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Theorem iserabs 10930
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.
StepHypRef 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 8820 . . . . . . 7  |-  ZZ  e.  _V
5 uzssz 9099 . . . . . . 7  |-  ( ZZ>= `  M )  C_  ZZ
64, 5ssexi 3983 . . . . . 6  |-  ( ZZ>= `  M )  e.  _V
71, 6eqeltri 2161 . . . . 5  |-  Z  e. 
_V
87mptex 5537 . . . 4  |-  ( m  e.  Z  |->  ( abs `  (  seq M (  +  ,  F ) `
 m ) ) )  e.  _V
98a1i 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 )
111, 2, 10serf 9961 . . . 4  |-  ( ph  ->  seq M (  +  ,  F ) : Z --> CC )
1211ffvelrnda 5448 . . 3  |-  ( (
ph  /\  n  e.  Z )  ->  (  seq M (  +  ,  F ) `  n
)  e.  CC )
13 simpr 109 . . . 4  |-  ( (
ph  /\  n  e.  Z )  ->  n  e.  Z )
1412abscld 10675 . . . 4  |-  ( (
ph  /\  n  e.  Z )  ->  ( abs `  (  seq M
(  +  ,  F
) `  n )
)  e.  RR )
15 2fveq3 5323 . . . . 5  |-  ( m  =  n  ->  ( abs `  (  seq M
(  +  ,  F
) `  m )
)  =  ( abs `  (  seq M (  +  ,  F ) `
 n ) ) )
16 eqid 2089 . . . . 5  |-  ( m  e.  Z  |->  ( abs `  (  seq M (  +  ,  F ) `
 m ) ) )  =  ( m  e.  Z  |->  ( abs `  (  seq M (  +  ,  F ) `
 m ) ) )
1715, 16fvmptg 5393 . . . 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
) ) )
1813, 14, 17syl2anc 404 . . 3  |-  ( (
ph  /\  n  e.  Z )  ->  (
( m  e.  Z  |->  ( abs `  (  seq M (  +  ,  F ) `  m
) ) ) `  n )  =  ( abs `  (  seq M (  +  ,  F ) `  n
) ) )
191, 3, 9, 2, 12, 18climabs 10769 . 2  |-  ( ph  ->  ( m  e.  Z  |->  ( abs `  (  seq M (  +  ,  F ) `  m
) ) )  ~~>  ( abs `  A ) )
20 iserabs.3 . 2  |-  ( ph  ->  seq M (  +  ,  G )  ~~>  B )
2118, 14eqeltrd 2165 . 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 )
) )
2310abscld 10675 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( F `  k ) )  e.  RR )
2422, 23eqeltrd 2165 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  RR )
251, 2, 24serfre 9962 . . 3  |-  ( ph  ->  seq M (  +  ,  G ) : Z --> RR )
2625ffvelrnda 5448 . 2  |-  ( (
ph  /\  n  e.  Z )  ->  (  seq M (  +  ,  G ) `  n
)  e.  RR )
272adantr 271 . . . . . 6  |-  ( (
ph  /\  n  e.  Z )  ->  M  e.  ZZ )
28 eluzelz 9089 . . . . . . . 8  |-  ( n  e.  ( ZZ>= `  M
)  ->  n  e.  ZZ )
2928, 1eleq2s 2183 . . . . . . 7  |-  ( n  e.  Z  ->  n  e.  ZZ )
3029adantl 272 . . . . . 6  |-  ( (
ph  /\  n  e.  Z )  ->  n  e.  ZZ )
3127, 30fzfigd 9899 . . . . 5  |-  ( (
ph  /\  n  e.  Z )  ->  ( M ... n )  e. 
Fin )
32 elfzuz 9497 . . . . . . . 8  |-  ( k  e.  ( M ... n )  ->  k  e.  ( ZZ>= `  M )
)
3332, 1syl6eleqr 2182 . . . . . . 7  |-  ( k  e.  ( M ... n )  ->  k  e.  Z )
3433, 10sylan2 281 . . . . . 6  |-  ( (
ph  /\  k  e.  ( M ... n ) )  ->  ( F `  k )  e.  CC )
3534adantlr 462 . . . . 5  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  ( M ... n
) )  ->  ( F `  k )  e.  CC )
3631, 35fsumabs 10920 . . . 4  |-  ( (
ph  /\  n  e.  Z )  ->  ( abs `  sum_ k  e.  ( M ... n ) ( F `  k
) )  <_  sum_ k  e.  ( M ... n
) ( abs `  ( F `  k )
) )
37 eqidd 2090 . . . . . 6  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  =  ( F `  k ) )
381eleq2i 2155 . . . . . . . 8  |-  ( n  e.  Z  <->  n  e.  ( ZZ>= `  M )
)
3938biimpi 119 . . . . . . 7  |-  ( n  e.  Z  ->  n  e.  ( ZZ>= `  M )
)
4039adantl 272 . . . . . 6  |-  ( (
ph  /\  n  e.  Z )  ->  n  e.  ( ZZ>= `  M )
)
411eleq2i 2155 . . . . . . . 8  |-  ( k  e.  Z  <->  k  e.  ( ZZ>= `  M )
)
4241, 10sylan2br 283 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
4342adantlr 462 . . . . . 6  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
4437, 40, 43fsum3ser 10852 . . . . 5  |-  ( (
ph  /\  n  e.  Z )  ->  sum_ k  e.  ( M ... n
) ( F `  k )  =  (  seq M (  +  ,  F ) `  n ) )
4544fveq2d 5322 . . . 4  |-  ( (
ph  /\  n  e.  Z )  ->  ( abs `  sum_ k  e.  ( M ... n ) ( F `  k
) )  =  ( abs `  (  seq M (  +  ,  F ) `  n
) ) )
4622adantlr 462 . . . . . 6  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  Z )  ->  ( G `  k )  =  ( abs `  ( F `  k )
) )
4741, 46sylan2br 283 . . . . 5  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( G `  k )  =  ( abs `  ( F `
 k ) ) )
4823adantlr 462 . . . . . . 7  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  Z )  ->  ( abs `  ( F `  k ) )  e.  RR )
4941, 48sylan2br 283 . . . . . 6  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( abs `  ( F `  k
) )  e.  RR )
5049recnd 7577 . . . . 5  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( abs `  ( F `  k
) )  e.  CC )
5147, 40, 50fsum3ser 10852 . . . 4  |-  ( (
ph  /\  n  e.  Z )  ->  sum_ k  e.  ( M ... n
) ( abs `  ( F `  k )
)  =  (  seq M (  +  ,  G ) `  n
) )
5236, 45, 513brtr3d 3880 . . 3  |-  ( (
ph  /\  n  e.  Z )  ->  ( abs `  (  seq M
(  +  ,  F
) `  n )
)  <_  (  seq M (  +  ,  G ) `  n
) )
5318, 52eqbrtrd 3871 . 2  |-  ( (
ph  /\  n  e.  Z )  ->  (
( m  e.  Z  |->  ( abs `  (  seq M (  +  ,  F ) `  m
) ) ) `  n )  <_  (  seq M (  +  ,  G ) `  n
) )
541, 2, 19, 20, 21, 26, 53climle 10783 1  |-  ( ph  ->  ( abs `  A
)  <_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1290    e. wcel 1439   _Vcvv 2620   class class class wbr 3851    |-> cmpt 3905   ` cfv 5028  (class class class)co 5666   CCcc 7409   RRcr 7410    + caddc 7414    <_ cle 7584   ZZcz 8811   ZZ>=cuz 9080   ...cfz 9485    seqcseq 9913   abscabs 10491    ~~> cli 10727   sum_csu 10803
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416  ax-cnex 7497  ax-resscn 7498  ax-1cn 7499  ax-1re 7500  ax-icn 7501  ax-addcl 7502  ax-addrcl 7503  ax-mulcl 7504  ax-mulrcl 7505  ax-addcom 7506  ax-mulcom 7507  ax-addass 7508  ax-mulass 7509  ax-distr 7510  ax-i2m1 7511  ax-0lt1 7512  ax-1rid 7513  ax-0id 7514  ax-rnegex 7515  ax-precex 7516  ax-cnre 7517  ax-pre-ltirr 7518  ax-pre-ltwlin 7519  ax-pre-lttrn 7520  ax-pre-apti 7521  ax-pre-ltadd 7522  ax-pre-mulgt0 7523  ax-pre-mulext 7524  ax-arch 7525  ax-caucvg 7526
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-if 3398  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-ilim 4205  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-isom 5037  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-irdg 6149  df-frec 6170  df-1o 6195  df-oadd 6199  df-er 6306  df-en 6512  df-dom 6513  df-fin 6514  df-pnf 7585  df-mnf 7586  df-xr 7587  df-ltxr 7588  df-le 7589  df-sub 7716  df-neg 7717  df-reap 8113  df-ap 8120  df-div 8201  df-inn 8484  df-2 8542  df-3 8543  df-4 8544  df-n0 8735  df-z 8812  df-uz 9081  df-q 9166  df-rp 9196  df-fz 9486  df-fzo 9615  df-iseq 9914  df-seq3 9915  df-exp 10016  df-ihash 10245  df-cj 10337  df-re 10338  df-im 10339  df-rsqrt 10492  df-abs 10493  df-clim 10728  df-isum 10804
This theorem is referenced by:  eftlub  11041
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