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Theorem iserabs 11256
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 9075 . . . . . . 7  |-  ZZ  e.  _V
5 uzssz 9357 . . . . . . 7  |-  ( ZZ>= `  M )  C_  ZZ
64, 5ssexi 4066 . . . . . 6  |-  ( ZZ>= `  M )  e.  _V
71, 6eqeltri 2212 . . . . 5  |-  Z  e. 
_V
87mptex 5646 . . . 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 10259 . . . 4  |-  ( ph  ->  seq M (  +  ,  F ) : Z --> CC )
1211ffvelrnda 5555 . . 3  |-  ( (
ph  /\  n  e.  Z )  ->  (  seq M (  +  ,  F ) `  n
)  e.  CC )
13 simpr 109 . . . 4  |-  ( (
ph  /\  n  e.  Z )  ->  n  e.  Z )
1412abscld 10965 . . . 4  |-  ( (
ph  /\  n  e.  Z )  ->  ( abs `  (  seq M
(  +  ,  F
) `  n )
)  e.  RR )
15 2fveq3 5426 . . . . 5  |-  ( m  =  n  ->  ( abs `  (  seq M
(  +  ,  F
) `  m )
)  =  ( abs `  (  seq M (  +  ,  F ) `
 n ) ) )
16 eqid 2139 . . . . 5  |-  ( m  e.  Z  |->  ( abs `  (  seq M (  +  ,  F ) `
 m ) ) )  =  ( m  e.  Z  |->  ( abs `  (  seq M (  +  ,  F ) `
 m ) ) )
1715, 16fvmptg 5497 . . . 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 408 . . 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 11101 . 2  |-  ( ph  ->  ( m  e.  Z  |->  ( abs `  (  seq M (  +  ,  F ) `  m
) ) )  ~~>  ( abs `  A ) )
20 iserabs.3 . 2  |-  ( ph  ->  seq M (  +  ,  G )  ~~>  B )
2118, 14eqeltrd 2216 . 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 10965 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( F `  k ) )  e.  RR )
2422, 23eqeltrd 2216 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  RR )
251, 2, 24serfre 10260 . . 3  |-  ( ph  ->  seq M (  +  ,  G ) : Z --> RR )
2625ffvelrnda 5555 . 2  |-  ( (
ph  /\  n  e.  Z )  ->  (  seq M (  +  ,  G ) `  n
)  e.  RR )
272adantr 274 . . . . . 6  |-  ( (
ph  /\  n  e.  Z )  ->  M  e.  ZZ )
28 eluzelz 9347 . . . . . . . 8  |-  ( n  e.  ( ZZ>= `  M
)  ->  n  e.  ZZ )
2928, 1eleq2s 2234 . . . . . . 7  |-  ( n  e.  Z  ->  n  e.  ZZ )
3029adantl 275 . . . . . 6  |-  ( (
ph  /\  n  e.  Z )  ->  n  e.  ZZ )
3127, 30fzfigd 10216 . . . . 5  |-  ( (
ph  /\  n  e.  Z )  ->  ( M ... n )  e. 
Fin )
32 elfzuz 9814 . . . . . . . 8  |-  ( k  e.  ( M ... n )  ->  k  e.  ( ZZ>= `  M )
)
3332, 1eleqtrrdi 2233 . . . . . . 7  |-  ( k  e.  ( M ... n )  ->  k  e.  Z )
3433, 10sylan2 284 . . . . . 6  |-  ( (
ph  /\  k  e.  ( M ... n ) )  ->  ( F `  k )  e.  CC )
3534adantlr 468 . . . . 5  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  ( M ... n
) )  ->  ( F `  k )  e.  CC )
3631, 35fsumabs 11246 . . . 4  |-  ( (
ph  /\  n  e.  Z )  ->  ( abs `  sum_ k  e.  ( M ... n ) ( F `  k
) )  <_  sum_ k  e.  ( M ... n
) ( abs `  ( F `  k )
) )
37 eqidd 2140 . . . . . 6  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  =  ( F `  k ) )
381eleq2i 2206 . . . . . . . 8  |-  ( n  e.  Z  <->  n  e.  ( ZZ>= `  M )
)
3938biimpi 119 . . . . . . 7  |-  ( n  e.  Z  ->  n  e.  ( ZZ>= `  M )
)
4039adantl 275 . . . . . 6  |-  ( (
ph  /\  n  e.  Z )  ->  n  e.  ( ZZ>= `  M )
)
411eleq2i 2206 . . . . . . . 8  |-  ( k  e.  Z  <->  k  e.  ( ZZ>= `  M )
)
4241, 10sylan2br 286 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
4342adantlr 468 . . . . . 6  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
4437, 40, 43fsum3ser 11178 . . . . 5  |-  ( (
ph  /\  n  e.  Z )  ->  sum_ k  e.  ( M ... n
) ( F `  k )  =  (  seq M (  +  ,  F ) `  n ) )
4544fveq2d 5425 . . . 4  |-  ( (
ph  /\  n  e.  Z )  ->  ( abs `  sum_ k  e.  ( M ... n ) ( F `  k
) )  =  ( abs `  (  seq M (  +  ,  F ) `  n
) ) )
4622adantlr 468 . . . . . 6  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  Z )  ->  ( G `  k )  =  ( abs `  ( F `  k )
) )
4741, 46sylan2br 286 . . . . 5  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( G `  k )  =  ( abs `  ( F `
 k ) ) )
4823adantlr 468 . . . . . . 7  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  Z )  ->  ( abs `  ( F `  k ) )  e.  RR )
4941, 48sylan2br 286 . . . . . 6  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( abs `  ( F `  k
) )  e.  RR )
5049recnd 7806 . . . . 5  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( abs `  ( F `  k
) )  e.  CC )
5147, 40, 50fsum3ser 11178 . . . 4  |-  ( (
ph  /\  n  e.  Z )  ->  sum_ k  e.  ( M ... n
) ( abs `  ( F `  k )
)  =  (  seq M (  +  ,  G ) `  n
) )
5236, 45, 513brtr3d 3959 . . 3  |-  ( (
ph  /\  n  e.  Z )  ->  ( abs `  (  seq M
(  +  ,  F
) `  n )
)  <_  (  seq M (  +  ,  G ) `  n
) )
5318, 52eqbrtrd 3950 . 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 11115 1  |-  ( ph  ->  ( abs `  A
)  <_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   _Vcvv 2686   class class class wbr 3929    |-> cmpt 3989   ` cfv 5123  (class class class)co 5774   CCcc 7630   RRcr 7631    + caddc 7635    <_ cle 7813   ZZcz 9066   ZZ>=cuz 9338   ...cfz 9802    seqcseq 10230   abscabs 10781    ~~> cli 11059   sum_csu 11134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-mulrcl 7731  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-precex 7742  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748  ax-pre-mulgt0 7749  ax-pre-mulext 7750  ax-arch 7751  ax-caucvg 7752
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-frec 6288  df-1o 6313  df-oadd 6317  df-er 6429  df-en 6635  df-dom 6636  df-fin 6637  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-reap 8349  df-ap 8356  df-div 8445  df-inn 8733  df-2 8791  df-3 8792  df-4 8793  df-n0 8990  df-z 9067  df-uz 9339  df-q 9424  df-rp 9454  df-fz 9803  df-fzo 9932  df-seqfrec 10231  df-exp 10305  df-ihash 10534  df-cj 10626  df-re 10627  df-im 10628  df-rsqrt 10782  df-abs 10783  df-clim 11060  df-sumdc 11135
This theorem is referenced by:  eftlub  11408
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