Theorem isumrpcl 11890

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 2299	. . . 4 |- ( ph -> N e. ( ZZ>= ` M ) )
5		eluzelz 9687	. . . 4 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
6	4, 5	syl 14	. . 3 |- ( ph -> N e. ZZ )
7		uzss 9699	. . . . . . 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 3240	. . . . 5 |- ( ph -> W C_ Z )
10	9	sselda 3197	. . . 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 9848	. . . 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 2283	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR+ )
18	17	rpcnd 9850	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
19	3, 2, 18	iserex 11735	. . . 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 11825	. 2 |- ( ph -> sum_ k e. W A e. RR )
22		fveq2 5594	. . . 4 |- ( k = N -> ( F ` k ) = ( F ` N ) )
23	22	eleq1d 2275	. . 3 |- ( k = N -> ( ( F ` k ) e. RR+ <-> ( F ` N ) e. RR+ ) )
24	17	ralrimiva 2580	. . 3 |- ( ph -> A. k e. Z ( F ` k ) e. RR+ )
25	23, 24, 2	rspcdva 2886	. 2 |- ( ph -> ( F ` N ) e. RR+ )
26	8	sselda 3197	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` N ) ) -> k e. ( ZZ>= ` M ) )
27	26, 3	eleqtrrdi 2300	. . . . 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 9829	. . . . 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 10639	. . 3 |- ( ph -> ( seq N ( + , F ) ` N ) = ( F ` N ) )
32		uzid 9692	. . . . . 6 |- ( N e. ZZ -> N e. ( ZZ>= ` N ) )
33	6, 32	syl 14	. . . . 5 |- ( ph -> N e. ( ZZ>= ` N ) )
34	33, 1	eleqtrrdi 2300	. . . 4 |- ( ph -> N e. W )
35	15	recnd 8131	. . . . 5 |- ( ( ph /\ k e. W ) -> A e. CC )
36	1, 6, 12, 35, 20	isumclim2 11818	. . . 4 |- ( ph -> seq N ( + , F ) ~~> sum_ k e. W A )
37	9	sseld 3196	. . . . . . 7 |- ( ph -> ( m e. W -> m e. Z ) )
38		fveq2 5594	. . . . . . . . 9 |- ( k = m -> ( F ` k ) = ( F ` m ) )
39	38	eleq1d 2275	. . . . . . . 8 |- ( k = m -> ( ( F ` k ) e. RR+ <-> ( F ` m ) e. RR+ ) )
40	39	rspcv 2877	. . . . . . 7 |- ( m e. Z -> ( A. k e. Z ( F ` k ) e. RR+ -> ( F ` m ) e. RR+ ) )
41	37, 24, 40	syl6ci 1466	. . . . . 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 9848	. . . 4 |- ( ( ph /\ m e. W ) -> ( F ` m ) e. RR )
44	42	rpge0d 9852	. . . 4 |- ( ( ph /\ m e. W ) -> 0 <_ ( F ` m ) )
45	1, 34, 36, 43, 44	climserle 11741	. . 3 |- ( ph -> ( seq N ( + , F ) ` N ) <_ sum_ k e. W A )
46	31, 45	eqbrtrrd 4078	. 2 |- ( ph -> ( F ` N ) <_ sum_ k e. W A )
47	21, 25, 46	rpgecld 9888	1 |- ( ph -> sum_ k e. W A e. RR+ )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2177   A.wral 2485   C_ wss 3170   dom cdm 4688   ` cfv 5285  (class class class)co 5962   RRcr 7954   + caddc 7958   <_ cle 8138   ZZcz 9402   ZZ>=cuz 9678   RR+crp 9805   seqcseq 10624   ~~> cli 11674   sum_csu 11749

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4170  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-iinf 4649  ax-cnex 8046  ax-resscn 8047  ax-1cn 8048  ax-1re 8049  ax-icn 8050  ax-addcl 8051  ax-addrcl 8052  ax-mulcl 8053  ax-mulrcl 8054  ax-addcom 8055  ax-mulcom 8056  ax-addass 8057  ax-mulass 8058  ax-distr 8059  ax-i2m1 8060  ax-0lt1 8061  ax-1rid 8062  ax-0id 8063  ax-rnegex 8064  ax-precex 8065  ax-cnre 8066  ax-pre-ltirr 8067  ax-pre-ltwlin 8068  ax-pre-lttrn 8069  ax-pre-apti 8070  ax-pre-ltadd 8071  ax-pre-mulgt0 8072  ax-pre-mulext 8073  ax-arch 8074  ax-caucvg 8075

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-tr 4154  df-id 4353  df-po 4356  df-iso 4357  df-iord 4426  df-on 4428  df-ilim 4429  df-suc 4431  df-iom 4652  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-isom 5294  df-riota 5917  df-ov 5965  df-oprab 5966  df-mpo 5967  df-1st 6244  df-2nd 6245  df-recs 6409  df-irdg 6474  df-frec 6495  df-1o 6520  df-oadd 6524  df-er 6638  df-en 6846  df-dom 6847  df-fin 6848  df-pnf 8139  df-mnf 8140  df-xr 8141  df-ltxr 8142  df-le 8143  df-sub 8275  df-neg 8276  df-reap 8678  df-ap 8685  df-div 8776  df-inn 9067  df-2 9125  df-3 9126  df-4 9127  df-n0 9326  df-z 9403  df-uz 9679  df-q 9771  df-rp 9806  df-fz 10161  df-fzo 10295  df-seqfrec 10625  df-exp 10716  df-ihash 10953  df-cj 11238  df-re 11239  df-im 11240  df-rsqrt 11394  df-abs 11395  df-clim 11675  df-sumdc 11750

This theorem is referenced by:  effsumlt  12088  eirraplem  12173