Theorem iserabs 10930

Description: Generalized triangle inequality: the absolute value of an infinite sum is less than or equal to the sum of absolute values. (Contributed by Paul Chapman, 10-Sep-2007.) (Revised by Jim Kingdon, 14-Dec-2022.)

Hypotheses

Ref	Expression
iserabs.1	|- Z = ( ZZ>= ` M )
iserabs.2	|- ( ph -> seq M ( + , F ) ~~> A )
iserabs.3	|- ( ph -> seq M ( + , G ) ~~> B )
iserabs.5	|- ( ph -> M e. ZZ )
iserabs.6	|- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
iserabs.7	|- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( abs ` ( F ` k ) ) )

Assertion

Ref	Expression
iserabs	|- ( ph -> ( abs ` A ) <_ B )

Distinct variable groups:   k, F   k, G   k, M   ph, k   k, Z

Allowed substitution hints:   A( k)   B( k)

Proof of Theorem iserabs

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

Step	Hyp	Ref	Expression
1		iserabs.1	. 2 |- Z = ( ZZ>= ` M )
2		iserabs.5	. 2 |- ( ph -> M e. ZZ )
3		iserabs.2	. . 3 |- ( ph -> seq M ( + , F ) ~~> A )
4		zex 8820	. . . . . . 7 |- ZZ e. _V
5		uzssz 9099	. . . . . . 7 |- ( ZZ>= ` M ) C_ ZZ
6	4, 5	ssexi 3983	. . . . . 6 |- ( ZZ>= ` M ) e. _V
7	1, 6	eqeltri 2161	. . . . 5 |- Z e. _V
8	7	mptex 5537	. . . 4 |- ( m e. Z |-> ( abs ` ( seq M ( + , F ) ` m ) ) ) e. _V
9	8	a1i 9	. . 3 |- ( ph -> ( m e. Z |-> ( abs ` ( seq M ( + , F ) ` m ) ) ) e. _V )
10		iserabs.6	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
11	1, 2, 10	serf 9961	. . . 4 |- ( ph -> seq M ( + , F ) : Z --> CC )
12	11	ffvelrnda 5448	. . 3 |- ( ( ph /\ n e. Z ) -> ( seq M ( + , F ) ` n ) e. CC )
13		simpr 109	. . . 4 |- ( ( ph /\ n e. Z ) -> n e. Z )
14	12	abscld 10675	. . . 4 |- ( ( ph /\ n e. Z ) -> ( abs ` ( seq M ( + , F ) ` n ) ) e. RR )
15		2fveq3 5323	. . . . 5 |- ( m = n -> ( abs ` ( seq M ( + , F ) ` m ) ) = ( abs ` ( seq M ( + , F ) ` n ) ) )
16		eqid 2089	. . . . 5 |- ( m e. Z |-> ( abs ` ( seq M ( + , F ) ` m ) ) ) = ( m e. Z |-> ( abs ` ( seq M ( + , F ) ` m ) ) )
17	15, 16	fvmptg 5393	. . . 4 |- ( ( n e. Z /\ ( abs ` ( seq M ( + , F ) ` n ) ) e. RR ) -> ( ( m e. Z |-> ( abs ` ( seq M ( + , F ) ` m ) ) ) ` n ) = ( abs ` ( seq M ( + , F ) ` n ) ) )
18	13, 14, 17	syl2anc 404	. . 3 |- ( ( ph /\ n e. Z ) -> ( ( m e. Z |-> ( abs ` ( seq M ( + , F ) ` m ) ) ) ` n ) = ( abs ` ( seq M ( + , F ) ` n ) ) )
19	1, 3, 9, 2, 12, 18	climabs 10769	. 2 |- ( ph -> ( m e. Z |-> ( abs ` ( seq M ( + , F ) ` m ) ) ) ~~> ( abs ` A ) )
20		iserabs.3	. 2 |- ( ph -> seq M ( + , G ) ~~> B )
21	18, 14	eqeltrd 2165	. 2 |- ( ( ph /\ n e. Z ) -> ( ( m e. Z |-> ( abs ` ( seq M ( + , F ) ` m ) ) ) ` n ) e. RR )
22		iserabs.7	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( abs ` ( F ` k ) ) )
23	10	abscld 10675	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( abs ` ( F ` k ) ) e. RR )
24	22, 23	eqeltrd 2165	. . . 4 |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. RR )
25	1, 2, 24	serfre 9962	. . 3 |- ( ph -> seq M ( + , G ) : Z --> RR )
26	25	ffvelrnda 5448	. 2 |- ( ( ph /\ n e. Z ) -> ( seq M ( + , G ) ` n ) e. RR )
27	2	adantr 271	. . . . . 6 |- ( ( ph /\ n e. Z ) -> M e. ZZ )
28		eluzelz 9089	. . . . . . . 8 |- ( n e. ( ZZ>= ` M ) -> n e. ZZ )
29	28, 1	eleq2s 2183	. . . . . . 7 |- ( n e. Z -> n e. ZZ )
30	29	adantl 272	. . . . . 6 |- ( ( ph /\ n e. Z ) -> n e. ZZ )
31	27, 30	fzfigd 9899	. . . . 5 |- ( ( ph /\ n e. Z ) -> ( M ... n ) e. Fin )
32		elfzuz 9497	. . . . . . . 8 |- ( k e. ( M ... n ) -> k e. ( ZZ>= ` M ) )
33	32, 1	syl6eleqr 2182	. . . . . . 7 |- ( k e. ( M ... n ) -> k e. Z )
34	33, 10	sylan2 281	. . . . . 6 |- ( ( ph /\ k e. ( M ... n ) ) -> ( F ` k ) e. CC )
35	34	adantlr 462	. . . . 5 |- ( ( ( ph /\ n e. Z ) /\ k e. ( M ... n ) ) -> ( F ` k ) e. CC )
36	31, 35	fsumabs 10920	. . . 4 |- ( ( ph /\ n e. Z ) -> ( abs ` sum_ k e. ( M ... n ) ( F ` k ) ) <_ sum_ k e. ( M ... n ) ( abs ` ( F ` k ) ) )
37		eqidd 2090	. . . . . 6 |- ( ( ( ph /\ n e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = ( F ` k ) )
38	1	eleq2i 2155	. . . . . . . 8 |- ( n e. Z <-> n e. ( ZZ>= ` M ) )
39	38	biimpi 119	. . . . . . 7 |- ( n e. Z -> n e. ( ZZ>= ` M ) )
40	39	adantl 272	. . . . . 6 |- ( ( ph /\ n e. Z ) -> n e. ( ZZ>= ` M ) )
41	1	eleq2i 2155	. . . . . . . 8 |- ( k e. Z <-> k e. ( ZZ>= ` M ) )
42	41, 10	sylan2br 283	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
43	42	adantlr 462	. . . . . 6 |- ( ( ( ph /\ n e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
44	37, 40, 43	fsum3ser 10852	. . . . 5 |- ( ( ph /\ n e. Z ) -> sum_ k e. ( M ... n ) ( F ` k ) = ( seq M ( + , F ) ` n ) )
45	44	fveq2d 5322	. . . 4 |- ( ( ph /\ n e. Z ) -> ( abs ` sum_ k e. ( M ... n ) ( F ` k ) ) = ( abs ` ( seq M ( + , F ) ` n ) ) )
46	22	adantlr 462	. . . . . 6 |- ( ( ( ph /\ n e. Z ) /\ k e. Z ) -> ( G ` k ) = ( abs ` ( F ` k ) ) )
47	41, 46	sylan2br 283	. . . . 5 |- ( ( ( ph /\ n e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) = ( abs ` ( F ` k ) ) )
48	23	adantlr 462	. . . . . . 7 |- ( ( ( ph /\ n e. Z ) /\ k e. Z ) -> ( abs ` ( F ` k ) ) e. RR )
49	41, 48	sylan2br 283	. . . . . 6 |- ( ( ( ph /\ n e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( abs ` ( F ` k ) ) e. RR )
50	49	recnd 7577	. . . . 5 |- ( ( ( ph /\ n e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( abs ` ( F ` k ) ) e. CC )
51	47, 40, 50	fsum3ser 10852	. . . 4 |- ( ( ph /\ n e. Z ) -> sum_ k e. ( M ... n ) ( abs ` ( F ` k ) ) = ( seq M ( + , G ) ` n ) )
52	36, 45, 51	3brtr3d 3880	. . 3 |- ( ( ph /\ n e. Z ) -> ( abs ` ( seq M ( + , F ) ` n ) ) <_ ( seq M ( + , G ) ` n ) )
53	18, 52	eqbrtrd 3871	. 2 |- ( ( ph /\ n e. Z ) -> ( ( m e. Z |-> ( abs ` ( seq M ( + , F ) ` m ) ) ) ` n ) <_ ( seq M ( + , G ) ` n ) )
54	1, 2, 19, 20, 21, 26, 53	climle 10783	1 |- ( ph -> ( abs ` A ) <_ B )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1290   e. wcel 1439   _Vcvv 2620   class class class wbr 3851   |-> cmpt 3905   ` cfv 5028  (class class class)co 5666   CCcc 7409   RRcr 7410   + caddc 7414   <_ cle 7584   ZZcz 8811   ZZ>=cuz 9080   ...cfz 9485   seqcseq 9913   abscabs 10491   ~~> cli 10727   sum_csu 10803

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416  ax-cnex 7497  ax-resscn 7498  ax-1cn 7499  ax-1re 7500  ax-icn 7501  ax-addcl 7502  ax-addrcl 7503  ax-mulcl 7504  ax-mulrcl 7505  ax-addcom 7506  ax-mulcom 7507  ax-addass 7508  ax-mulass 7509  ax-distr 7510  ax-i2m1 7511  ax-0lt1 7512  ax-1rid 7513  ax-0id 7514  ax-rnegex 7515  ax-precex 7516  ax-cnre 7517  ax-pre-ltirr 7518  ax-pre-ltwlin 7519  ax-pre-lttrn 7520  ax-pre-apti 7521  ax-pre-ltadd 7522  ax-pre-mulgt0 7523  ax-pre-mulext 7524  ax-arch 7525  ax-caucvg 7526

This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-if 3398  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-ilim 4205  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-isom 5037  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-irdg 6149  df-frec 6170  df-1o 6195  df-oadd 6199  df-er 6306  df-en 6512  df-dom 6513  df-fin 6514  df-pnf 7585  df-mnf 7586  df-xr 7587  df-ltxr 7588  df-le 7589  df-sub 7716  df-neg 7717  df-reap 8113  df-ap 8120  df-div 8201  df-inn 8484  df-2 8542  df-3 8543  df-4 8544  df-n0 8735  df-z 8812  df-uz 9081  df-q 9166  df-rp 9196  df-fz 9486  df-fzo 9615  df-iseq 9914  df-seq3 9915  df-exp 10016  df-ihash 10245  df-cj 10337  df-re 10338  df-im 10339  df-rsqrt 10492  df-abs 10493  df-clim 10728  df-isum 10804

This theorem is referenced by:  eftlub  11041