ x ) )
5	1, 4	sylib 208	. . . . . . . 8 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( x e. RR /\ 2 <_ x ) )
6	5	simpld 475	. . . . . . 7 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> x e. RR )
7		0red 10041	. . . . . . . 8 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> 0 e. RR )
8	2	a1i 11	. . . . . . . 8 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> 2 e. RR )
9		2pos 11112	. . . . . . . . 9 |- 0 < 2
10	9	a1i 11	. . . . . . . 8 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> 0 < 2 )
11	5	simprd 479	. . . . . . . 8 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> 2 <_ x )
12	7, 8, 6, 10, 11	ltletrd 10197	. . . . . . 7 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> 0 < x )
13	6, 12	elrpd 11869	. . . . . 6 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> x e. RR+ )
14		chtppilim.1	. . . . . . . 8 |- ( ph -> A e. RR+ )
15	14	rpred 11872	. . . . . . 7 |- ( ph -> A e. RR )
16	15	adantr 481	. . . . . 6 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> A e. RR )
17	13, 16	rpcxpcld 24476	. . . . 5 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( x ^c A ) e. RR+ )
18		ppinncl 24900	. . . . . . 7 |- ( ( x e. RR /\ 2 <_ x ) -> (π ` x ) e. NN )
19	5, 18	syl 17	. . . . . 6 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> (π ` x ) e. NN )
20	19	nnrpd 11870	. . . . 5 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> (π ` x ) e. RR+ )
21	17, 20	rpdivcld 11889	. . . 4 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( ( x ^c A ) / (π ` x ) ) e. RR+ )
22	21	ralrimiva 2966	. . 3 |- ( ph -> A. x e. ( 2 [,) +oo ) ( ( x ^c A ) / (π ` x ) ) e. RR+ )
23		chtppilim.2	. . . 4 |- ( ph -> A < 1 )
24		1re 10039	. . . . 5 |- 1 e. RR
25		difrp 11868	. . . . 5 |- ( ( A e. RR /\ 1 e. RR ) -> ( A < 1 <-> ( 1 - A ) e. RR+ ) )
26	15, 24, 25	sylancl 694	. . . 4 |- ( ph -> ( A < 1 <-> ( 1 - A ) e. RR+ ) )
27	23, 26	mpbid 222	. . 3 |- ( ph -> ( 1 - A ) e. RR+ )
28		ovexd 6680	. . . . . 6 |- ( ph -> ( 2 [,) +oo ) e. _V )
29	24	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> 1 e. RR )
30		1lt2 11194	. . . . . . . . . . 11 |- 1 < 2
31	30	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> 1 < 2 )
32	29, 8, 6, 31, 11	ltletrd 10197	. . . . . . . . 9 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> 1 < x )
33	6, 32	rplogcld 24375	. . . . . . . 8 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( log ` x ) e. RR+ )
34	13, 33	rpdivcld 11889	. . . . . . 7 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( x / ( log ` x ) ) e. RR+ )
35	34, 20	rpdivcld 11889	. . . . . 6 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( ( x / ( log ` x ) ) / (π ` x ) ) e. RR+ )
36	27	adantr 481	. . . . . . . . 9 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( 1 - A ) e. RR+ )
37	36	rpred 11872	. . . . . . . 8 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( 1 - A ) e. RR )
38	13, 37	rpcxpcld 24476	. . . . . . 7 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( x ^c ( 1 - A ) ) e. RR+ )
39	33, 38	rpdivcld 11889	. . . . . 6 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( ( log ` x ) / ( x ^c ( 1 - A ) ) ) e. RR+ )
40		eqidd 2623	. . . . . 6 |- ( ph -> ( x e. ( 2 [,) +oo ) |-> ( ( x / ( log ` x ) ) / (π ` x ) ) ) = ( x e. ( 2 [,) +oo ) |-> ( ( x / ( log ` x ) ) / (π ` x ) ) ) )
41		eqidd 2623	. . . . . 6 |- ( ph -> ( x e. ( 2 [,) +oo ) |-> ( ( log ` x ) / ( x ^c ( 1 - A ) ) ) ) = ( x e. ( 2 [,) +oo ) |-> ( ( log ` x ) / ( x ^c ( 1 - A ) ) ) ) )
42	28, 35, 39, 40, 41	offval2 6914	. . . . 5 |- ( ph -> ( ( x e. ( 2 [,) +oo ) |-> ( ( x / ( log ` x ) ) / (π ` x ) ) ) oF x. ( x e. ( 2 [,) +oo ) |-> ( ( log ` x ) / ( x ^c ( 1 - A ) ) ) ) ) = ( x e. ( 2 [,) +oo ) |-> ( ( ( x / ( log ` x ) ) / (π ` x ) ) x. ( ( log ` x ) / ( x ^c ( 1 - A ) ) ) ) ) )
43	34	rpcnd 11874	. . . . . . . 8 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( x / ( log ` x ) ) e. CC )
44	39	rpcnd 11874	. . . . . . . 8 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( ( log ` x ) / ( x ^c ( 1 - A ) ) ) e. CC )
45	20	rpcnne0d 11881	. . . . . . . 8 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( (π ` x ) e. CC /\ (π ` x ) =/= 0 ) )
46		div23 10704	. . . . . . . 8 |- ( ( ( x / ( log ` x ) ) e. CC /\ ( ( log ` x ) / ( x ^c ( 1 - A ) ) ) e. CC /\ ( (π ` x ) e. CC /\ (π ` x ) =/= 0 ) ) -> ( ( ( x / ( log ` x ) ) x. ( ( log ` x ) / ( x ^c ( 1 - A ) ) ) ) / (π ` x ) ) = ( ( ( x / ( log ` x ) ) / (π ` x ) ) x. ( ( log ` x ) / ( x ^c ( 1 - A ) ) ) ) )
47	43, 44, 45, 46	syl3anc 1326	. . . . . . 7 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( ( ( x / ( log ` x ) ) x. ( ( log ` x ) / ( x ^c ( 1 - A ) ) ) ) / (π ` x ) ) = ( ( ( x / ( log ` x ) ) / (π ` x ) ) x. ( ( log ` x ) / ( x ^c ( 1 - A ) ) ) ) )
48	33	rpcnne0d 11881	. . . . . . . . . 10 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( ( log ` x ) e. CC /\ ( log ` x ) =/= 0 ) )
49	38	rpcnne0d 11881	. . . . . . . . . 10 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( ( x ^c ( 1 - A ) ) e. CC /\ ( x ^c ( 1 - A ) ) =/= 0 ) )
50	6	recnd 10068	. . . . . . . . . 10 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> x e. CC )
51		dmdcan 10735	. . . . . . . . . 10 |- ( ( ( ( log ` x ) e. CC /\ ( log ` x ) =/= 0 ) /\ ( ( x ^c ( 1 - A ) ) e. CC /\ ( x ^c ( 1 - A ) ) =/= 0 ) /\ x e. CC ) -> ( ( ( log ` x ) / ( x ^c ( 1 - A ) ) ) x. ( x / ( log ` x ) ) ) = ( x / ( x ^c ( 1 - A ) ) ) )
52	48, 49, 50, 51	syl3anc 1326	. . . . . . . . 9 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( ( ( log ` x ) / ( x ^c ( 1 - A ) ) ) x. ( x / ( log ` x ) ) ) = ( x / ( x ^c ( 1 - A ) ) ) )
53	43, 44	mulcomd 10061	. . . . . . . . 9 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( ( x / ( log ` x ) ) x. ( ( log ` x ) / ( x ^c ( 1 - A ) ) ) ) = ( ( ( log ` x ) / ( x ^c ( 1 - A ) ) ) x. ( x / ( log ` x ) ) ) )
54	13	rpcnne0d 11881	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( x e. CC /\ x =/= 0 ) )
55		ax-1cn 9994	. . . . . . . . . . . . 13 |- 1 e. CC
56	55	a1i 11	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> 1 e. CC )
57	36	rpcnd 11874	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( 1 - A ) e. CC )
58		cxpsub 24428	. . . . . . . . . . . 12 |- ( ( ( x e. CC /\ x =/= 0 ) /\ 1 e. CC /\ ( 1 - A ) e. CC ) -> ( x ^c ( 1 - ( 1 - A ) ) ) = ( ( x ^c 1 ) / ( x ^c ( 1 - A ) ) ) )
59	54, 56, 57, 58	syl3anc 1326	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( x ^c ( 1 - ( 1 - A ) ) ) = ( ( x ^c 1 ) / ( x ^c ( 1 - A ) ) ) )
60	16	recnd 10068	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> A e. CC )
61		nncan 10310	. . . . . . . . . . . . 13 |- ( ( 1 e. CC /\ A e. CC ) -> ( 1 - ( 1 - A ) ) = A )
62	55, 60, 61	sylancr 695	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( 1 - ( 1 - A ) ) = A )
63	62	oveq2d 6666	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( x ^c ( 1 - ( 1 - A ) ) ) = ( x ^c A ) )
64	59, 63	eqtr3d 2658	. . . . . . . . . 10 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( ( x ^c 1 ) / ( x ^c ( 1 - A ) ) ) = ( x ^c A ) )
65	50	cxp1d 24452	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( x ^c 1 ) = x )
66	65	oveq1d 6665	. . . . . . . . . 10 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( ( x ^c 1 ) / ( x ^c ( 1 - A ) ) ) = ( x / ( x ^c ( 1 - A ) ) ) )
67	64, 66	eqtr3d 2658	. . . . . . . . 9 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( x ^c A ) = ( x / ( x ^c ( 1 - A ) ) ) )
68	52, 53, 67	3eqtr4d 2666	. . . . . . . 8 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( ( x / ( log ` x ) ) x. ( ( log ` x ) / ( x ^c ( 1 - A ) ) ) ) = ( x ^c A ) )
69	68	oveq1d 6665	. . . . . . 7 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( ( ( x / ( log ` x ) ) x. ( ( log ` x ) / ( x ^c ( 1 - A ) ) ) ) / (π ` x ) ) = ( ( x ^c A ) / (π ` x ) ) )
70	47, 69	eqtr3d 2658	. . . . . 6 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( ( ( x / ( log ` x ) ) / (π ` x ) ) x. ( ( log ` x ) / ( x ^c ( 1 - A ) ) ) ) = ( ( x ^c A ) / (π ` x ) ) )
71	70	mpteq2dva 4744	. . . . 5 |- ( ph -> ( x e. ( 2 [,) +oo ) |-> ( ( ( x / ( log ` x ) ) / (π ` x ) ) x. ( ( log ` x ) / ( x ^c ( 1 - A ) ) ) ) ) = ( x e. ( 2 [,) +oo ) |-> ( ( x ^c A ) / (π ` x ) ) ) )
72	42, 71	eqtrd 2656	. . . 4 |- ( ph -> ( ( x e. ( 2 [,) +oo ) |-> ( ( x / ( log ` x ) ) / (π ` x ) ) ) oF x. ( x e. ( 2 [,) +oo ) |-> ( ( log ` x ) / ( x ^c ( 1 - A ) ) ) ) ) = ( x e. ( 2 [,) +oo ) |-> ( ( x ^c A ) / (π ` x ) ) ) )
73		chebbnd1 25161	. . . . 5 |- ( x e. ( 2 [,) +oo ) |-> ( ( x / ( log ` x ) ) / (π ` x ) ) ) e. O(1)
74	13	ex 450	. . . . . . 7 |- ( ph -> ( x e. ( 2 [,) +oo ) -> x e. RR+ ) )
75	74	ssrdv 3609	. . . . . 6 |- ( ph -> ( 2 [,) +oo ) C_ RR+ )
76		cxploglim 24704	. . . . . . 7 |- ( ( 1 - A ) e. RR+ -> ( x e. RR+ |-> ( ( log ` x ) / ( x ^c ( 1 - A ) ) ) ) ~~> r 0 )
77	27, 76	syl 17	. . . . . 6 |- ( ph -> ( x e. RR+ |-> ( ( log ` x ) / ( x ^c ( 1 - A ) ) ) ) ~~> r 0 )
78	75, 77	rlimres2 14292	. . . . 5 |- ( ph -> ( x e. ( 2 [,) +oo ) |-> ( ( log ` x ) / ( x ^c ( 1 - A ) ) ) ) ~~> r 0 )
79		o1rlimmul 14349	. . . . 5 |- ( ( ( x e. ( 2 [,) +oo ) |-> ( ( x / ( log ` x ) ) / (π ` x ) ) ) e. O(1) /\ ( x e. ( 2 [,) +oo ) |-> ( ( log ` x ) / ( x ^c ( 1 - A ) ) ) ) ~~> r 0 ) -> ( ( x e. ( 2 [,) +oo ) |-> ( ( x / ( log ` x ) ) / (π ` x ) ) ) oF x. ( x e. ( 2 [,) +oo ) |-> ( ( log ` x ) / ( x ^c ( 1 - A ) ) ) ) ) ~~> r 0 )
80	73, 78, 79	sylancr 695	. . . 4 |- ( ph -> ( ( x e. ( 2 [,) +oo ) |-> ( ( x / ( log ` x ) ) / (π ` x ) ) ) oF x. ( x e. ( 2 [,) +oo ) |-> ( ( log ` x ) / ( x ^c ( 1 - A ) ) ) ) ) ~~> r 0 )
81	72, 80	eqbrtrrd 4677	. . 3 |- ( ph -> ( x e. ( 2 [,) +oo ) |-> ( ( x ^c A ) / (π ` x ) ) ) ~~> r 0 )
82	22, 27, 81	rlimi 14244	. 2 |- ( ph -> E. z e. RR A. x e. ( 2 [,) +oo ) ( z <_ x -> ( abs ` ( ( ( x ^c A ) / (π ` x ) ) - 0 ) ) < ( 1 - A ) ) )
83	21	rpcnd 11874	. . . . . . . . . 10 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( ( x ^c A ) / (π ` x ) ) e. CC )
84	83	subid1d 10381	. . . . . . . . 9 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( ( ( x ^c A ) / (π ` x ) ) - 0 ) = ( ( x ^c A ) / (π ` x ) ) )
85	84	fveq2d 6195	. . . . . . . 8 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( abs ` ( ( ( x ^c A ) / (π ` x ) ) - 0 ) ) = ( abs ` ( ( x ^c A ) / (π ` x ) ) ) )
86	21	rpred 11872	. . . . . . . . 9 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( ( x ^c A ) / (π ` x ) ) e. RR )
87	21	rpge0d 11876	. . . . . . . . 9 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> 0 <_ ( ( x ^c A ) / (π ` x ) ) )
88	86, 87	absidd 14161	. . . . . . . 8 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( abs ` ( ( x ^c A ) / (π ` x ) ) ) = ( ( x ^c A ) / (π ` x ) ) )
89	85, 88	eqtrd 2656	. . . . . . 7 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( abs ` ( ( ( x ^c A ) / (π ` x ) ) - 0 ) ) = ( ( x ^c A ) / (π ` x ) ) )
90	89	breq1d 4663	. . . . . 6 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( ( abs ` ( ( ( x ^c A ) / (π ` x ) ) - 0 ) ) < ( 1 - A ) <-> ( ( x ^c A ) / (π ` x ) ) < ( 1 - A ) ) )
91	14	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( x e. ( 2 [,) +oo ) /\ ( ( x ^c A ) / (π ` x ) ) < ( 1 - A ) ) ) -> A e. RR+ )
92	23	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( x e. ( 2 [,) +oo ) /\ ( ( x ^c A ) / (π ` x ) ) < ( 1 - A ) ) ) -> A < 1 )
93		simprl 794	. . . . . . . 8 |- ( ( ph /\ ( x e. ( 2 [,) +oo ) /\ ( ( x ^c A ) / (π ` x ) ) < ( 1 - A ) ) ) -> x e. ( 2 [,) +oo ) )
94		simprr 796	. . . . . . . 8 |- ( ( ph /\ ( x e. ( 2 [,) +oo ) /\ ( ( x ^c A ) / (π ` x ) ) < ( 1 - A ) ) ) -> ( ( x ^c A ) / (π ` x ) ) < ( 1 - A ) )
95	91, 92, 93, 94	chtppilimlem1 25162	. . . . . . 7 |- ( ( ph /\ ( x e. ( 2 [,) +oo ) /\ ( ( x ^c A ) / (π ` x ) ) < ( 1 - A ) ) ) -> ( ( A ^ 2 ) x. ( (π ` x ) x. ( log ` x ) ) ) < ( theta ` x ) )
96	95	expr 643	. . . . . 6 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( ( ( x ^c A ) / (π ` x ) ) < ( 1 - A ) -> ( ( A ^ 2 ) x. ( (π ` x ) x. ( log ` x ) ) ) < ( theta ` x ) ) )
97	90, 96	sylbid 230	. . . . 5 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( ( abs ` ( ( ( x ^c A ) / (π ` x ) ) - 0 ) ) < ( 1 - A ) -> ( ( A ^ 2 ) x. ( (π ` x ) x. ( log ` x ) ) ) < ( theta ` x ) ) )
98	97	imim2d 57	. . . 4 |- ( ( ph /\ x e. ( 2 [,) +oo ) ) -> ( ( z <_ x -> ( abs ` ( ( ( x ^c A ) / (π ` x ) ) - 0 ) ) < ( 1 - A ) ) -> ( z <_ x -> ( ( A ^ 2 ) x. ( (π ` x ) x. ( log ` x ) ) ) < ( theta ` x ) ) ) )
99	98	ralimdva 2962	. . 3 |- ( ph -> ( A. x e. ( 2 [,) +oo ) ( z <_ x -> ( abs ` ( ( ( x ^c A ) / (π ` x ) ) - 0 ) ) < ( 1 - A ) ) -> A. x e. ( 2 [,) +oo ) ( z <_ x -> ( ( A ^ 2 ) x. ( (π ` x ) x. ( log ` x ) ) ) < ( theta ` x ) ) ) )
100	99	reximdv 3016	. 2 |- ( ph -> ( E. z e. RR A. x e. ( 2 [,) +oo ) ( z <_ x -> ( abs ` ( ( ( x ^c A ) / (π ` x ) ) - 0 ) ) < ( 1 - A ) ) -> E. z e. RR A. x e. ( 2 [,) +oo ) ( z <_ x -> ( ( A ^ 2 ) x. ( (π ` x ) x. ( log ` x ) ) ) < ( theta ` x ) ) ) )
101	82, 100	mpd 15	1 |- ( ph -> E. z e. RR A. x e. ( 2 [,) +oo ) ( z <_ x -> ( ( A ^ 2 ) x. ( (π ` x ) x. ( log ` x ) ) ) < ( theta ` x ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   +oocpnf 10071   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   RR+crp 11832   [,)cico 12177   ^cexp 12860   abscabs 13974   ~~> r crli 14216   O(1)co1 14217   logclog 24301   ^c ccxp 24302   thetaccht 24817  πcppi 24820

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-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-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-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  df-cxp 24304  df-cht 24823  df-ppi 24826

This theorem is referenced by:  chtppilim  25164