Theorem isumrpcl 12180

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 2325	. . . 4 |- ( ph -> N e. ( ZZ>= ` M ) )
5		eluzelz 9863	. . . 4 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
6	4, 5	syl 14	. . 3 |- ( ph -> N e. ZZ )
7		uzss 9875	. . . . . . 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 3281	. . . . 5 |- ( ph -> W C_ Z )
10	9	sselda 3238	. . . 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 10029	. . . 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 2309	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR+ )
18	17	rpcnd 10031	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
19	3, 2, 18	iserex 12024	. . . 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 12115	. 2 |- ( ph -> sum_ k e. W A e. RR )
22		fveq2 5670	. . . 4 |- ( k = N -> ( F ` k ) = ( F ` N ) )
23	22	eleq1d 2301	. . 3 |- ( k = N -> ( ( F ` k ) e. RR+ <-> ( F ` N ) e. RR+ ) )
24	17	ralrimiva 2615	. . 3 |- ( ph -> A. k e. Z ( F ` k ) e. RR+ )
25	23, 24, 2	rspcdva 2926	. 2 |- ( ph -> ( F ` N ) e. RR+ )
26	8	sselda 3238	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` N ) ) -> k e. ( ZZ>= ` M ) )
27	26, 3	eleqtrrdi 2326	. . . . 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 10010	. . . . 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 10824	. . 3 |- ( ph -> ( seq N ( + , F ) ` N ) = ( F ` N ) )
32		uzid 9868	. . . . . 6 |- ( N e. ZZ -> N e. ( ZZ>= ` N ) )
33	6, 32	syl 14	. . . . 5 |- ( ph -> N e. ( ZZ>= ` N ) )
34	33, 1	eleqtrrdi 2326	. . . 4 |- ( ph -> N e. W )
35	15	recnd 8302	. . . . 5 |- ( ( ph /\ k e. W ) -> A e. CC )
36	1, 6, 12, 35, 20	isumclim2 12108	. . . 4 |- ( ph -> seq N ( + , F ) ~~> sum_ k e. W A )
37	9	sseld 3237	. . . . . . 7 |- ( ph -> ( m e. W -> m e. Z ) )
38		fveq2 5670	. . . . . . . . 9 |- ( k = m -> ( F ` k ) = ( F ` m ) )
39	38	eleq1d 2301	. . . . . . . 8 |- ( k = m -> ( ( F ` k ) e. RR+ <-> ( F ` m ) e. RR+ ) )
40	39	rspcv 2917	. . . . . . 7 |- ( m e. Z -> ( A. k e. Z ( F ` k ) e. RR+ -> ( F ` m ) e. RR+ ) )
41	37, 24, 40	syl6ci 1491	. . . . . 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 10029	. . . 4 |- ( ( ph /\ m e. W ) -> ( F ` m ) e. RR )
44	42	rpge0d 10033	. . . 4 |- ( ( ph /\ m e. W ) -> 0 <_ ( F ` m ) )
45	1, 34, 36, 43, 44	climserle 12030	. . 3 |- ( ph -> ( seq N ( + , F ) ` N ) <_ sum_ k e. W A )
46	31, 45	eqbrtrrd 4133	. 2 |- ( ph -> ( F ` N ) <_ sum_ k e. W A )
47	21, 25, 46	rpgecld 10069	1 |- ( ph -> sum_ k e. W A e. RR+ )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   A.wral 2520   C_ wss 3211   dom cdm 4749   ` cfv 5352  (class class class)co 6050   RRcr 8126   + caddc 8130   <_ cle 8309   ZZcz 9577   ZZ>=cuz 9853   RR+crp 9986   seqcseq 10809   ~~> cli 11963   sum_csu 12038

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-isom 5361  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-frec 6622  df-1o 6647  df-oadd 6651  df-er 6767  df-en 6976  df-dom 6977  df-fin 6978  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-n0 9497  df-z 9578  df-uz 9854  df-q 9952  df-rp 9987  df-fz 10343  df-fzo 10477  df-seqfrec 10810  df-exp 10901  df-ihash 11139  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-clim 11964  df-sumdc 12039

This theorem is referenced by:  effsumlt  12378  eirraplem  12463