Theorem isumrpcl 10884

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	syl6eleq 2180	. . . 4 |- ( ph -> N e. ( ZZ>= ` M ) )
5		eluzelz 9026	. . . 4 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
6	4, 5	syl 14	. . 3 |- ( ph -> N e. ZZ )
7		uzss 9037	. . . . . . 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 3067	. . . . 5 |- ( ph -> W C_ Z )
10	9	sselda 3025	. . . 4 |- ( ( ph /\ k e. W ) -> k e. Z )
11		isumrpcl.4	. . . 4 |- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )
12	10, 11	syldan 276	. . 3 |- ( ( ph /\ k e. W ) -> ( F ` k ) = A )
13		isumrpcl.5	. . . . 5 |- ( ( ph /\ k e. Z ) -> A e. RR+ )
14	13	rpred 9171	. . . 4 |- ( ( ph /\ k e. Z ) -> A e. RR )
15	10, 14	syldan 276	. . 3 |- ( ( ph /\ k e. W ) -> A e. RR )
16		isumrpcl.6	. . . 4 |- ( ph -> seq M ( + , F ) e. dom ~~> )
17	11, 13	eqeltrd 2164	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR+ )
18	17	rpcnd 9173	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
19	3, 2, 18	iserex 10723	. . . 4 |- ( ph -> ( seq M ( + , F ) e. dom ~~> <-> seq N ( + , F ) e. dom ~~> ) )
20	16, 19	mpbid 145	. . 3 |- ( ph -> seq N ( + , F ) e. dom ~~> )
21	1, 6, 12, 15, 20	isumrecl 10819	. 2 |- ( ph -> sum_ k e. W A e. RR )
22		fveq2 5305	. . . 4 |- ( k = N -> ( F ` k ) = ( F ` N ) )
23	22	eleq1d 2156	. . 3 |- ( k = N -> ( ( F ` k ) e. RR+ <-> ( F ` N ) e. RR+ ) )
24	17	ralrimiva 2446	. . 3 |- ( ph -> A. k e. Z ( F ` k ) e. RR+ )
25	23, 24, 2	rspcdva 2727	. 2 |- ( ph -> ( F ` N ) e. RR+ )
26	8	sselda 3025	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` N ) ) -> k e. ( ZZ>= ` M ) )
27	26, 3	syl6eleqr 2181	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` N ) ) -> k e. Z )
28	27, 17	syldan 276	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` N ) ) -> ( F ` k ) e. RR+ )
29		rpaddcl 9155	. . . . 5 |- ( ( k e. RR+ /\ y e. RR+ ) -> ( k + y ) e. RR+ )
30	29	adantl 271	. . . 4 |- ( ( ph /\ ( k e. RR+ /\ y e. RR+ ) ) -> ( k + y ) e. RR+ )
31	6, 28, 30	seq3-1 9873	. . 3 |- ( ph -> ( seq N ( + , F ) ` N ) = ( F ` N ) )
32		uzid 9031	. . . . . 6 |- ( N e. ZZ -> N e. ( ZZ>= ` N ) )
33	6, 32	syl 14	. . . . 5 |- ( ph -> N e. ( ZZ>= ` N ) )
34	33, 1	syl6eleqr 2181	. . . 4 |- ( ph -> N e. W )
35	15	recnd 7514	. . . . 5 |- ( ( ph /\ k e. W ) -> A e. CC )
36	1, 6, 12, 35, 20	isumclim2 10812	. . . 4 |- ( ph -> seq N ( + , F ) ~~> sum_ k e. W A )
37	9	sseld 3024	. . . . . . 7 |- ( ph -> ( m e. W -> m e. Z ) )
38		fveq2 5305	. . . . . . . . 9 |- ( k = m -> ( F ` k ) = ( F ` m ) )
39	38	eleq1d 2156	. . . . . . . 8 |- ( k = m -> ( ( F ` k ) e. RR+ <-> ( F ` m ) e. RR+ ) )
40	39	rspcv 2718	. . . . . . 7 |- ( m e. Z -> ( A. k e. Z ( F ` k ) e. RR+ -> ( F ` m ) e. RR+ ) )
41	37, 24, 40	syl6ci 1379	. . . . . 6 |- ( ph -> ( m e. W -> ( F ` m ) e. RR+ ) )
42	41	imp 122	. . . . 5 |- ( ( ph /\ m e. W ) -> ( F ` m ) e. RR+ )
43	42	rpred 9171	. . . 4 |- ( ( ph /\ m e. W ) -> ( F ` m ) e. RR )
44	42	rpge0d 9175	. . . 4 |- ( ( ph /\ m e. W ) -> 0 <_ ( F ` m ) )
45	1, 34, 36, 43, 44	climserle 10730	. . 3 |- ( ph -> ( seq N ( + , F ) ` N ) <_ sum_ k e. W A )
46	31, 45	eqbrtrrd 3867	. 2 |- ( ph -> ( F ` N ) <_ sum_ k e. W A )
47	21, 25, 46	rpgecld 9211	1 |- ( ph -> sum_ k e. W A e. RR+ )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1289   e. wcel 1438   A.wral 2359   C_ wss 2999   dom cdm 4438   ` cfv 5015  (class class class)co 5652   RRcr 7347   + caddc 7351   <_ cle 7521   ZZcz 8748   ZZ>=cuz 9017   RR+crp 9132   seqcseq 9848   ~~> cli 10662   sum_csu 10738

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7434  ax-resscn 7435  ax-1cn 7436  ax-1re 7437  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-mulrcl 7442  ax-addcom 7443  ax-mulcom 7444  ax-addass 7445  ax-mulass 7446  ax-distr 7447  ax-i2m1 7448  ax-0lt1 7449  ax-1rid 7450  ax-0id 7451  ax-rnegex 7452  ax-precex 7453  ax-cnre 7454  ax-pre-ltirr 7455  ax-pre-ltwlin 7456  ax-pre-lttrn 7457  ax-pre-apti 7458  ax-pre-ltadd 7459  ax-pre-mulgt0 7460  ax-pre-mulext 7461  ax-arch 7462  ax-caucvg 7463

This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-isom 5024  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-frec 6156  df-1o 6181  df-oadd 6185  df-er 6290  df-en 6456  df-dom 6457  df-fin 6458  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525  df-le 7526  df-sub 7653  df-neg 7654  df-reap 8050  df-ap 8057  df-div 8138  df-inn 8421  df-2 8479  df-3 8480  df-4 8481  df-n0 8672  df-z 8749  df-uz 9018  df-q 9103  df-rp 9133  df-fz 9423  df-fzo 9550  df-iseq 9849  df-seq3 9850  df-exp 9951  df-ihash 10180  df-cj 10272  df-re 10273  df-im 10274  df-rsqrt 10427  df-abs 10428  df-clim 10663  df-isum 10739

This theorem is referenced by:  effsumlt  10978  eirraplem  11060