selbergr - Metamath Proof Explorer

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Theorem selbergr 25257

Description: Selberg's symmetry formula, using the residual of the second Chebyshev function. Equation 10.6.2 of [Shapiro], p. 428. (Contributed by Mario Carneiro, 16-Apr-2016.)

**Hypothesis**
Ref	Expression
pntrval.r	ψ

**Assertion**
Ref	Expression
selbergr	Λ

Distinct variable groups:

Allowed substitution hint:

(

)

**Proof of Theorem selbergr**
Step	Hyp	Ref	Expression
1		reex 10027	. . . . . . 7
2		rpssre 11843	. . . . . . 7
3	1, 2	ssexi 4803	. . . . . 6
4	3	a1i 11	. . . . 5
5		ovexd 6680	. . . . 5 $/\$ ψ Λ ψ
6		ovexd 6680	. . . . 5 $/\$ Λ
7		eqidd 2623	. . . . 5 ψ Λ ψ ψ Λ ψ
8		eqidd 2623	. . . . 5 Λ Λ
9	4, 5, 6, 7, 8	offval2 6914	. . . 4 ψ Λ ψ Λ ψ Λ ψ Λ
10	9	trud 1493	. . 3 ψ Λ ψ Λ ψ Λ ψ Λ
11		pntrval.r	. . . . . . . . . . . 12 ψ
12	11	pntrf 25252	. . . . . . . . . . 11
13	12	ffvelrni 6358	. . . . . . . . . 10
14	13	recnd 10068	. . . . . . . . 9
15		relogcl 24322	. . . . . . . . . 10
16	15	recnd 10068	. . . . . . . . 9
17	14, 16	mulcld 10060	. . . . . . . 8
18		fzfid 12772	. . . . . . . . 9
19		elfznn 12370	. . . . . . . . . . . . 13
20	19	adantl 482	. . . . . . . . . . . 12 $/\$
21		vmacl 24844	. . . . . . . . . . . 12 Λ
22	20, 21	syl 17	. . . . . . . . . . 11 $/\$ Λ
23	22	recnd 10068	. . . . . . . . . 10 $/\$ Λ
24		rpre 11839	. . . . . . . . . . . . 13
25		nndivre 11056	. . . . . . . . . . . . 13 $/\$
26	24, 19, 25	syl2an 494	. . . . . . . . . . . 12 $/\$
27		chpcl 24850	. . . . . . . . . . . 12 ψ
28	26, 27	syl 17	. . . . . . . . . . 11 $/\$ ψ
29	28	recnd 10068	. . . . . . . . . 10 $/\$ ψ
30	23, 29	mulcld 10060	. . . . . . . . 9 $/\$ Λ ψ
31	18, 30	fsumcl 14464	. . . . . . . 8 Λ ψ
32	17, 31	addcld 10059	. . . . . . 7 Λ ψ
33		rpcn 11841	. . . . . . 7
34		rpne0 11848	. . . . . . 7
35	32, 33, 34	divcld 10801	. . . . . 6 Λ ψ
36	22, 20	nndivred 11069	. . . . . . . 8 $/\$ Λ
37	36	recnd 10068	. . . . . . 7 $/\$ Λ
38	18, 37	fsumcl 14464	. . . . . 6 Λ
39	35, 38, 16	nnncan2d 10427	. . . . 5 Λ ψ Λ Λ ψ Λ
40		chpcl 24850	. . . . . . . . . . . . 13 ψ
41	24, 40	syl 17	. . . . . . . . . . . 12 ψ
42	41	recnd 10068	. . . . . . . . . . 11 ψ
43	42, 16	mulcld 10060	. . . . . . . . . 10 ψ
44	43, 31	addcld 10059	. . . . . . . . 9 ψ Λ ψ
45	44, 33, 34	divcld 10801	. . . . . . . 8 ψ Λ ψ
46	45, 16, 16	subsub4d 10423	. . . . . . 7 ψ Λ ψ ψ Λ ψ
47	11	pntrval 25251	. . . . . . . . . . . . . 14 ψ
48	47	oveq1d 6665	. . . . . . . . . . . . 13 ψ
49	42, 33, 16	subdird 10487	. . . . . . . . . . . . 13 ψ ψ
50	48, 49	eqtrd 2656	. . . . . . . . . . . 12 ψ
51	50	oveq1d 6665	. . . . . . . . . . 11 Λ ψ ψ Λ ψ
52	33, 16	mulcld 10060	. . . . . . . . . . . 12
53	43, 31, 52	addsubd 10413	. . . . . . . . . . 11 ψ Λ ψ ψ Λ ψ
54	51, 53	eqtr4d 2659	. . . . . . . . . 10 Λ ψ ψ Λ ψ
55	54	oveq1d 6665	. . . . . . . . 9 Λ ψ ψ Λ ψ
56		rpcnne0 11850	. . . . . . . . . 10 $/\$
57		divsubdir 10721	. . . . . . . . . 10 ψ Λ ψ $/\$ $/\$ $/\$ ψ Λ ψ ψ Λ ψ
58	44, 52, 56, 57	syl3anc 1326	. . . . . . . . 9 ψ Λ ψ ψ Λ ψ
59	16, 33, 34	divcan3d 10806	. . . . . . . . . 10
60	59	oveq2d 6666	. . . . . . . . 9 ψ Λ ψ ψ Λ ψ
61	55, 58, 60	3eqtrd 2660	. . . . . . . 8 Λ ψ ψ Λ ψ
62	61	oveq1d 6665	. . . . . . 7 Λ ψ ψ Λ ψ
63	16	2timesd 11275	. . . . . . . 8
64	63	oveq2d 6666	. . . . . . 7 ψ Λ ψ ψ Λ ψ
65	46, 62, 64	3eqtr4d 2666	. . . . . 6 Λ ψ ψ Λ ψ
66	65	oveq1d 6665	. . . . 5 Λ ψ Λ ψ Λ ψ Λ
67	33, 38	mulcld 10060	. . . . . . 7 Λ
68		divsubdir 10721	. . . . . . 7 Λ ψ $/\$ Λ $/\$ $/\$ Λ ψ Λ Λ ψ Λ
69	32, 67, 56, 68	syl3anc 1326	. . . . . 6 Λ ψ Λ Λ ψ Λ
70	17, 31, 67	addsubassd 10412	. . . . . . . 8 Λ ψ Λ Λ ψ Λ
71	33	adantr 481	. . . . . . . . . . . 12 $/\$
72	71, 37	mulcld 10060	. . . . . . . . . . 11 $/\$ Λ
73	18, 30, 72	fsumsub 14520	. . . . . . . . . 10 Λ ψ Λ Λ ψ Λ
74	26	recnd 10068	. . . . . . . . . . . . 13 $/\$
75	23, 29, 74	subdid 10486	. . . . . . . . . . . 12 $/\$ Λ ψ Λ ψ Λ
76	19	nnrpd 11870	. . . . . . . . . . . . . . 15
77		rpdivcl 11856	. . . . . . . . . . . . . . 15 $/\$
78	76, 77	sylan2 491	. . . . . . . . . . . . . 14 $/\$
79	11	pntrval 25251	. . . . . . . . . . . . . 14 ψ
80	78, 79	syl 17	. . . . . . . . . . . . 13 $/\$ ψ
81	80	oveq2d 6666	. . . . . . . . . . . 12 $/\$ Λ Λ ψ
82	20	nnrpd 11870	. . . . . . . . . . . . . . 15 $/\$
83		rpcnne0 11850	. . . . . . . . . . . . . . 15 $/\$
84	82, 83	syl 17	. . . . . . . . . . . . . 14 $/\$ $/\$
85		div12 10707	. . . . . . . . . . . . . 14 $/\$ Λ $/\$ $/\$ Λ Λ
86	71, 23, 84, 85	syl3anc 1326	. . . . . . . . . . . . 13 $/\$ Λ Λ
87	86	oveq2d 6666	. . . . . . . . . . . 12 $/\$ Λ ψ Λ Λ ψ Λ
88	75, 81, 87	3eqtr4d 2666	. . . . . . . . . . 11 $/\$ Λ Λ ψ Λ
89	88	sumeq2dv 14433	. . . . . . . . . 10 Λ Λ ψ Λ
90	18, 33, 37	fsummulc2 14516	. . . . . . . . . . 11 Λ Λ
91	90	oveq2d 6666	. . . . . . . . . 10 Λ ψ Λ Λ ψ Λ
92	73, 89, 91	3eqtr4rd 2667	. . . . . . . . 9 Λ ψ Λ Λ
93	92	oveq2d 6666	. . . . . . . 8 Λ ψ Λ Λ
94	70, 93	eqtrd 2656	. . . . . . 7 Λ ψ Λ Λ
95	94	oveq1d 6665	. . . . . 6 Λ ψ Λ Λ
96	38, 33, 34	divcan3d 10806	. . . . . . 7 Λ Λ
97	96	oveq2d 6666	. . . . . 6 Λ ψ Λ Λ ψ Λ
98	69, 95, 97	3eqtr3rd 2665	. . . . 5 Λ ψ Λ Λ
99	39, 66, 98	3eqtr3d 2664	. . . 4 ψ Λ ψ Λ Λ
100	99	mpteq2ia 4740	. . 3 ψ Λ ψ Λ Λ
101	10, 100	eqtri 2644	. 2 ψ Λ ψ Λ Λ
102		selberg2 25240	. . 3 ψ Λ ψ
103		vmadivsum 25171	. . 3 Λ
104		o1sub 14346	. . 3 ψ Λ ψ $/\$ Λ ψ Λ ψ Λ
105	102, 103, 104	mp2an 708	. 2 ψ Λ ψ Λ
106	101, 105	eqeltrri 2698	1 Λ

Colors of variables: wff setvar class

Syntax hints: $/\$ wa 384

wceq 1483

wtru 1484

wcel 1990

wne 2794

cvv 3200

cmpt 4729

cfv 5888 (class class class)co 6650

cof 6895

cc 9934

cr 9935

cc0 9936

c1 9937

caddc 9939

cmul 9941

cmin 10266

cdiv 10684

cn 11020

c2 11070

crp 11832

cfz 12326

cfl 12591

co1 14217

csu 14416

clog 24301 Λcvma 24818 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-disj 4621 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-cmp 21190 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-em 24719 df-cht 24823 df-vma 24824 df-chp 24825 df-ppi 24826 df-mu 24827

This theorem is referenced by: (None)

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