isumsplit - Metamath Proof Explorer

	Metamath Proof Explorer	< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > isumsplit		Unicode version

Theorem isumsplit 13652

Description: Split off the first terms of an infinite sum. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 24-Apr-2014.)

**Hypotheses**
Ref	Expression
isumsplit.1
isumsplit.2
isumsplit.3
isumsplit.4
isumsplit.5
isumsplit.6

**Assertion**
Ref	Expression
isumsplit

Distinct variable groups: , ,M , , ,N ,

Proof of Theorem isumsplit Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isumsplit.1	. 2
2		isumsplit.3	. . . 4
3	2, 1	syl6eleq 2555	. . 3
4		eluzel2 11115	. . 3
5	3, 4	syl 16	. 2
6		isumsplit.4	. 2
7		isumsplit.5	. 2
8		isumsplit.2	. . 3
9		eluzelz 11119	. . . 4
10	3, 9	syl 16	. . 3
11		uzss 11130	. . . . . . . 8
12	3, 11	syl 16	. . . . . . 7
13	12, 8, 1	3sstr4g 3544	. . . . . 6
14	13	sselda 3503	. . . . 5
15	14, 6	syldan 470	. . . 4
16	14, 7	syldan 470	. . . 4
17		isumsplit.6	. . . . 5
18	6, 7	eqeltrd 2545	. . . . . 6
19	1, 2, 18	iserex 13479	. . . . 5
20	17, 19	mpbid 210	. . . 4
21	8, 10, 15, 16, 20	isumclim2 13573	. . 3
22		fzfid 12083	. . . 4
23		elfzuz 11713	. . . . . 6
24	23, 1	syl6eleqr 2556	. . . . 5
25	24, 7	sylan2 474	. . . 4
26	22, 25	fsumcl 13555	. . 3
27	14, 18	syldan 470	. . . . 5
28	8, 10, 27	serf 12135	. . . 4
29	28	ffvelrnda 6031	. . 3
30	5	zred 10994	. . . . . . . . . . . 12
31	30	ltm1d 10503	. . . . . . . . . . 11
32		peano2zm 10932	. . . . . . . . . . . . 13
33	5, 32	syl 16	. . . . . . . . . . . 12
34		fzn 11731	. . . . . . . . . . . 12
35	5, 33, 34	syl2anc 661	. . . . . . . . . . 11
36	31, 35	mpbid 210	. . . . . . . . . 10
37	36	sumeq1d 13523	. . . . . . . . 9
38	37	adantr 465	. . . . . . . 8
39		sum0 13543	. . . . . . . 8
40	38, 39	syl6eq 2514	. . . . . . 7
41	40	oveq1d 6311	. . . . . 6
42	13	sselda 3503	. . . . . . . 8
43	1, 5, 18	serf 12135	. . . . . . . . 9
44	43	ffvelrnda 6031	. . . . . . . 8
45	42, 44	syldan 470	. . . . . . 7
46	45	addid2d 9802	. . . . . 6
47	41, 46	eqtr2d 2499	. . . . 5
48		oveq1 6303	. . . . . . . . 9
49	48	oveq2d 6312	. . . . . . . 8
50	49	sumeq1d 13523	. . . . . . 7
51		seqeq1 12110	. . . . . . . 8
52	51	fveq1d 5873	. . . . . . 7
53	50, 52	oveq12d 6314	. . . . . 6
54	53	eqeq2d 2471	. . . . 5
55	47, 54	syl5ibrcom 222	. . . 4
56		addcl 9595	. . . . . . . 8
57	56	adantl 466	. . . . . . 7
58		addass 9600	. . . . . . . 8
59	58	adantl 466	. . . . . . 7
60		simplr 755	. . . . . . . 8
61		simpll 753	. . . . . . . . . . 11
62	10	zcnd 10995	. . . . . . . . . . . . 13
63		ax-1cn 9571	. . . . . . . . . . . . 13
64		npcan 9852	. . . . . . . . . . . . 13
65	62, 63, 64	sylancl 662	. . . . . . . . . . . 12
66	65	eqcomd 2465	. . . . . . . . . . 11
67	61, 66	syl 16	. . . . . . . . . 10
68	67	fveq2d 5875	. . . . . . . . 9
69	8, 68	syl5eq 2510	. . . . . . . 8
70	60, 69	eleqtrd 2547	. . . . . . 7
71	5	adantr 465	. . . . . . . 8
72		eluzp1m1 11133	. . . . . . . 8
73	71, 72	sylan 471	. . . . . . 7
74		elfzuz 11713	. . . . . . . . 9
75	74, 1	syl6eleqr 2556	. . . . . . . 8
76	61, 75, 18	syl2an 477	. . . . . . 7
77	57, 59, 70, 73, 76	seqsplit 12140	. . . . . 6
78	61, 24, 6	syl2an 477	. . . . . . . 8
79	61, 24, 7	syl2an 477	. . . . . . . 8
80	78, 73, 79	fsumser 13552	. . . . . . 7
81	67	seqeq1d 12113	. . . . . . . 8
82	81	fveq1d 5873	. . . . . . 7
83	80, 82	oveq12d 6314	. . . . . 6
84	77, 83	eqtr4d 2501	. . . . 5
85	84	ex 434	. . . 4
86		uzp1 11143	. . . . . 6
87	3, 86	syl 16	. . . . 5
88	87	adantr 465	. . . 4
89	55, 85, 88	mpjaod 381	. . 3
90	8, 10, 21, 26, 17, 29, 89	climaddc2 13458	. 2
91	1, 5, 6, 7, 90	isumclim 13572	1

Colors of variables: wff setvar class

Syntax hints: ->wi 4 <->wb 184 \/wo 368 /\wa 369 /\w3a 973 =wceq 1395 e.wcel 1818 C_wss 3475 c0 3784 class class class wbr 4452 domcdm 5004 `cfv 5593 (class class class)co 6296 cc 9511 0cc0 9513 1c1 9514 caddc 9516 clt 9649 cmin 9828 cz 10889 cuz 11110 cfz 11701 seqcseq 12107 cli 13307 sum_csu 13508

This theorem is referenced by: isum1p 13653 geolim2 13680 mertenslem2 13694 mertens 13695 effsumlt 13846 eirrlem 13937 rpnnen2lem8 13955 prmreclem6 14439 aaliou3lem7 22745 abelthlem7 22833 log2tlbnd 23276 subfaclim 28632 binomcxplemnn0 31254 stirlinglem12 31867

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1618 ax-4 1631 ax-5 1704 ax-6 1747 ax-7 1790 ax-8 1820 ax-9 1822 ax-10 1837 ax-11 1842 ax-12 1854 ax-13 1999 ax-ext 2435 ax-rep 4563 ax-sep 4573 ax-nul 4581 ax-pow 4630 ax-pr 4691 ax-un 6592 ax-inf2 8079 ax-cnex 9569 ax-resscn 9570 ax-1cn 9571 ax-icn 9572 ax-addcl 9573 ax-addrcl 9574 ax-mulcl 9575 ax-mulrcl 9576 ax-mulcom 9577 ax-addass 9578 ax-mulass 9579 ax-distr 9580 ax-i2m1 9581 ax-1ne0 9582 ax-1rid 9583 ax-rnegex 9584 ax-rrecex 9585 ax-cnre 9586 ax-pre-lttri 9587 ax-pre-lttrn 9588 ax-pre-ltadd 9589 ax-pre-mulgt0 9590 ax-pre-sup 9591

This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 974 df-3an 975 df-tru 1398 df-fal 1401 df-ex 1613 df-nf 1617 df-sb 1740 df-eu 2286 df-mo 2287 df-clab 2443 df-cleq 2449 df-clel 2452 df-nfc 2607 df-ne 2654 df-nel 2655 df-ral 2812 df-rex 2813 df-reu 2814 df-rmo 2815 df-rab 2816 df-v 3111 df-sbc 3328 df-csb 3435 df-dif 3478 df-un 3480 df-in 3482 df-ss 3489 df-pss 3491 df-nul 3785 df-if 3942 df-pw 4014 df-sn 4030 df-pr 4032 df-tp 4034 df-op 4036 df-uni 4250 df-int 4287 df-iun 4332 df-br 4453 df-opab 4511 df-mpt 4512 df-tr 4546 df-eprel 4796 df-id 4800 df-po 4805 df-so 4806 df-fr 4843 df-se 4844 df-we 4845 df-ord 4886 df-on 4887 df-lim 4888 df-suc 4889 df-xp 5010 df-rel 5011 df-cnv 5012 df-co 5013 df-dm 5014 df-rn 5015 df-res 5016 df-ima 5017 df-iota 5556 df-fun 5595 df-fn 5596 df-f 5597 df-f1 5598 df-fo 5599 df-f1o 5600 df-fv 5601 df-isom 5602 df-riota 6257 df-ov 6299 df-oprab 6300 df-mpt2 6301 df-om 6701 df-1st 6800 df-2nd 6801 df-recs 7061 df-rdg 7095 df-1o 7149 df-oadd 7153 df-er 7330 df-en 7537 df-dom 7538 df-sdom 7539 df-fin 7540 df-sup 7921 df-oi 7956 df-card 8341 df-pnf 9651 df-mnf 9652 df-xr 9653 df-ltxr 9654 df-le 9655 df-sub 9830 df-neg 9831 df-div 10232 df-nn 10562 df-2 10619 df-3 10620 df-n0 10821 df-z 10890 df-uz 11111 df-rp 11250 df-fz 11702 df-fzo 11825 df-seq 12108 df-exp 12167 df-hash 12406 df-cj 12932 df-re 12933 df-im 12934 df-sqrt 13068 df-abs 13069 df-clim 13311 df-sum 13509

W3C validator