chpchtsum - Metamath Proof Explorer

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Theorem chpchtsum 24944

Description: The second Chebyshev function is the sum of the theta function at arguments quickly approaching zero. (This is usually stated as an infinite sum, but after a certain point, the terms are all zero, and it is easier for us to use an explicit finite sum.) (Contributed by Mario Carneiro, 7-Apr-2016.)

**Assertion**
Ref	Expression
chpchtsum	ψ

Distinct variable group:

Proof of Theorem chpchtsum Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzfid 12772	. . . . 5 $/\$
2		inss2 3834	. . . . . . . . . 10
3		simpr 477	. . . . . . . . . 10 $/\$
4	2, 3	sseldi 3601	. . . . . . . . 9 $/\$
5		prmnn 15388	. . . . . . . . 9
6	4, 5	syl 17	. . . . . . . 8 $/\$
7	6	nnrpd 11870	. . . . . . 7 $/\$
8	7	relogcld 24369	. . . . . 6 $/\$
9	8	recnd 10068	. . . . 5 $/\$
10		fsumconst 14522	. . . . 5 $/\$
11	1, 9, 10	syl2anc 693	. . . 4 $/\$
12		simpl 473	. . . . . . . . . 10 $/\$
13		1red 10055	. . . . . . . . . . 11 $/\$
14	6	nnred 11035	. . . . . . . . . . 11 $/\$
15		prmuz2 15408	. . . . . . . . . . . . 13
16	4, 15	syl 17	. . . . . . . . . . . 12 $/\$
17		eluz2b2 11761	. . . . . . . . . . . . 13 $/\$
18	17	simprbi 480	. . . . . . . . . . . 12
19	16, 18	syl 17	. . . . . . . . . . 11 $/\$
20		inss1 3833	. . . . . . . . . . . . . 14
21	20, 3	sseldi 3601	. . . . . . . . . . . . 13 $/\$
22		0re 10040	. . . . . . . . . . . . . 14
23		elicc2 12238	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
24	22, 12, 23	sylancr 695	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
25	21, 24	mpbid 222	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
26	25	simp3d 1075	. . . . . . . . . . 11 $/\$
27	13, 14, 12, 19, 26	ltletrd 10197	. . . . . . . . . 10 $/\$
28	12, 27	rplogcld 24375	. . . . . . . . 9 $/\$
29	14, 19	rplogcld 24375	. . . . . . . . 9 $/\$
30	28, 29	rpdivcld 11889	. . . . . . . 8 $/\$
31	30	rpred 11872	. . . . . . 7 $/\$
32	30	rpge0d 11876	. . . . . . 7 $/\$
33		flge0nn0 12621	. . . . . . 7 $/\$
34	31, 32, 33	syl2anc 693	. . . . . 6 $/\$
35		hashfz1 13134	. . . . . 6
36	34, 35	syl 17	. . . . 5 $/\$
37	36	oveq1d 6665	. . . 4 $/\$
38	31	flcld 12599	. . . . . 6 $/\$
39	38	zcnd 11483	. . . . 5 $/\$
40	39, 9	mulcomd 10061	. . . 4 $/\$
41	11, 37, 40	3eqtrrd 2661	. . 3 $/\$
42	41	sumeq2dv 14433	. 2
43		chpval2 24943	. 2 ψ
44		simpl 473	. . . . . 6 $/\$
45		0red 10041	. . . . . . 7 $/\$
46		1red 10055	. . . . . . . 8 $/\$
47		0lt1 10550	. . . . . . . . 9
48	47	a1i 11	. . . . . . . 8 $/\$
49		elfzuz2 12346	. . . . . . . . 9
50		eluzle 11700	. . . . . . . . . . 11
51	50	adantl 482	. . . . . . . . . 10 $/\$
52		simpl 473	. . . . . . . . . . 11 $/\$
53		1z 11407	. . . . . . . . . . 11
54		flge 12606	. . . . . . . . . . 11 $/\$
55	52, 53, 54	sylancl 694	. . . . . . . . . 10 $/\$
56	51, 55	mpbird 247	. . . . . . . . 9 $/\$
57	49, 56	sylan2 491	. . . . . . . 8 $/\$
58	45, 46, 44, 48, 57	ltletrd 10197	. . . . . . 7 $/\$
59	45, 44, 58	ltled 10185	. . . . . 6 $/\$
60		elfznn 12370	. . . . . . . 8
61	60	adantl 482	. . . . . . 7 $/\$
62	61	nnrecred 11066	. . . . . 6 $/\$
63	44, 59, 62	recxpcld 24469	. . . . 5 $/\$
64		chtval 24836	. . . . 5
65	63, 64	syl 17	. . . 4 $/\$
66	65	sumeq2dv 14433	. . 3
67		ppifi 24832	. . . 4
68		fzfid 12772	. . . 4
69	2	sseli 3599	. . . . . . . 8
70		elfznn 12370	. . . . . . . 8
71	69, 70	anim12i 590	. . . . . . 7 $/\$ $/\$
72	71	a1i 11	. . . . . 6 $/\$ $/\$
73		0red 10041	. . . . . . . . 9 $/\$
74	2	a1i 11	. . . . . . . . . . . 12
75	74	sselda 3603	. . . . . . . . . . 11 $/\$
76	75, 5	syl 17	. . . . . . . . . 10 $/\$
77	76	nnred 11035	. . . . . . . . 9 $/\$
78	76	nngt0d 11064	. . . . . . . . 9 $/\$
79	73, 77, 12, 78, 26	ltletrd 10197	. . . . . . . 8 $/\$
80	79	ex 450	. . . . . . 7
81	80	adantrd 484	. . . . . 6 $/\$
82	72, 81	jcad 555	. . . . 5 $/\$ $/\$ $/\$
83		inss2 3834	. . . . . . . . 9
84	83	sseli 3599	. . . . . . . 8
85	60, 84	anim12ci 591	. . . . . . 7 $/\$ $/\$
86	85	a1i 11	. . . . . 6 $/\$ $/\$
87	58	ex 450	. . . . . . 7
88	87	adantrd 484	. . . . . 6 $/\$
89	86, 88	jcad 555	. . . . 5 $/\$ $/\$ $/\$
90		elin 3796	. . . . . . . . 9 $/\$
91		simprll 802	. . . . . . . . . . 11 $/\$ $/\$ $/\$
92	91	biantrud 528	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
93		0red 10041	. . . . . . . . . . 11 $/\$ $/\$ $/\$
94		simpl 473	. . . . . . . . . . 11 $/\$ $/\$ $/\$
95	91, 5	syl 17	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
96	95	nnred 11035	. . . . . . . . . . 11 $/\$ $/\$ $/\$
97	95	nnnn0d 11351	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
98	97	nn0ge0d 11354	. . . . . . . . . . 11 $/\$ $/\$ $/\$
99		df-3an 1039	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
100	23, 99	syl6bb 276	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
101	100	baibd 948	. . . . . . . . . . 11 $/\$ $/\$ $/\$
102	93, 94, 96, 98, 101	syl22anc 1327	. . . . . . . . . 10 $/\$ $/\$ $/\$
103	92, 102	bitr3d 270	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
104	90, 103	syl5bb 272	. . . . . . . 8 $/\$ $/\$ $/\$
105		simprr 796	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
106	94, 105	elrpd 11869	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
107	106	relogcld 24369	. . . . . . . . . . 11 $/\$ $/\$ $/\$
108	91, 15	syl 17	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
109	108, 18	syl 17	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
110	96, 109	rplogcld 24375	. . . . . . . . . . 11 $/\$ $/\$ $/\$
111	107, 110	rerpdivcld 11903	. . . . . . . . . 10 $/\$ $/\$ $/\$
112		simprlr 803	. . . . . . . . . . 11 $/\$ $/\$ $/\$
113	112	nnzd 11481	. . . . . . . . . 10 $/\$ $/\$ $/\$
114		flge 12606	. . . . . . . . . 10 $/\$
115	111, 113, 114	syl2anc 693	. . . . . . . . 9 $/\$ $/\$ $/\$
116	112	nnnn0d 11351	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
117	95, 116	nnexpcld 13030	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
118	117	nnrpd 11870	. . . . . . . . . . 11 $/\$ $/\$ $/\$
119	118, 106	logled 24373	. . . . . . . . . 10 $/\$ $/\$ $/\$
120	95	nnrpd 11870	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
121		relogexp 24342	. . . . . . . . . . . 12 $/\$
122	120, 113, 121	syl2anc 693	. . . . . . . . . . 11 $/\$ $/\$ $/\$
123	122	breq1d 4663	. . . . . . . . . 10 $/\$ $/\$ $/\$
124	112	nnred 11035	. . . . . . . . . . 11 $/\$ $/\$ $/\$
125	124, 107, 110	lemuldivd 11921	. . . . . . . . . 10 $/\$ $/\$ $/\$
126	119, 123, 125	3bitrd 294	. . . . . . . . 9 $/\$ $/\$ $/\$
127		nnuz 11723	. . . . . . . . . . 11
128	112, 127	syl6eleq 2711	. . . . . . . . . 10 $/\$ $/\$ $/\$
129	111	flcld 12599	. . . . . . . . . 10 $/\$ $/\$ $/\$
130		elfz5 12334	. . . . . . . . . 10 $/\$
131	128, 129, 130	syl2anc 693	. . . . . . . . 9 $/\$ $/\$ $/\$
132	115, 126, 131	3bitr4rd 301	. . . . . . . 8 $/\$ $/\$ $/\$
133	104, 132	anbi12d 747	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
134	94	flcld 12599	. . . . . . . . . . 11 $/\$ $/\$ $/\$
135		elfz5 12334	. . . . . . . . . . 11 $/\$
136	128, 134, 135	syl2anc 693	. . . . . . . . . 10 $/\$ $/\$ $/\$
137		flge 12606	. . . . . . . . . . 11 $/\$
138	94, 113, 137	syl2anc 693	. . . . . . . . . 10 $/\$ $/\$ $/\$
139	136, 138	bitr4d 271	. . . . . . . . 9 $/\$ $/\$ $/\$
140		elin 3796	. . . . . . . . . 10 $/\$
141	91	biantrud 528	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
142	106	rpge0d 11876	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
143	112	nnrecred 11066	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
144	94, 142, 143	recxpcld 24469	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
145		elicc2 12238	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
146		df-3an 1039	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
147	145, 146	syl6bb 276	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
148	147	baibd 948	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
149	93, 144, 96, 98, 148	syl22anc 1327	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
150	141, 149	bitr3d 270	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
151	94, 142, 143	cxpge0d 24470	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
152	112	nnrpd 11870	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
153	96, 98, 144, 151, 152	cxple2d 24473	. . . . . . . . . . 11 $/\$ $/\$ $/\$
154	95	nncnd 11036	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
155		cxpexp 24414	. . . . . . . . . . . . 13 $/\$
156	154, 116, 155	syl2anc 693	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
157	112	nncnd 11036	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
158	112	nnne0d 11065	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
159	157, 158	recid2d 10797	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
160	159	oveq2d 6666	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
161	106, 143, 157	cxpmuld 24480	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
162	94	recnd 10068	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
163	162	cxp1d 24452	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
164	160, 161, 163	3eqtr3d 2664	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
165	156, 164	breq12d 4666	. . . . . . . . . . 11 $/\$ $/\$ $/\$
166	150, 153, 165	3bitrd 294	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
167	140, 166	syl5bb 272	. . . . . . . . 9 $/\$ $/\$ $/\$
168	139, 167	anbi12d 747	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
169	117	nnred 11035	. . . . . . . . . . 11 $/\$ $/\$ $/\$
170		bernneq3 12992	. . . . . . . . . . . 12 $/\$
171	108, 116, 170	syl2anc 693	. . . . . . . . . . 11 $/\$ $/\$ $/\$
172	124, 169, 171	ltled 10185	. . . . . . . . . 10 $/\$ $/\$ $/\$
173		letr 10131	. . . . . . . . . . 11 $/\$ $/\$ $/\$
174	124, 169, 94, 173	syl3anc 1326	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
175	172, 174	mpand 711	. . . . . . . . 9 $/\$ $/\$ $/\$
176	175	pm4.71rd 667	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
177	154	exp1d 13003	. . . . . . . . . . 11 $/\$ $/\$ $/\$
178	95	nnge1d 11063	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
179	96, 178, 128	leexp2ad 13041	. . . . . . . . . . 11 $/\$ $/\$ $/\$
180	177, 179	eqbrtrrd 4677	. . . . . . . . . 10 $/\$ $/\$ $/\$
181		letr 10131	. . . . . . . . . . 11 $/\$ $/\$ $/\$
182	96, 169, 94, 181	syl3anc 1326	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
183	180, 182	mpand 711	. . . . . . . . 9 $/\$ $/\$ $/\$
184	183	pm4.71rd 667	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
185	168, 176, 184	3bitr2rd 297	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
186	133, 185	bitrd 268	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
187	186	ex 450	. . . . 5 $/\$ $/\$ $/\$ $/\$
188	82, 89, 187	pm5.21ndd 369	. . . 4 $/\$ $/\$
189	9	adantrr 753	. . . 4 $/\$ $/\$
190	67, 68, 1, 188, 189	fsumcom2 14505	. . 3
191	66, 190	eqtr4d 2659	. 2
192	42, 43, 191	3eqtr4d 2666	1 ψ

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

cin 3573

wss 3574 class class class wbr 4653

cfv 5888 (class class class)co 6650

cfn 7955

cc 9934

cr 9935

cc0 9936

c1 9937

cmul 9941

clt 10074

cle 10075

cdiv 10684

cn 11020

c2 11070

cn0 11292

cz 11377

cuz 11687

crp 11832

cicc 12178

cfz 12326

cfl 12591

cexp 12860

chash 13117

csu 14416

cprime 15385

clog 24301

ccxp 24302

ccht 24817 ψcchp 24819

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 df-cxp 24304 df-cht 24823 df-vma 24824 df-chp 24825

This theorem is referenced by: (None)

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