Theorem iserabs 14547

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 Mario Carneiro, 27-May-2014.)

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		fvex 6201	. . . . . 6 |- ( ZZ>= ` M ) e. _V
5	1, 4	eqeltri 2697	. . . . 5 |- Z e. _V
6	5	mptex 6486	. . . 4 |- ( m e. Z |-> ( abs ` ( seq M ( + , F ) ` m ) ) ) e. _V
7	6	a1i 11	. . 3 |- ( ph -> ( m e. Z |-> ( abs ` ( seq M ( + , F ) ` m ) ) ) e. _V )
8		iserabs.6	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
9	1, 2, 8	serf 12829	. . . 4 |- ( ph -> seq M ( + , F ) : Z --> CC )
10	9	ffvelrnda 6359	. . 3 |- ( ( ph /\ n e. Z ) -> ( seq M ( + , F ) ` n ) e. CC )
11		fveq2 6191	. . . . . 6 |- ( m = n -> ( seq M ( + , F ) ` m ) = ( seq M ( + , F ) ` n ) )
12	11	fveq2d 6195	. . . . 5 |- ( m = n -> ( abs ` ( seq M ( + , F ) ` m ) ) = ( abs ` ( seq M ( + , F ) ` n ) ) )
13		eqid 2622	. . . . 5 |- ( m e. Z |-> ( abs ` ( seq M ( + , F ) ` m ) ) ) = ( m e. Z |-> ( abs ` ( seq M ( + , F ) ` m ) ) )
14		fvex 6201	. . . . 5 |- ( abs ` ( seq M ( + , F ) ` n ) ) e. _V
15	12, 13, 14	fvmpt 6282	. . . 4 |- ( n e. Z -> ( ( m e. Z |-> ( abs ` ( seq M ( + , F ) ` m ) ) ) ` n ) = ( abs ` ( seq M ( + , F ) ` n ) ) )
16	15	adantl 482	. . 3 |- ( ( ph /\ n e. Z ) -> ( ( m e. Z |-> ( abs ` ( seq M ( + , F ) ` m ) ) ) ` n ) = ( abs ` ( seq M ( + , F ) ` n ) ) )
17	1, 3, 7, 2, 10, 16	climabs 14334	. 2 |- ( ph -> ( m e. Z |-> ( abs ` ( seq M ( + , F ) ` m ) ) ) ~~> ( abs ` A ) )
18		iserabs.3	. 2 |- ( ph -> seq M ( + , G ) ~~> B )
19	10	abscld 14175	. . 3 |- ( ( ph /\ n e. Z ) -> ( abs ` ( seq M ( + , F ) ` n ) ) e. RR )
20	16, 19	eqeltrd 2701	. 2 |- ( ( ph /\ n e. Z ) -> ( ( m e. Z |-> ( abs ` ( seq M ( + , F ) ` m ) ) ) ` n ) e. RR )
21		iserabs.7	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( abs ` ( F ` k ) ) )
22	8	abscld 14175	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( abs ` ( F ` k ) ) e. RR )
23	21, 22	eqeltrd 2701	. . . 4 |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. RR )
24	1, 2, 23	serfre 12830	. . 3 |- ( ph -> seq M ( + , G ) : Z --> RR )
25	24	ffvelrnda 6359	. 2 |- ( ( ph /\ n e. Z ) -> ( seq M ( + , G ) ` n ) e. RR )
26		simpr 477	. . . . 5 |- ( ( ph /\ n e. Z ) -> n e. Z )
27	26, 1	syl6eleq 2711	. . . 4 |- ( ( ph /\ n e. Z ) -> n e. ( ZZ>= ` M ) )
28		elfzuz 12338	. . . . . . 7 |- ( k e. ( M ... n ) -> k e. ( ZZ>= ` M ) )
29	28, 1	syl6eleqr 2712	. . . . . 6 |- ( k e. ( M ... n ) -> k e. Z )
30	29, 8	sylan2 491	. . . . 5 |- ( ( ph /\ k e. ( M ... n ) ) -> ( F ` k ) e. CC )
31	30	adantlr 751	. . . 4 |- ( ( ( ph /\ n e. Z ) /\ k e. ( M ... n ) ) -> ( F ` k ) e. CC )
32	29, 21	sylan2 491	. . . . 5 |- ( ( ph /\ k e. ( M ... n ) ) -> ( G ` k ) = ( abs ` ( F ` k ) ) )
33	32	adantlr 751	. . . 4 |- ( ( ( ph /\ n e. Z ) /\ k e. ( M ... n ) ) -> ( G ` k ) = ( abs ` ( F ` k ) ) )
34	27, 31, 33	seqabs 14546	. . 3 |- ( ( ph /\ n e. Z ) -> ( abs ` ( seq M ( + , F ) ` n ) ) <_ ( seq M ( + , G ) ` n ) )
35	16, 34	eqbrtrd 4675	. 2 |- ( ( ph /\ n e. Z ) -> ( ( m e. Z |-> ( abs ` ( seq M ( + , F ) ` m ) ) ) ` n ) <_ ( seq M ( + , G ) ` n ) )
36	1, 2, 17, 18, 20, 25, 35	climle 14370	1 |- ( ph -> ( abs ` A ) <_ B )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   + caddc 9939   <_ cle 10075   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801   abscabs 13974   ~~> cli 14215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417

This theorem is referenced by:  eftlub  14839  abelthlem7  24192