Theorem iserabs 11237

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 9056	. . . . . . 7 |- ZZ e. _V
5		uzssz 9338	. . . . . . 7 |- ( ZZ>= ` M ) C_ ZZ
6	4, 5	ssexi 4061	. . . . . 6 |- ( ZZ>= ` M ) e. _V
7	1, 6	eqeltri 2210	. . . . 5 |- Z e. _V
8	7	mptex 5639	. . . 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 10240	. . . 4 |- ( ph -> seq M ( + , F ) : Z --> CC )
12	11	ffvelrnda 5548	. . 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 10946	. . . 4 |- ( ( ph /\ n e. Z ) -> ( abs ` ( seq M ( + , F ) ` n ) ) e. RR )
15		2fveq3 5419	. . . . 5 |- ( m = n -> ( abs ` ( seq M ( + , F ) ` m ) ) = ( abs ` ( seq M ( + , F ) ` n ) ) )
16		eqid 2137	. . . . 5 |- ( m e. Z |-> ( abs ` ( seq M ( + , F ) ` m ) ) ) = ( m e. Z |-> ( abs ` ( seq M ( + , F ) ` m ) ) )
17	15, 16	fvmptg 5490	. . . 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 408	. . 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 11082	. 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 2214	. 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 10946	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( abs ` ( F ` k ) ) e. RR )
24	22, 23	eqeltrd 2214	. . . 4 |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. RR )
25	1, 2, 24	serfre 10241	. . 3 |- ( ph -> seq M ( + , G ) : Z --> RR )
26	25	ffvelrnda 5548	. 2 |- ( ( ph /\ n e. Z ) -> ( seq M ( + , G ) ` n ) e. RR )
27	2	adantr 274	. . . . . 6 |- ( ( ph /\ n e. Z ) -> M e. ZZ )
28		eluzelz 9328	. . . . . . . 8 |- ( n e. ( ZZ>= ` M ) -> n e. ZZ )
29	28, 1	eleq2s 2232	. . . . . . 7 |- ( n e. Z -> n e. ZZ )
30	29	adantl 275	. . . . . 6 |- ( ( ph /\ n e. Z ) -> n e. ZZ )
31	27, 30	fzfigd 10197	. . . . 5 |- ( ( ph /\ n e. Z ) -> ( M ... n ) e. Fin )
32		elfzuz 9795	. . . . . . . 8 |- ( k e. ( M ... n ) -> k e. ( ZZ>= ` M ) )
33	32, 1	eleqtrrdi 2231	. . . . . . 7 |- ( k e. ( M ... n ) -> k e. Z )
34	33, 10	sylan2 284	. . . . . 6 |- ( ( ph /\ k e. ( M ... n ) ) -> ( F ` k ) e. CC )
35	34	adantlr 468	. . . . 5 |- ( ( ( ph /\ n e. Z ) /\ k e. ( M ... n ) ) -> ( F ` k ) e. CC )
36	31, 35	fsumabs 11227	. . . 4 |- ( ( ph /\ n e. Z ) -> ( abs ` sum_ k e. ( M ... n ) ( F ` k ) ) <_ sum_ k e. ( M ... n ) ( abs ` ( F ` k ) ) )
37		eqidd 2138	. . . . . 6 |- ( ( ( ph /\ n e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = ( F ` k ) )
38	1	eleq2i 2204	. . . . . . . 8 |- ( n e. Z <-> n e. ( ZZ>= ` M ) )
39	38	biimpi 119	. . . . . . 7 |- ( n e. Z -> n e. ( ZZ>= ` M ) )
40	39	adantl 275	. . . . . 6 |- ( ( ph /\ n e. Z ) -> n e. ( ZZ>= ` M ) )
41	1	eleq2i 2204	. . . . . . . 8 |- ( k e. Z <-> k e. ( ZZ>= ` M ) )
42	41, 10	sylan2br 286	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
43	42	adantlr 468	. . . . . 6 |- ( ( ( ph /\ n e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
44	37, 40, 43	fsum3ser 11159	. . . . 5 |- ( ( ph /\ n e. Z ) -> sum_ k e. ( M ... n ) ( F ` k ) = ( seq M ( + , F ) ` n ) )
45	44	fveq2d 5418	. . . 4 |- ( ( ph /\ n e. Z ) -> ( abs ` sum_ k e. ( M ... n ) ( F ` k ) ) = ( abs ` ( seq M ( + , F ) ` n ) ) )
46	22	adantlr 468	. . . . . 6 |- ( ( ( ph /\ n e. Z ) /\ k e. Z ) -> ( G ` k ) = ( abs ` ( F ` k ) ) )
47	41, 46	sylan2br 286	. . . . 5 |- ( ( ( ph /\ n e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) = ( abs ` ( F ` k ) ) )
48	23	adantlr 468	. . . . . . 7 |- ( ( ( ph /\ n e. Z ) /\ k e. Z ) -> ( abs ` ( F ` k ) ) e. RR )
49	41, 48	sylan2br 286	. . . . . 6 |- ( ( ( ph /\ n e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( abs ` ( F ` k ) ) e. RR )
50	49	recnd 7787	. . . . 5 |- ( ( ( ph /\ n e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( abs ` ( F ` k ) ) e. CC )
51	47, 40, 50	fsum3ser 11159	. . . 4 |- ( ( ph /\ n e. Z ) -> sum_ k e. ( M ... n ) ( abs ` ( F ` k ) ) = ( seq M ( + , G ) ` n ) )
52	36, 45, 51	3brtr3d 3954	. . 3 |- ( ( ph /\ n e. Z ) -> ( abs ` ( seq M ( + , F ) ` n ) ) <_ ( seq M ( + , G ) ` n ) )
53	18, 52	eqbrtrd 3945	. 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 11096	1 |- ( ph -> ( abs ` A ) <_ B )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   _Vcvv 2681   class class class wbr 3924   |-> cmpt 3984   ` cfv 5118  (class class class)co 5767   CCcc 7611   RRcr 7612   + caddc 7616   <_ cle 7794   ZZcz 9047   ZZ>=cuz 9319   ...cfz 9783   seqcseq 10211   abscabs 10762   ~~> cli 11040   sum_csu 11115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731  ax-arch 7732  ax-caucvg 7733

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-isom 5127  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-frec 6281  df-1o 6306  df-oadd 6310  df-er 6422  df-en 6628  df-dom 6629  df-fin 6630  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-2 8772  df-3 8773  df-4 8774  df-n0 8971  df-z 9048  df-uz 9320  df-q 9405  df-rp 9435  df-fz 9784  df-fzo 9913  df-seqfrec 10212  df-exp 10286  df-ihash 10515  df-cj 10607  df-re 10608  df-im 10609  df-rsqrt 10763  df-abs 10764  df-clim 11041  df-sumdc 11116

This theorem is referenced by:  eftlub  11385