Theorem iserabs 12029

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 9481	. . . . . . 7 |- ZZ e. _V
5		uzssz 9769	. . . . . . 7 |- ( ZZ>= ` M ) C_ ZZ
6	4, 5	ssexi 4225	. . . . . 6 |- ( ZZ>= ` M ) e. _V
7	1, 6	eqeltri 2302	. . . . 5 |- Z e. _V
8	7	mptex 5875	. . . 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 10738	. . . 4 |- ( ph -> seq M ( + , F ) : Z --> CC )
12	11	ffvelcdmda 5778	. . 3 |- ( ( ph /\ n e. Z ) -> ( seq M ( + , F ) ` n ) e. CC )
13		simpr 110	. . . 4 |- ( ( ph /\ n e. Z ) -> n e. Z )
14	12	abscld 11735	. . . 4 |- ( ( ph /\ n e. Z ) -> ( abs ` ( seq M ( + , F ) ` n ) ) e. RR )
15		2fveq3 5640	. . . . 5 |- ( m = n -> ( abs ` ( seq M ( + , F ) ` m ) ) = ( abs ` ( seq M ( + , F ) ` n ) ) )
16		eqid 2229	. . . . 5 |- ( m e. Z |-> ( abs ` ( seq M ( + , F ) ` m ) ) ) = ( m e. Z |-> ( abs ` ( seq M ( + , F ) ` m ) ) )
17	15, 16	fvmptg 5718	. . . 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 411	. . 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 11874	. 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 2306	. 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 11735	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( abs ` ( F ` k ) ) e. RR )
24	22, 23	eqeltrd 2306	. . . 4 |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. RR )
25	1, 2, 24	serfre 10739	. . 3 |- ( ph -> seq M ( + , G ) : Z --> RR )
26	25	ffvelcdmda 5778	. 2 |- ( ( ph /\ n e. Z ) -> ( seq M ( + , G ) ` n ) e. RR )
27	2	adantr 276	. . . . . 6 |- ( ( ph /\ n e. Z ) -> M e. ZZ )
28		eluzelz 9758	. . . . . . . 8 |- ( n e. ( ZZ>= ` M ) -> n e. ZZ )
29	28, 1	eleq2s 2324	. . . . . . 7 |- ( n e. Z -> n e. ZZ )
30	29	adantl 277	. . . . . 6 |- ( ( ph /\ n e. Z ) -> n e. ZZ )
31	27, 30	fzfigd 10686	. . . . 5 |- ( ( ph /\ n e. Z ) -> ( M ... n ) e. Fin )
32		elfzuz 10249	. . . . . . . 8 |- ( k e. ( M ... n ) -> k e. ( ZZ>= ` M ) )
33	32, 1	eleqtrrdi 2323	. . . . . . 7 |- ( k e. ( M ... n ) -> k e. Z )
34	33, 10	sylan2 286	. . . . . 6 |- ( ( ph /\ k e. ( M ... n ) ) -> ( F ` k ) e. CC )
35	34	adantlr 477	. . . . 5 |- ( ( ( ph /\ n e. Z ) /\ k e. ( M ... n ) ) -> ( F ` k ) e. CC )
36	31, 35	fsumabs 12019	. . . 4 |- ( ( ph /\ n e. Z ) -> ( abs ` sum_ k e. ( M ... n ) ( F ` k ) ) <_ sum_ k e. ( M ... n ) ( abs ` ( F ` k ) ) )
37		eqidd 2230	. . . . . 6 |- ( ( ( ph /\ n e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = ( F ` k ) )
38	1	eleq2i 2296	. . . . . . . 8 |- ( n e. Z <-> n e. ( ZZ>= ` M ) )
39	38	biimpi 120	. . . . . . 7 |- ( n e. Z -> n e. ( ZZ>= ` M ) )
40	39	adantl 277	. . . . . 6 |- ( ( ph /\ n e. Z ) -> n e. ( ZZ>= ` M ) )
41	1	eleq2i 2296	. . . . . . . 8 |- ( k e. Z <-> k e. ( ZZ>= ` M ) )
42	41, 10	sylan2br 288	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
43	42	adantlr 477	. . . . . 6 |- ( ( ( ph /\ n e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
44	37, 40, 43	fsum3ser 11951	. . . . 5 |- ( ( ph /\ n e. Z ) -> sum_ k e. ( M ... n ) ( F ` k ) = ( seq M ( + , F ) ` n ) )
45	44	fveq2d 5639	. . . 4 |- ( ( ph /\ n e. Z ) -> ( abs ` sum_ k e. ( M ... n ) ( F ` k ) ) = ( abs ` ( seq M ( + , F ) ` n ) ) )
46	22	adantlr 477	. . . . . 6 |- ( ( ( ph /\ n e. Z ) /\ k e. Z ) -> ( G ` k ) = ( abs ` ( F ` k ) ) )
47	41, 46	sylan2br 288	. . . . 5 |- ( ( ( ph /\ n e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) = ( abs ` ( F ` k ) ) )
48	23	adantlr 477	. . . . . . 7 |- ( ( ( ph /\ n e. Z ) /\ k e. Z ) -> ( abs ` ( F ` k ) ) e. RR )
49	41, 48	sylan2br 288	. . . . . 6 |- ( ( ( ph /\ n e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( abs ` ( F ` k ) ) e. RR )
50	49	recnd 8201	. . . . 5 |- ( ( ( ph /\ n e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( abs ` ( F ` k ) ) e. CC )
51	47, 40, 50	fsum3ser 11951	. . . 4 |- ( ( ph /\ n e. Z ) -> sum_ k e. ( M ... n ) ( abs ` ( F ` k ) ) = ( seq M ( + , G ) ` n ) )
52	36, 45, 51	3brtr3d 4117	. . 3 |- ( ( ph /\ n e. Z ) -> ( abs ` ( seq M ( + , F ) ` n ) ) <_ ( seq M ( + , G ) ` n ) )
53	18, 52	eqbrtrd 4108	. 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 11888	1 |- ( ph -> ( abs ` A ) <_ B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   _Vcvv 2800   class class class wbr 4086   |-> cmpt 4148   ` cfv 5324  (class class class)co 6013   CCcc 8023   RRcr 8024   + caddc 8028   <_ cle 8208   ZZcz 9472   ZZ>=cuz 9748   ...cfz 10236   seqcseq 10702   abscabs 11551   ~~> cli 11832   sum_csu 11907

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-mulrcl 8124  ax-addcom 8125  ax-mulcom 8126  ax-addass 8127  ax-mulass 8128  ax-distr 8129  ax-i2m1 8130  ax-0lt1 8131  ax-1rid 8132  ax-0id 8133  ax-rnegex 8134  ax-precex 8135  ax-cnre 8136  ax-pre-ltirr 8137  ax-pre-ltwlin 8138  ax-pre-lttrn 8139  ax-pre-apti 8140  ax-pre-ltadd 8141  ax-pre-mulgt0 8142  ax-pre-mulext 8143  ax-arch 8144  ax-caucvg 8145

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-isom 5333  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-frec 6552  df-1o 6577  df-oadd 6581  df-er 6697  df-en 6905  df-dom 6906  df-fin 6907  df-pnf 8209  df-mnf 8210  df-xr 8211  df-ltxr 8212  df-le 8213  df-sub 8345  df-neg 8346  df-reap 8748  df-ap 8755  df-div 8846  df-inn 9137  df-2 9195  df-3 9196  df-4 9197  df-n0 9396  df-z 9473  df-uz 9749  df-q 9847  df-rp 9882  df-fz 10237  df-fzo 10371  df-seqfrec 10703  df-exp 10794  df-ihash 11031  df-cj 11396  df-re 11397  df-im 11398  df-rsqrt 11552  df-abs 11553  df-clim 11833  df-sumdc 11908

This theorem is referenced by:  eftlub  12244