Theorem iserabs 11467

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 9251	. . . . . . 7 |- ZZ e. _V
5		uzssz 9536	. . . . . . 7 |- ( ZZ>= ` M ) C_ ZZ
6	4, 5	ssexi 4138	. . . . . 6 |- ( ZZ>= ` M ) e. _V
7	1, 6	eqeltri 2250	. . . . 5 |- Z e. _V
8	7	mptex 5738	. . . 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 10460	. . . 4 |- ( ph -> seq M ( + , F ) : Z --> CC )
12	11	ffvelcdmda 5647	. . 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 11174	. . . 4 |- ( ( ph /\ n e. Z ) -> ( abs ` ( seq M ( + , F ) ` n ) ) e. RR )
15		2fveq3 5516	. . . . 5 |- ( m = n -> ( abs ` ( seq M ( + , F ) ` m ) ) = ( abs ` ( seq M ( + , F ) ` n ) ) )
16		eqid 2177	. . . . 5 |- ( m e. Z |-> ( abs ` ( seq M ( + , F ) ` m ) ) ) = ( m e. Z |-> ( abs ` ( seq M ( + , F ) ` m ) ) )
17	15, 16	fvmptg 5588	. . . 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 11312	. 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 2254	. 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 11174	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( abs ` ( F ` k ) ) e. RR )
24	22, 23	eqeltrd 2254	. . . 4 |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. RR )
25	1, 2, 24	serfre 10461	. . 3 |- ( ph -> seq M ( + , G ) : Z --> RR )
26	25	ffvelcdmda 5647	. 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 9526	. . . . . . . 8 |- ( n e. ( ZZ>= ` M ) -> n e. ZZ )
29	28, 1	eleq2s 2272	. . . . . . 7 |- ( n e. Z -> n e. ZZ )
30	29	adantl 277	. . . . . 6 |- ( ( ph /\ n e. Z ) -> n e. ZZ )
31	27, 30	fzfigd 10417	. . . . 5 |- ( ( ph /\ n e. Z ) -> ( M ... n ) e. Fin )
32		elfzuz 10007	. . . . . . . 8 |- ( k e. ( M ... n ) -> k e. ( ZZ>= ` M ) )
33	32, 1	eleqtrrdi 2271	. . . . . . 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 11457	. . . 4 |- ( ( ph /\ n e. Z ) -> ( abs ` sum_ k e. ( M ... n ) ( F ` k ) ) <_ sum_ k e. ( M ... n ) ( abs ` ( F ` k ) ) )
37		eqidd 2178	. . . . . 6 |- ( ( ( ph /\ n e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = ( F ` k ) )
38	1	eleq2i 2244	. . . . . . . 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 2244	. . . . . . . 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 11389	. . . . 5 |- ( ( ph /\ n e. Z ) -> sum_ k e. ( M ... n ) ( F ` k ) = ( seq M ( + , F ) ` n ) )
45	44	fveq2d 5515	. . . 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 7976	. . . . 5 |- ( ( ( ph /\ n e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( abs ` ( F ` k ) ) e. CC )
51	47, 40, 50	fsum3ser 11389	. . . 4 |- ( ( ph /\ n e. Z ) -> sum_ k e. ( M ... n ) ( abs ` ( F ` k ) ) = ( seq M ( + , G ) ` n ) )
52	36, 45, 51	3brtr3d 4031	. . 3 |- ( ( ph /\ n e. Z ) -> ( abs ` ( seq M ( + , F ) ` n ) ) <_ ( seq M ( + , G ) ` n ) )
53	18, 52	eqbrtrd 4022	. 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 11326	1 |- ( ph -> ( abs ` A ) <_ B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   _Vcvv 2737   class class class wbr 4000   |-> cmpt 4061   ` cfv 5212  (class class class)co 5869   CCcc 7800   RRcr 7801   + caddc 7805   <_ cle 7983   ZZcz 9242   ZZ>=cuz 9517   ...cfz 9995   seqcseq 10431   abscabs 10990   ~~> cli 11270   sum_csu 11345

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-frec 6386  df-1o 6411  df-oadd 6415  df-er 6529  df-en 6735  df-dom 6736  df-fin 6737  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fz 9996  df-fzo 10129  df-seqfrec 10432  df-exp 10506  df-ihash 10740  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-clim 11271  df-sumdc 11346

This theorem is referenced by:  eftlub  11682