Theorem isumrpcl 11776

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 2297	. . . 4 |- ( ph -> N e. ( ZZ>= ` M ) )
5		eluzelz 9656	. . . 4 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
6	4, 5	syl 14	. . 3 |- ( ph -> N e. ZZ )
7		uzss 9668	. . . . . . 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 3235	. . . . 5 |- ( ph -> W C_ Z )
10	9	sselda 3192	. . . 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 9817	. . . 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 2281	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR+ )
18	17	rpcnd 9819	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
19	3, 2, 18	iserex 11621	. . . 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 11711	. 2 |- ( ph -> sum_ k e. W A e. RR )
22		fveq2 5575	. . . 4 |- ( k = N -> ( F ` k ) = ( F ` N ) )
23	22	eleq1d 2273	. . 3 |- ( k = N -> ( ( F ` k ) e. RR+ <-> ( F ` N ) e. RR+ ) )
24	17	ralrimiva 2578	. . 3 |- ( ph -> A. k e. Z ( F ` k ) e. RR+ )
25	23, 24, 2	rspcdva 2881	. 2 |- ( ph -> ( F ` N ) e. RR+ )
26	8	sselda 3192	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` N ) ) -> k e. ( ZZ>= ` M ) )
27	26, 3	eleqtrrdi 2298	. . . . 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 9798	. . . . 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 10605	. . 3 |- ( ph -> ( seq N ( + , F ) ` N ) = ( F ` N ) )
32		uzid 9661	. . . . . 6 |- ( N e. ZZ -> N e. ( ZZ>= ` N ) )
33	6, 32	syl 14	. . . . 5 |- ( ph -> N e. ( ZZ>= ` N ) )
34	33, 1	eleqtrrdi 2298	. . . 4 |- ( ph -> N e. W )
35	15	recnd 8100	. . . . 5 |- ( ( ph /\ k e. W ) -> A e. CC )
36	1, 6, 12, 35, 20	isumclim2 11704	. . . 4 |- ( ph -> seq N ( + , F ) ~~> sum_ k e. W A )
37	9	sseld 3191	. . . . . . 7 |- ( ph -> ( m e. W -> m e. Z ) )
38		fveq2 5575	. . . . . . . . 9 |- ( k = m -> ( F ` k ) = ( F ` m ) )
39	38	eleq1d 2273	. . . . . . . 8 |- ( k = m -> ( ( F ` k ) e. RR+ <-> ( F ` m ) e. RR+ ) )
40	39	rspcv 2872	. . . . . . 7 |- ( m e. Z -> ( A. k e. Z ( F ` k ) e. RR+ -> ( F ` m ) e. RR+ ) )
41	37, 24, 40	syl6ci 1464	. . . . . 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 9817	. . . 4 |- ( ( ph /\ m e. W ) -> ( F ` m ) e. RR )
44	42	rpge0d 9821	. . . 4 |- ( ( ph /\ m e. W ) -> 0 <_ ( F ` m ) )
45	1, 34, 36, 43, 44	climserle 11627	. . 3 |- ( ph -> ( seq N ( + , F ) ` N ) <_ sum_ k e. W A )
46	31, 45	eqbrtrrd 4067	. 2 |- ( ph -> ( F ` N ) <_ sum_ k e. W A )
47	21, 25, 46	rpgecld 9857	1 |- ( ph -> sum_ k e. W A e. RR+ )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   e. wcel 2175   A.wral 2483   C_ wss 3165   dom cdm 4674   ` cfv 5270  (class class class)co 5943   RRcr 7923   + caddc 7927   <_ cle 8107   ZZcz 9371   ZZ>=cuz 9647   RR+crp 9774   seqcseq 10590   ~~> cli 11560   sum_csu 11635

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-mulrcl 8023  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-mulass 8027  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-1rid 8031  ax-0id 8032  ax-rnegex 8033  ax-precex 8034  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040  ax-pre-mulgt0 8041  ax-pre-mulext 8042  ax-arch 8043  ax-caucvg 8044

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-ilim 4415  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-isom 5279  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-irdg 6455  df-frec 6476  df-1o 6501  df-oadd 6505  df-er 6619  df-en 6827  df-dom 6828  df-fin 6829  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-reap 8647  df-ap 8654  df-div 8745  df-inn 9036  df-2 9094  df-3 9095  df-4 9096  df-n0 9295  df-z 9372  df-uz 9648  df-q 9740  df-rp 9775  df-fz 10130  df-fzo 10264  df-seqfrec 10591  df-exp 10682  df-ihash 10919  df-cj 11124  df-re 11125  df-im 11126  df-rsqrt 11280  df-abs 11281  df-clim 11561  df-sumdc 11636

This theorem is referenced by:  effsumlt  11974  eirraplem  12059