Theorem isumrpcl 11676

Description: The infinite sum of positive reals is positive. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 24-Apr-2014.)

Hypotheses

Ref	Expression
isumrpcl.1	|- Z = ( ZZ>= ` M )
isumrpcl.2	|- W = ( ZZ>= ` N )
isumrpcl.3	|- ( ph -> N e. Z )
isumrpcl.4	|- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )
isumrpcl.5	|- ( ( ph /\ k e. Z ) -> A e. RR+ )
isumrpcl.6	|- ( ph -> seq M ( + , F ) e. dom ~~> )

Assertion

Ref	Expression
isumrpcl	|- ( ph -> sum_ k e. W A e. RR+ )

Distinct variable groups:   k, F   k, M   k, N   ph, k   k, W   k, Z

Allowed substitution hint:   A( k)

Proof of Theorem isumrpcl

Dummy variables m y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isumrpcl.2	. . 3 |- W = ( ZZ>= ` N )
2		isumrpcl.3	. . . . 5 |- ( ph -> N e. Z )
3		isumrpcl.1	. . . . 5 |- Z = ( ZZ>= ` M )
4	2, 3	eleqtrdi 2289	. . . 4 |- ( ph -> N e. ( ZZ>= ` M ) )
5		eluzelz 9627	. . . 4 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
6	4, 5	syl 14	. . 3 |- ( ph -> N e. ZZ )
7		uzss 9639	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> ( ZZ>= ` N ) C_ ( ZZ>= ` M ) )
8	4, 7	syl 14	. . . . . 6 |- ( ph -> ( ZZ>= ` N ) C_ ( ZZ>= ` M ) )
9	8, 1, 3	3sstr4g 3227	. . . . 5 |- ( ph -> W C_ Z )
10	9	sselda 3184	. . . 4 |- ( ( ph /\ k e. W ) -> k e. Z )
11		isumrpcl.4	. . . 4 |- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )
12	10, 11	syldan 282	. . 3 |- ( ( ph /\ k e. W ) -> ( F ` k ) = A )
13		isumrpcl.5	. . . . 5 |- ( ( ph /\ k e. Z ) -> A e. RR+ )
14	13	rpred 9788	. . . 4 |- ( ( ph /\ k e. Z ) -> A e. RR )
15	10, 14	syldan 282	. . 3 |- ( ( ph /\ k e. W ) -> A e. RR )
16		isumrpcl.6	. . . 4 |- ( ph -> seq M ( + , F ) e. dom ~~> )
17	11, 13	eqeltrd 2273	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR+ )
18	17	rpcnd 9790	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
19	3, 2, 18	iserex 11521	. . . 4 |- ( ph -> ( seq M ( + , F ) e. dom ~~> <-> seq N ( + , F ) e. dom ~~> ) )
20	16, 19	mpbid 147	. . 3 |- ( ph -> seq N ( + , F ) e. dom ~~> )
21	1, 6, 12, 15, 20	isumrecl 11611	. 2 |- ( ph -> sum_ k e. W A e. RR )
22		fveq2 5561	. . . 4 |- ( k = N -> ( F ` k ) = ( F ` N ) )
23	22	eleq1d 2265	. . 3 |- ( k = N -> ( ( F ` k ) e. RR+ <-> ( F ` N ) e. RR+ ) )
24	17	ralrimiva 2570	. . 3 |- ( ph -> A. k e. Z ( F ` k ) e. RR+ )
25	23, 24, 2	rspcdva 2873	. 2 |- ( ph -> ( F ` N ) e. RR+ )
26	8	sselda 3184	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` N ) ) -> k e. ( ZZ>= ` M ) )
27	26, 3	eleqtrrdi 2290	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` N ) ) -> k e. Z )
28	27, 17	syldan 282	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` N ) ) -> ( F ` k ) e. RR+ )
29		rpaddcl 9769	. . . . 5 |- ( ( k e. RR+ /\ y e. RR+ ) -> ( k + y ) e. RR+ )
30	29	adantl 277	. . . 4 |- ( ( ph /\ ( k e. RR+ /\ y e. RR+ ) ) -> ( k + y ) e. RR+ )
31	6, 28, 30	seq3-1 10571	. . 3 |- ( ph -> ( seq N ( + , F ) ` N ) = ( F ` N ) )
32		uzid 9632	. . . . . 6 |- ( N e. ZZ -> N e. ( ZZ>= ` N ) )
33	6, 32	syl 14	. . . . 5 |- ( ph -> N e. ( ZZ>= ` N ) )
34	33, 1	eleqtrrdi 2290	. . . 4 |- ( ph -> N e. W )
35	15	recnd 8072	. . . . 5 |- ( ( ph /\ k e. W ) -> A e. CC )
36	1, 6, 12, 35, 20	isumclim2 11604	. . . 4 |- ( ph -> seq N ( + , F ) ~~> sum_ k e. W A )
37	9	sseld 3183	. . . . . . 7 |- ( ph -> ( m e. W -> m e. Z ) )
38		fveq2 5561	. . . . . . . . 9 |- ( k = m -> ( F ` k ) = ( F ` m ) )
39	38	eleq1d 2265	. . . . . . . 8 |- ( k = m -> ( ( F ` k ) e. RR+ <-> ( F ` m ) e. RR+ ) )
40	39	rspcv 2864	. . . . . . 7 |- ( m e. Z -> ( A. k e. Z ( F ` k ) e. RR+ -> ( F ` m ) e. RR+ ) )
41	37, 24, 40	syl6ci 1456	. . . . . 6 |- ( ph -> ( m e. W -> ( F ` m ) e. RR+ ) )
42	41	imp 124	. . . . 5 |- ( ( ph /\ m e. W ) -> ( F ` m ) e. RR+ )
43	42	rpred 9788	. . . 4 |- ( ( ph /\ m e. W ) -> ( F ` m ) e. RR )
44	42	rpge0d 9792	. . . 4 |- ( ( ph /\ m e. W ) -> 0 <_ ( F ` m ) )
45	1, 34, 36, 43, 44	climserle 11527	. . 3 |- ( ph -> ( seq N ( + , F ) ` N ) <_ sum_ k e. W A )
46	31, 45	eqbrtrrd 4058	. 2 |- ( ph -> ( F ` N ) <_ sum_ k e. W A )
47	21, 25, 46	rpgecld 9828	1 |- ( ph -> sum_ k e. W A e. RR+ )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   A.wral 2475   C_ wss 3157   dom cdm 4664   ` cfv 5259  (class class class)co 5925   RRcr 7895   + caddc 7899   <_ cle 8079   ZZcz 9343   ZZ>=cuz 9618   RR+crp 9745   seqcseq 10556   ~~> cli 11460   sum_csu 11535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-frec 6458  df-1o 6483  df-oadd 6487  df-er 6601  df-en 6809  df-dom 6810  df-fin 6811  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-n0 9267  df-z 9344  df-uz 9619  df-q 9711  df-rp 9746  df-fz 10101  df-fzo 10235  df-seqfrec 10557  df-exp 10648  df-ihash 10885  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181  df-clim 11461  df-sumdc 11536

This theorem is referenced by:  effsumlt  11874  eirraplem  11959