chtnprm - Metamath Proof Explorer

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Theorem chtnprm 24880

Description: The Chebyshev function at a non-prime. (Contributed by Mario Carneiro, 19-Sep-2014.)

**Assertion**
Ref	Expression
chtnprm	$/\$

Proof of Theorem chtnprm Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss2 3834	. . . . . . . . . . . . 13
2		simprr 796	. . . . . . . . . . . . 13 $/\$ $/\$
3	1, 2	sseldi 3601	. . . . . . . . . . . 12 $/\$ $/\$
4		simprl 794	. . . . . . . . . . . 12 $/\$ $/\$
5		nelne2 2891	. . . . . . . . . . . 12 $/\$
6	3, 4, 5	syl2anc 693	. . . . . . . . . . 11 $/\$ $/\$
7		velsn 4193	. . . . . . . . . . . 12
8	7	necon3bbii 2841	. . . . . . . . . . 11
9	6, 8	sylibr 224	. . . . . . . . . 10 $/\$ $/\$
10		inss1 3833	. . . . . . . . . . . . . 14
11	10, 2	sseldi 3601	. . . . . . . . . . . . 13 $/\$ $/\$
12		2z 11409	. . . . . . . . . . . . . 14
13		zcn 11382	. . . . . . . . . . . . . . . . . 18
14	13	adantr 481	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
15		ax-1cn 9994	. . . . . . . . . . . . . . . . 17
16		pncan 10287	. . . . . . . . . . . . . . . . 17 $/\$
17	14, 15, 16	sylancl 694	. . . . . . . . . . . . . . . 16 $/\$ $/\$
18		elfzuz2 12346	. . . . . . . . . . . . . . . . 17
19		uz2m1nn 11763	. . . . . . . . . . . . . . . . 17
20	11, 18, 19	3syl 18	. . . . . . . . . . . . . . . 16 $/\$ $/\$
21	17, 20	eqeltrrd 2702	. . . . . . . . . . . . . . 15 $/\$ $/\$
22		nnuz 11723	. . . . . . . . . . . . . . . 16
23		2m1e1 11135	. . . . . . . . . . . . . . . . 17
24	23	fveq2i 6194	. . . . . . . . . . . . . . . 16
25	22, 24	eqtr4i 2647	. . . . . . . . . . . . . . 15
26	21, 25	syl6eleq 2711	. . . . . . . . . . . . . 14 $/\$ $/\$
27		fzsuc2 12398	. . . . . . . . . . . . . 14 $/\$
28	12, 26, 27	sylancr 695	. . . . . . . . . . . . 13 $/\$ $/\$
29	11, 28	eleqtrd 2703	. . . . . . . . . . . 12 $/\$ $/\$
30		elun 3753	. . . . . . . . . . . 12 $\/$
31	29, 30	sylib 208	. . . . . . . . . . 11 $/\$ $/\$ $\/$
32	31	ord 392	. . . . . . . . . 10 $/\$ $/\$
33	9, 32	mt3d 140	. . . . . . . . 9 $/\$ $/\$
34	33, 3	elind 3798	. . . . . . . 8 $/\$ $/\$
35	34	expr 643	. . . . . . 7 $/\$
36	35	ssrdv 3609	. . . . . 6 $/\$
37		uzid 11702	. . . . . . . 8
38	37	adantr 481	. . . . . . 7 $/\$
39		peano2uz 11741	. . . . . . 7
40		fzss2 12381	. . . . . . 7
41		ssrin 3838	. . . . . . 7
42	38, 39, 40, 41	4syl 19	. . . . . 6 $/\$
43	36, 42	eqssd 3620	. . . . 5 $/\$
44		peano2z 11418	. . . . . . . . 9
45	44	adantr 481	. . . . . . . 8 $/\$
46		flid 12609	. . . . . . . 8
47	45, 46	syl 17	. . . . . . 7 $/\$
48	47	oveq2d 6666	. . . . . 6 $/\$
49	48	ineq1d 3813	. . . . 5 $/\$
50		flid 12609	. . . . . . . 8
51	50	adantr 481	. . . . . . 7 $/\$
52	51	oveq2d 6666	. . . . . 6 $/\$
53	52	ineq1d 3813	. . . . 5 $/\$
54	43, 49, 53	3eqtr4d 2666	. . . 4 $/\$
55		zre 11381	. . . . . 6
56	55	adantr 481	. . . . 5 $/\$
57		peano2re 10209	. . . . 5
58		ppisval 24830	. . . . 5
59	56, 57, 58	3syl 18	. . . 4 $/\$
60		ppisval 24830	. . . . 5
61	56, 60	syl 17	. . . 4 $/\$
62	54, 59, 61	3eqtr4d 2666	. . 3 $/\$
63	62	sumeq1d 14431	. 2 $/\$
64		chtval 24836	. . 3
65	56, 57, 64	3syl 18	. 2 $/\$
66		chtval 24836	. . 3
67	56, 66	syl 17	. 2 $/\$
68	63, 65, 67	3eqtr4d 2666	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4 $\/$ wo 383 $/\$ wa 384

wceq 1483

wcel 1990

wne 2794

cun 3572

cin 3573

wss 3574

csn 4177

cfv 5888 (class class class)co 6650

cc 9934

cr 9935

cc0 9936

c1 9937

caddc 9939

cmin 10266

cn 11020

c2 11070

cz 11377

cuz 11687

cicc 12178

cfz 12326

cfl 12591

csu 14416

cprime 15385

clog 24301

ccht 24817

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-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 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

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 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-iun 4522 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-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-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 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-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-icc 12182 df-fz 12327 df-fl 12593 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-sum 14417 df-dvds 14984 df-prm 15386 df-cht 24823

This theorem is referenced by: chtub 24937

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