Theorem chtppilimlem1 25162

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 )
chtppilim.3	|- ( ph -> N e. ( 2 [,) +oo ) )
chtppilim.4	|- ( ph -> ( ( N ^c A ) / (π ` N ) ) < ( 1 - A ) )

Assertion

Ref	Expression
chtppilimlem1	|- ( ph -> ( ( A ^ 2 ) x. ( (π ` N ) x. ( log ` N ) ) ) < ( theta ` N ) )

Proof of Theorem chtppilimlem1

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		chtppilim.1	. . . . . . 7 |- ( ph -> A e. RR+ )
2	1	rpred 11872	. . . . . 6 |- ( ph -> A e. RR )
3	2	recnd 10068	. . . . 5 |- ( ph -> A e. CC )
4	3	sqvald 13005	. . . 4 |- ( ph -> ( A ^ 2 ) = ( A x. A ) )
5	4	oveq1d 6665	. . 3 |- ( ph -> ( ( A ^ 2 ) x. ( (π ` N ) x. ( log ` N ) ) ) = ( ( A x. A ) x. ( (π ` N ) x. ( log ` N ) ) ) )
6		chtppilim.3	. . . . . . . . 9 |- ( ph -> N e. ( 2 [,) +oo ) )
7		2re 11090	. . . . . . . . . 10 |- 2 e. RR
8		elicopnf 12269	. . . . . . . . . 10 |- ( 2 e. RR -> ( N e. ( 2 [,) +oo ) <-> ( N e. RR /\ 2 <_ N ) ) )
9	7, 8	ax-mp 5	. . . . . . . . 9 |- ( N e. ( 2 [,) +oo ) <-> ( N e. RR /\ 2 <_ N ) )
10	6, 9	sylib 208	. . . . . . . 8 |- ( ph -> ( N e. RR /\ 2 <_ N ) )
11	10	simpld 475	. . . . . . 7 |- ( ph -> N e. RR )
12		ppicl 24857	. . . . . . 7 |- ( N e. RR -> (π ` N ) e. NN0 )
13	11, 12	syl 17	. . . . . 6 |- ( ph -> (π ` N ) e. NN0 )
14	13	nn0red 11352	. . . . 5 |- ( ph -> (π ` N ) e. RR )
15	14	recnd 10068	. . . 4 |- ( ph -> (π ` N ) e. CC )
16		0red 10041	. . . . . . . 8 |- ( ph -> 0 e. RR )
17	7	a1i 11	. . . . . . . 8 |- ( ph -> 2 e. RR )
18		2pos 11112	. . . . . . . . 9 |- 0 < 2
19	18	a1i 11	. . . . . . . 8 |- ( ph -> 0 < 2 )
20	10	simprd 479	. . . . . . . 8 |- ( ph -> 2 <_ N )
21	16, 17, 11, 19, 20	ltletrd 10197	. . . . . . 7 |- ( ph -> 0 < N )
22	11, 21	elrpd 11869	. . . . . 6 |- ( ph -> N e. RR+ )
23	22	relogcld 24369	. . . . 5 |- ( ph -> ( log ` N ) e. RR )
24	23	recnd 10068	. . . 4 |- ( ph -> ( log ` N ) e. CC )
25	3, 3, 15, 24	mul4d 10248	. . 3 |- ( ph -> ( ( A x. A ) x. ( (π ` N ) x. ( log ` N ) ) ) = ( ( A x. (π ` N ) ) x. ( A x. ( log ` N ) ) ) )
26	5, 25	eqtrd 2656	. 2 |- ( ph -> ( ( A ^ 2 ) x. ( (π ` N ) x. ( log ` N ) ) ) = ( ( A x. (π ` N ) ) x. ( A x. ( log ` N ) ) ) )
27	2, 14	remulcld 10070	. . . 4 |- ( ph -> ( A x. (π ` N ) ) e. RR )
28	2, 23	remulcld 10070	. . . 4 |- ( ph -> ( A x. ( log ` N ) ) e. RR )
29	27, 28	remulcld 10070	. . 3 |- ( ph -> ( ( A x. (π ` N ) ) x. ( A x. ( log ` N ) ) ) e. RR )
30	22, 2	rpcxpcld 24476	. . . . . . . 8 |- ( ph -> ( N ^c A ) e. RR+ )
31	30	rpred 11872	. . . . . . 7 |- ( ph -> ( N ^c A ) e. RR )
32		ppicl 24857	. . . . . . 7 |- ( ( N ^c A ) e. RR -> (π ` ( N ^c A ) ) e. NN0 )
33	31, 32	syl 17	. . . . . 6 |- ( ph -> (π ` ( N ^c A ) ) e. NN0 )
34	33	nn0red 11352	. . . . 5 |- ( ph -> (π ` ( N ^c A ) ) e. RR )
35	14, 34	resubcld 10458	. . . 4 |- ( ph -> ( (π ` N ) - (π ` ( N ^c A ) ) ) e. RR )
36	35, 28	remulcld 10070	. . 3 |- ( ph -> ( ( (π ` N ) - (π ` ( N ^c A ) ) ) x. ( A x. ( log ` N ) ) ) e. RR )
37		chtcl 24835	. . . 4 |- ( N e. RR -> ( theta ` N ) e. RR )
38	11, 37	syl 17	. . 3 |- ( ph -> ( theta ` N ) e. RR )
39		1red 10055	. . . . . . 7 |- ( ph -> 1 e. RR )
40		1lt2 11194	. . . . . . . 8 |- 1 < 2
41	40	a1i 11	. . . . . . 7 |- ( ph -> 1 < 2 )
42	39, 17, 11, 41, 20	ltletrd 10197	. . . . . 6 |- ( ph -> 1 < N )
43	11, 42	rplogcld 24375	. . . . 5 |- ( ph -> ( log ` N ) e. RR+ )
44	1, 43	rpmulcld 11888	. . . 4 |- ( ph -> ( A x. ( log ` N ) ) e. RR+ )
45	14, 31	resubcld 10458	. . . . 5 |- ( ph -> ( (π ` N ) - ( N ^c A ) ) e. RR )
46		ppinncl 24900	. . . . . . . . . 10 |- ( ( N e. RR /\ 2 <_ N ) -> (π ` N ) e. NN )
47	10, 46	syl 17	. . . . . . . . 9 |- ( ph -> (π ` N ) e. NN )
48	31, 47	nndivred 11069	. . . . . . . 8 |- ( ph -> ( ( N ^c A ) / (π ` N ) ) e. RR )
49		chtppilim.4	. . . . . . . 8 |- ( ph -> ( ( N ^c A ) / (π ` N ) ) < ( 1 - A ) )
50	48, 39, 2, 49	ltsub13d 10633	. . . . . . 7 |- ( ph -> A < ( 1 - ( ( N ^c A ) / (π ` N ) ) ) )
51	31	recnd 10068	. . . . . . . . 9 |- ( ph -> ( N ^c A ) e. CC )
52	47	nnrpd 11870	. . . . . . . . . 10 |- ( ph -> (π ` N ) e. RR+ )
53	52	rpcnne0d 11881	. . . . . . . . 9 |- ( ph -> ( (π ` N ) e. CC /\ (π ` N ) =/= 0 ) )
54		divsubdir 10721	. . . . . . . . 9 |- ( ( (π ` N ) e. CC /\ ( N ^c A ) e. CC /\ ( (π ` N ) e. CC /\ (π ` N ) =/= 0 ) ) -> ( ( (π ` N ) - ( N ^c A ) ) / (π ` N ) ) = ( ( (π ` N ) / (π ` N ) ) - ( ( N ^c A ) / (π ` N ) ) ) )
55	15, 51, 53, 54	syl3anc 1326	. . . . . . . 8 |- ( ph -> ( ( (π ` N ) - ( N ^c A ) ) / (π ` N ) ) = ( ( (π ` N ) / (π ` N ) ) - ( ( N ^c A ) / (π ` N ) ) ) )
56		divid 10714	. . . . . . . . . 10 |- ( ( (π ` N ) e. CC /\ (π ` N ) =/= 0 ) -> ( (π ` N ) / (π ` N ) ) = 1 )
57	53, 56	syl 17	. . . . . . . . 9 |- ( ph -> ( (π ` N ) / (π ` N ) ) = 1 )
58	57	oveq1d 6665	. . . . . . . 8 |- ( ph -> ( ( (π ` N ) / (π ` N ) ) - ( ( N ^c A ) / (π ` N ) ) ) = ( 1 - ( ( N ^c A ) / (π ` N ) ) ) )
59	55, 58	eqtrd 2656	. . . . . . 7 |- ( ph -> ( ( (π ` N ) - ( N ^c A ) ) / (π ` N ) ) = ( 1 - ( ( N ^c A ) / (π ` N ) ) ) )
60	50, 59	breqtrrd 4681	. . . . . 6 |- ( ph -> A < ( ( (π ` N ) - ( N ^c A ) ) / (π ` N ) ) )
61	2, 45, 52	ltmuldivd 11919	. . . . . 6 |- ( ph -> ( ( A x. (π ` N ) ) < ( (π ` N ) - ( N ^c A ) ) <-> A < ( ( (π ` N ) - ( N ^c A ) ) / (π ` N ) ) ) )
62	60, 61	mpbird 247	. . . . 5 |- ( ph -> ( A x. (π ` N ) ) < ( (π ` N ) - ( N ^c A ) ) )
63		ppiltx 24903	. . . . . . 7 |- ( ( N ^c A ) e. RR+ -> (π ` ( N ^c A ) ) < ( N ^c A ) )
64	30, 63	syl 17	. . . . . 6 |- ( ph -> (π ` ( N ^c A ) ) < ( N ^c A ) )
65	34, 31, 14, 64	ltsub2dd 10640	. . . . 5 |- ( ph -> ( (π ` N ) - ( N ^c A ) ) < ( (π ` N ) - (π ` ( N ^c A ) ) ) )
66	27, 45, 35, 62, 65	lttrd 10198	. . . 4 |- ( ph -> ( A x. (π ` N ) ) < ( (π ` N ) - (π ` ( N ^c A ) ) ) )
67	27, 35, 44, 66	ltmul1dd 11927	. . 3 |- ( ph -> ( ( A x. (π ` N ) ) x. ( A x. ( log ` N ) ) ) < ( ( (π ` N ) - (π ` ( N ^c A ) ) ) x. ( A x. ( log ` N ) ) ) )
68		fzfid 12772	. . . . . 6 |- ( ph -> ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) e. Fin )
69		inss1 3833	. . . . . 6 |- ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) C_ ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) )
70		ssfi 8180	. . . . . 6 |- ( ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) e. Fin /\ ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) C_ ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) ) -> ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) e. Fin )
71	68, 69, 70	sylancl 694	. . . . 5 |- ( ph -> ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) e. Fin )
72		inss2 3834	. . . . . . . 8 |- ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) C_ Prime
73		simpr 477	. . . . . . . 8 |- ( ( ph /\ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) -> p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) )
74	72, 73	sseldi 3601	. . . . . . 7 |- ( ( ph /\ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) -> p e. Prime )
75		prmnn 15388	. . . . . . . 8 |- ( p e. Prime -> p e. NN )
76	75	nnrpd 11870	. . . . . . 7 |- ( p e. Prime -> p e. RR+ )
77	74, 76	syl 17	. . . . . 6 |- ( ( ph /\ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) -> p e. RR+ )
78	77	relogcld 24369	. . . . 5 |- ( ( ph /\ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) -> ( log ` p ) e. RR )
79	71, 78	fsumrecl 14465	. . . 4 |- ( ph -> sum_ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ( log ` p ) e. RR )
80	28	recnd 10068	. . . . . . 7 |- ( ph -> ( A x. ( log ` N ) ) e. CC )
81		fsumconst 14522	. . . . . . 7 |- ( ( ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) e. Fin /\ ( A x. ( log ` N ) ) e. CC ) -> sum_ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ( A x. ( log ` N ) ) = ( ( # ` ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) x. ( A x. ( log ` N ) ) ) )
82	71, 80, 81	syl2anc 693	. . . . . 6 |- ( ph -> sum_ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ( A x. ( log ` N ) ) = ( ( # ` ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) x. ( A x. ( log ` N ) ) ) )
83		ppifl 24886	. . . . . . . . . 10 |- ( N e. RR -> (π ` ( |_ ` N ) ) = (π ` N ) )
84	11, 83	syl 17	. . . . . . . . 9 |- ( ph -> (π ` ( |_ ` N ) ) = (π ` N ) )
85		ppifl 24886	. . . . . . . . . 10 |- ( ( N ^c A ) e. RR -> (π ` ( |_ ` ( N ^c A ) ) ) = (π ` ( N ^c A ) ) )
86	31, 85	syl 17	. . . . . . . . 9 |- ( ph -> (π ` ( |_ ` ( N ^c A ) ) ) = (π ` ( N ^c A ) ) )
87	84, 86	oveq12d 6668	. . . . . . . 8 |- ( ph -> ( (π ` ( |_ ` N ) ) - (π ` ( |_ ` ( N ^c A ) ) ) ) = ( (π ` N ) - (π ` ( N ^c A ) ) ) )
88	39, 11, 42	ltled 10185	. . . . . . . . . . . 12 |- ( ph -> 1 <_ N )
89		chtppilim.2	. . . . . . . . . . . . 13 |- ( ph -> A < 1 )
90		1re 10039	. . . . . . . . . . . . . 14 |- 1 e. RR
91		ltle 10126	. . . . . . . . . . . . . 14 |- ( ( A e. RR /\ 1 e. RR ) -> ( A < 1 -> A <_ 1 ) )
92	2, 90, 91	sylancl 694	. . . . . . . . . . . . 13 |- ( ph -> ( A < 1 -> A <_ 1 ) )
93	89, 92	mpd 15	. . . . . . . . . . . 12 |- ( ph -> A <_ 1 )
94	11, 88, 2, 39, 93	cxplead 24467	. . . . . . . . . . 11 |- ( ph -> ( N ^c A ) <_ ( N ^c 1 ) )
95	11	recnd 10068	. . . . . . . . . . . 12 |- ( ph -> N e. CC )
96	95	cxp1d 24452	. . . . . . . . . . 11 |- ( ph -> ( N ^c 1 ) = N )
97	94, 96	breqtrd 4679	. . . . . . . . . 10 |- ( ph -> ( N ^c A ) <_ N )
98		flword2 12614	. . . . . . . . . 10 |- ( ( ( N ^c A ) e. RR /\ N e. RR /\ ( N ^c A ) <_ N ) -> ( |_ ` N ) e. ( ZZ>= ` ( |_ ` ( N ^c A ) ) ) )
99	31, 11, 97, 98	syl3anc 1326	. . . . . . . . 9 |- ( ph -> ( |_ ` N ) e. ( ZZ>= ` ( |_ ` ( N ^c A ) ) ) )
100		ppidif 24889	. . . . . . . . 9 |- ( ( |_ ` N ) e. ( ZZ>= ` ( |_ ` ( N ^c A ) ) ) -> ( (π ` ( |_ ` N ) ) - (π ` ( |_ ` ( N ^c A ) ) ) ) = ( # ` ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) )
101	99, 100	syl 17	. . . . . . . 8 |- ( ph -> ( (π ` ( |_ ` N ) ) - (π ` ( |_ ` ( N ^c A ) ) ) ) = ( # ` ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) )
102	87, 101	eqtr3d 2658	. . . . . . 7 |- ( ph -> ( (π ` N ) - (π ` ( N ^c A ) ) ) = ( # ` ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) )
103	102	oveq1d 6665	. . . . . 6 |- ( ph -> ( ( (π ` N ) - (π ` ( N ^c A ) ) ) x. ( A x. ( log ` N ) ) ) = ( ( # ` ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) x. ( A x. ( log ` N ) ) ) )
104	82, 103	eqtr4d 2659	. . . . 5 |- ( ph -> sum_ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ( A x. ( log ` N ) ) = ( ( (π ` N ) - (π ` ( N ^c A ) ) ) x. ( A x. ( log ` N ) ) ) )
105	28	adantr 481	. . . . . 6 |- ( ( ph /\ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) -> ( A x. ( log ` N ) ) e. RR )
106	31	adantr 481	. . . . . . . . 9 |- ( ( ph /\ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) -> ( N ^c A ) e. RR )
107		reflcl 12597	. . . . . . . . . . 11 |- ( ( N ^c A ) e. RR -> ( |_ ` ( N ^c A ) ) e. RR )
108		peano2re 10209	. . . . . . . . . . 11 |- ( ( |_ ` ( N ^c A ) ) e. RR -> ( ( |_ ` ( N ^c A ) ) + 1 ) e. RR )
109	31, 107, 108	3syl 18	. . . . . . . . . 10 |- ( ph -> ( ( |_ ` ( N ^c A ) ) + 1 ) e. RR )
110	109	adantr 481	. . . . . . . . 9 |- ( ( ph /\ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) -> ( ( |_ ` ( N ^c A ) ) + 1 ) e. RR )
111	77	rpred 11872	. . . . . . . . 9 |- ( ( ph /\ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) -> p e. RR )
112		fllep1 12602	. . . . . . . . . . 11 |- ( ( N ^c A ) e. RR -> ( N ^c A ) <_ ( ( |_ ` ( N ^c A ) ) + 1 ) )
113	31, 112	syl 17	. . . . . . . . . 10 |- ( ph -> ( N ^c A ) <_ ( ( |_ ` ( N ^c A ) ) + 1 ) )
114	113	adantr 481	. . . . . . . . 9 |- ( ( ph /\ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) -> ( N ^c A ) <_ ( ( |_ ` ( N ^c A ) ) + 1 ) )
115	69, 73	sseldi 3601	. . . . . . . . . 10 |- ( ( ph /\ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) -> p e. ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) )
116		elfzle1 12344	. . . . . . . . . 10 |- ( p e. ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) -> ( ( |_ ` ( N ^c A ) ) + 1 ) <_ p )
117	115, 116	syl 17	. . . . . . . . 9 |- ( ( ph /\ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) -> ( ( |_ ` ( N ^c A ) ) + 1 ) <_ p )
118	106, 110, 111, 114, 117	letrd 10194	. . . . . . . 8 |- ( ( ph /\ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) -> ( N ^c A ) <_ p )
119	22	rpne0d 11877	. . . . . . . . . . 11 |- ( ph -> N =/= 0 )
120	95, 119, 3	cxpefd 24458	. . . . . . . . . 10 |- ( ph -> ( N ^c A ) = ( exp ` ( A x. ( log ` N ) ) ) )
121	120	eqcomd 2628	. . . . . . . . 9 |- ( ph -> ( exp ` ( A x. ( log ` N ) ) ) = ( N ^c A ) )
122	121	adantr 481	. . . . . . . 8 |- ( ( ph /\ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) -> ( exp ` ( A x. ( log ` N ) ) ) = ( N ^c A ) )
123	77	reeflogd 24370	. . . . . . . 8 |- ( ( ph /\ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) -> ( exp ` ( log ` p ) ) = p )
124	118, 122, 123	3brtr4d 4685	. . . . . . 7 |- ( ( ph /\ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) -> ( exp ` ( A x. ( log ` N ) ) ) <_ ( exp ` ( log ` p ) ) )
125		efle 14848	. . . . . . . 8 |- ( ( ( A x. ( log ` N ) ) e. RR /\ ( log ` p ) e. RR ) -> ( ( A x. ( log ` N ) ) <_ ( log ` p ) <-> ( exp ` ( A x. ( log ` N ) ) ) <_ ( exp ` ( log ` p ) ) ) )
126	105, 78, 125	syl2anc 693	. . . . . . 7 |- ( ( ph /\ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) -> ( ( A x. ( log ` N ) ) <_ ( log ` p ) <-> ( exp ` ( A x. ( log ` N ) ) ) <_ ( exp ` ( log ` p ) ) ) )
127	124, 126	mpbird 247	. . . . . 6 |- ( ( ph /\ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) -> ( A x. ( log ` N ) ) <_ ( log ` p ) )
128	71, 105, 78, 127	fsumle 14531	. . . . 5 |- ( ph -> sum_ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ( A x. ( log ` N ) ) <_ sum_ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ( log ` p ) )
129	104, 128	eqbrtrrd 4677	. . . 4 |- ( ph -> ( ( (π ` N ) - (π ` ( N ^c A ) ) ) x. ( A x. ( log ` N ) ) ) <_ sum_ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ( log ` p ) )
130		fzfid 12772	. . . . . . 7 |- ( ph -> ( 1 ... ( |_ ` N ) ) e. Fin )
131		inss1 3833	. . . . . . 7 |- ( ( 1 ... ( |_ ` N ) ) i^i Prime ) C_ ( 1 ... ( |_ ` N ) )
132		ssfi 8180	. . . . . . 7 |- ( ( ( 1 ... ( |_ ` N ) ) e. Fin /\ ( ( 1 ... ( |_ ` N ) ) i^i Prime ) C_ ( 1 ... ( |_ ` N ) ) ) -> ( ( 1 ... ( |_ ` N ) ) i^i Prime ) e. Fin )
133	130, 131, 132	sylancl 694	. . . . . 6 |- ( ph -> ( ( 1 ... ( |_ ` N ) ) i^i Prime ) e. Fin )
134		inss2 3834	. . . . . . . . . . . . 13 |- ( ( 1 ... ( |_ ` N ) ) i^i Prime ) C_ Prime
135		simpr 477	. . . . . . . . . . . . 13 |- ( ( ph /\ p e. ( ( 1 ... ( |_ ` N ) ) i^i Prime ) ) -> p e. ( ( 1 ... ( |_ ` N ) ) i^i Prime ) )
136	134, 135	sseldi 3601	. . . . . . . . . . . 12 |- ( ( ph /\ p e. ( ( 1 ... ( |_ ` N ) ) i^i Prime ) ) -> p e. Prime )
137		prmuz2 15408	. . . . . . . . . . . 12 |- ( p e. Prime -> p e. ( ZZ>= ` 2 ) )
138	136, 137	syl 17	. . . . . . . . . . 11 |- ( ( ph /\ p e. ( ( 1 ... ( |_ ` N ) ) i^i Prime ) ) -> p e. ( ZZ>= ` 2 ) )
139		eluz2b2 11761	. . . . . . . . . . 11 |- ( p e. ( ZZ>= ` 2 ) <-> ( p e. NN /\ 1 < p ) )
140	138, 139	sylib 208	. . . . . . . . . 10 |- ( ( ph /\ p e. ( ( 1 ... ( |_ ` N ) ) i^i Prime ) ) -> ( p e. NN /\ 1 < p ) )
141	140	simpld 475	. . . . . . . . 9 |- ( ( ph /\ p e. ( ( 1 ... ( |_ ` N ) ) i^i Prime ) ) -> p e. NN )
142	141	nnred 11035	. . . . . . . 8 |- ( ( ph /\ p e. ( ( 1 ... ( |_ ` N ) ) i^i Prime ) ) -> p e. RR )
143	140	simprd 479	. . . . . . . 8 |- ( ( ph /\ p e. ( ( 1 ... ( |_ ` N ) ) i^i Prime ) ) -> 1 < p )
144	142, 143	rplogcld 24375	. . . . . . 7 |- ( ( ph /\ p e. ( ( 1 ... ( |_ ` N ) ) i^i Prime ) ) -> ( log ` p ) e. RR+ )
145	144	rpred 11872	. . . . . 6 |- ( ( ph /\ p e. ( ( 1 ... ( |_ ` N ) ) i^i Prime ) ) -> ( log ` p ) e. RR )
146	144	rpge0d 11876	. . . . . 6 |- ( ( ph /\ p e. ( ( 1 ... ( |_ ` N ) ) i^i Prime ) ) -> 0 <_ ( log ` p ) )
147	30	rpge0d 11876	. . . . . . . . . 10 |- ( ph -> 0 <_ ( N ^c A ) )
148		flge0nn0 12621	. . . . . . . . . 10 |- ( ( ( N ^c A ) e. RR /\ 0 <_ ( N ^c A ) ) -> ( |_ ` ( N ^c A ) ) e. NN0 )
149	31, 147, 148	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( |_ ` ( N ^c A ) ) e. NN0 )
150		nn0p1nn 11332	. . . . . . . . 9 |- ( ( |_ ` ( N ^c A ) ) e. NN0 -> ( ( |_ ` ( N ^c A ) ) + 1 ) e. NN )
151	149, 150	syl 17	. . . . . . . 8 |- ( ph -> ( ( |_ ` ( N ^c A ) ) + 1 ) e. NN )
152		nnuz 11723	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
153	151, 152	syl6eleq 2711	. . . . . . 7 |- ( ph -> ( ( |_ ` ( N ^c A ) ) + 1 ) e. ( ZZ>= ` 1 ) )
154		fzss1 12380	. . . . . . 7 |- ( ( ( |_ ` ( N ^c A ) ) + 1 ) e. ( ZZ>= ` 1 ) -> ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) C_ ( 1 ... ( |_ ` N ) ) )
155		ssrin 3838	. . . . . . 7 |- ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) C_ ( 1 ... ( |_ ` N ) ) -> ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) C_ ( ( 1 ... ( |_ ` N ) ) i^i Prime ) )
156	153, 154, 155	3syl 18	. . . . . 6 |- ( ph -> ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) C_ ( ( 1 ... ( |_ ` N ) ) i^i Prime ) )
157	133, 145, 146, 156	fsumless 14528	. . . . 5 |- ( ph -> sum_ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ( log ` p ) <_ sum_ p e. ( ( 1 ... ( |_ ` N ) ) i^i Prime ) ( log ` p ) )
158		chtval 24836	. . . . . . 7 |- ( N e. RR -> ( theta ` N ) = sum_ p e. ( ( 0 [,] N ) i^i Prime ) ( log ` p ) )
159	11, 158	syl 17	. . . . . 6 |- ( ph -> ( theta ` N ) = sum_ p e. ( ( 0 [,] N ) i^i Prime ) ( log ` p ) )
160		2eluzge1 11734	. . . . . . . 8 |- 2 e. ( ZZ>= ` 1 )
161		ppisval2 24831	. . . . . . . 8 |- ( ( N e. RR /\ 2 e. ( ZZ>= ` 1 ) ) -> ( ( 0 [,] N ) i^i Prime ) = ( ( 1 ... ( |_ ` N ) ) i^i Prime ) )
162	11, 160, 161	sylancl 694	. . . . . . 7 |- ( ph -> ( ( 0 [,] N ) i^i Prime ) = ( ( 1 ... ( |_ ` N ) ) i^i Prime ) )
163	162	sumeq1d 14431	. . . . . 6 |- ( ph -> sum_ p e. ( ( 0 [,] N ) i^i Prime ) ( log ` p ) = sum_ p e. ( ( 1 ... ( |_ ` N ) ) i^i Prime ) ( log ` p ) )
164	159, 163	eqtrd 2656	. . . . 5 |- ( ph -> ( theta ` N ) = sum_ p e. ( ( 1 ... ( |_ ` N ) ) i^i Prime ) ( log ` p ) )
165	157, 164	breqtrrd 4681	. . . 4 |- ( ph -> sum_ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ( log ` p ) <_ ( theta ` N ) )
166	36, 79, 38, 129, 165	letrd 10194	. . 3 |- ( ph -> ( ( (π ` N ) - (π ` ( N ^c A ) ) ) x. ( A x. ( log ` N ) ) ) <_ ( theta ` N ) )
167	29, 36, 38, 67, 166	ltletrd 10197	. 2 |- ( ph -> ( ( A x. (π ` N ) ) x. ( A x. ( log ` N ) ) ) < ( theta ` N ) )
168	26, 167	eqbrtrd 4675	1 |- ( ph -> ( ( A ^ 2 ) x. ( (π ` N ) x. ( log ` N ) ) ) < ( theta ` N ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   i^i cin 3573   C_ wss 3574   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   +oocpnf 10071   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   ZZ>=cuz 11687   RR+crp 11832   [,)cico 12177   [,]cicc 12178   ...cfz 12326   |_cfl 12591   ^cexp 12860   #chash 13117   sum_csu 14416   expce 14792   Primecprime 15385   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-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-dvds 14984  df-prm 15386  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:  chtppilimlem2  25163