Theorem iserabs 12225

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 9636	. . . . . . 7 |- ZZ e. _V
5		uzssz 9925	. . . . . . 7 |- ( ZZ>= ` M ) C_ ZZ
6	4, 5	ssexi 4269	. . . . . 6 |- ( ZZ>= ` M ) e. _V
7	1, 6	eqeltri 2311	. . . . 5 |- Z e. _V
8	7	mptex 5937	. . . 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 10903	. . . 4 |- ( ph -> seq M ( + , F ) : Z --> CC )
12	11	ffvelcdmda 5837	. . 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 11930	. . . 4 |- ( ( ph /\ n e. Z ) -> ( abs ` ( seq M ( + , F ) ` n ) ) e. RR )
15		2fveq3 5698	. . . . 5 |- ( m = n -> ( abs ` ( seq M ( + , F ) ` m ) ) = ( abs ` ( seq M ( + , F ) ` n ) ) )
16		eqid 2238	. . . . 5 |- ( m e. Z |-> ( abs ` ( seq M ( + , F ) ` m ) ) ) = ( m e. Z |-> ( abs ` ( seq M ( + , F ) ` m ) ) )
17	15, 16	fvmptg 5778	. . . 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 415	. . 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 12069	. 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 2315	. 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 11930	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( abs ` ( F ` k ) ) e. RR )
24	22, 23	eqeltrd 2315	. . . 4 |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. RR )
25	1, 2, 24	serfre 10904	. . 3 |- ( ph -> seq M ( + , G ) : Z --> RR )
26	25	ffvelcdmda 5837	. 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 9914	. . . . . . . 8 |- ( n e. ( ZZ>= ` M ) -> n e. ZZ )
29	28, 1	eleq2s 2333	. . . . . . 7 |- ( n e. Z -> n e. ZZ )
30	29	adantl 277	. . . . . 6 |- ( ( ph /\ n e. Z ) -> n e. ZZ )
31	27, 30	fzfigd 10851	. . . . 5 |- ( ( ph /\ n e. Z ) -> ( M ... n ) e. Fin )
32		elfzuz 10407	. . . . . . . 8 |- ( k e. ( M ... n ) -> k e. ( ZZ>= ` M ) )
33	32, 1	eleqtrrdi 2332	. . . . . . 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 481	. . . . 5 |- ( ( ( ph /\ n e. Z ) /\ k e. ( M ... n ) ) -> ( F ` k ) e. CC )
36	31, 35	fsumabs 12215	. . . 4 |- ( ( ph /\ n e. Z ) -> ( abs ` sum_ k e. ( M ... n ) ( F ` k ) ) <_ sum_ k e. ( M ... n ) ( abs ` ( F ` k ) ) )
37		eqidd 2239	. . . . . 6 |- ( ( ( ph /\ n e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) = ( F ` k ) )
38	1	eleq2i 2305	. . . . . . . 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 2305	. . . . . . . 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 481	. . . . . 6 |- ( ( ( ph /\ n e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
44	37, 40, 43	fsum3ser 12147	. . . . 5 |- ( ( ph /\ n e. Z ) -> sum_ k e. ( M ... n ) ( F ` k ) = ( seq M ( + , F ) ` n ) )
45	44	fveq2d 5697	. . . 4 |- ( ( ph /\ n e. Z ) -> ( abs ` sum_ k e. ( M ... n ) ( F ` k ) ) = ( abs ` ( seq M ( + , F ) ` n ) ) )
46	22	adantlr 481	. . . . . 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 481	. . . . . . 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 8348	. . . . 5 |- ( ( ( ph /\ n e. Z ) /\ k e. ( ZZ>= ` M ) ) -> ( abs ` ( F ` k ) ) e. CC )
51	47, 40, 50	fsum3ser 12147	. . . 4 |- ( ( ph /\ n e. Z ) -> sum_ k e. ( M ... n ) ( abs ` ( F ` k ) ) = ( seq M ( + , G ) ` n ) )
52	36, 45, 51	3brtr3d 4159	. . 3 |- ( ( ph /\ n e. Z ) -> ( abs ` ( seq M ( + , F ) ` n ) ) <_ ( seq M ( + , G ) ` n ) )
53	18, 52	eqbrtrd 4150	. 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 12083	1 |- ( ph -> ( abs ` A ) <_ B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   class class class wbr 4128   |-> cmpt 4190   ` cfv 5375  (class class class)co 6079   CCcc 8171   RRcr 8172   + caddc 8176   <_ cle 8355   ZZcz 9627   ZZ>=cuz 9904   ...cfz 10394   seqcseq 10867   abscabs 11746   ~~> cli 12027   sum_csu 12102

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291  ax-arch 8292  ax-caucvg 8293

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-isom 5384  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-irdg 6635  df-frec 6656  df-1o 6681  df-oadd 6685  df-er 6801  df-en 7017  df-dom 7018  df-fin 7019  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-n0 9547  df-z 9628  df-uz 9905  df-q 10003  df-rp 10038  df-fz 10395  df-fzo 10533  df-seqfrec 10868  df-exp 10959  df-ihash 11198  df-cj 11590  df-re 11591  df-im 11592  df-rsqrt 11747  df-abs 11748  df-clim 12028  df-sumdc 12103

This theorem is referenced by:  eftlub  12440