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Theorem iserabs 11515
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 9292 . . . . . . 7  |-  ZZ  e.  _V
5 uzssz 9577 . . . . . . 7  |-  ( ZZ>= `  M )  C_  ZZ
64, 5ssexi 4156 . . . . . 6  |-  ( ZZ>= `  M )  e.  _V
71, 6eqeltri 2262 . . . . 5  |-  Z  e. 
_V
87mptex 5763 . . . 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 10505 . . . 4  |-  ( ph  ->  seq M (  +  ,  F ) : Z --> CC )
1211ffvelcdmda 5672 . . 3  |-  ( (
ph  /\  n  e.  Z )  ->  (  seq M (  +  ,  F ) `  n
)  e.  CC )
13 simpr 110 . . . 4  |-  ( (
ph  /\  n  e.  Z )  ->  n  e.  Z )
1412abscld 11222 . . . 4  |-  ( (
ph  /\  n  e.  Z )  ->  ( abs `  (  seq M
(  +  ,  F
) `  n )
)  e.  RR )
15 2fveq3 5539 . . . . 5  |-  ( m  =  n  ->  ( abs `  (  seq M
(  +  ,  F
) `  m )
)  =  ( abs `  (  seq M (  +  ,  F ) `
 n ) ) )
16 eqid 2189 . . . . 5  |-  ( m  e.  Z  |->  ( abs `  (  seq M (  +  ,  F ) `
 m ) ) )  =  ( m  e.  Z  |->  ( abs `  (  seq M (  +  ,  F ) `
 m ) ) )
1715, 16fvmptg 5613 . . . 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 411 . . 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 11360 . 2  |-  ( ph  ->  ( m  e.  Z  |->  ( abs `  (  seq M (  +  ,  F ) `  m
) ) )  ~~>  ( abs `  A ) )
20 iserabs.3 . 2  |-  ( ph  ->  seq M (  +  ,  G )  ~~>  B )
2118, 14eqeltrd 2266 . 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 11222 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( F `  k ) )  e.  RR )
2422, 23eqeltrd 2266 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  RR )
251, 2, 24serfre 10506 . . 3  |-  ( ph  ->  seq M (  +  ,  G ) : Z --> RR )
2625ffvelcdmda 5672 . 2  |-  ( (
ph  /\  n  e.  Z )  ->  (  seq M (  +  ,  G ) `  n
)  e.  RR )
272adantr 276 . . . . . 6  |-  ( (
ph  /\  n  e.  Z )  ->  M  e.  ZZ )
28 eluzelz 9567 . . . . . . . 8  |-  ( n  e.  ( ZZ>= `  M
)  ->  n  e.  ZZ )
2928, 1eleq2s 2284 . . . . . . 7  |-  ( n  e.  Z  ->  n  e.  ZZ )
3029adantl 277 . . . . . 6  |-  ( (
ph  /\  n  e.  Z )  ->  n  e.  ZZ )
3127, 30fzfigd 10462 . . . . 5  |-  ( (
ph  /\  n  e.  Z )  ->  ( M ... n )  e. 
Fin )
32 elfzuz 10051 . . . . . . . 8  |-  ( k  e.  ( M ... n )  ->  k  e.  ( ZZ>= `  M )
)
3332, 1eleqtrrdi 2283 . . . . . . 7  |-  ( k  e.  ( M ... n )  ->  k  e.  Z )
3433, 10sylan2 286 . . . . . 6  |-  ( (
ph  /\  k  e.  ( M ... n ) )  ->  ( F `  k )  e.  CC )
3534adantlr 477 . . . . 5  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  ( M ... n
) )  ->  ( F `  k )  e.  CC )
3631, 35fsumabs 11505 . . . 4  |-  ( (
ph  /\  n  e.  Z )  ->  ( abs `  sum_ k  e.  ( M ... n ) ( F `  k
) )  <_  sum_ k  e.  ( M ... n
) ( abs `  ( F `  k )
) )
37 eqidd 2190 . . . . . 6  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  =  ( F `  k ) )
381eleq2i 2256 . . . . . . . 8  |-  ( n  e.  Z  <->  n  e.  ( ZZ>= `  M )
)
3938biimpi 120 . . . . . . 7  |-  ( n  e.  Z  ->  n  e.  ( ZZ>= `  M )
)
4039adantl 277 . . . . . 6  |-  ( (
ph  /\  n  e.  Z )  ->  n  e.  ( ZZ>= `  M )
)
411eleq2i 2256 . . . . . . . 8  |-  ( k  e.  Z  <->  k  e.  ( ZZ>= `  M )
)
4241, 10sylan2br 288 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
4342adantlr 477 . . . . . 6  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
4437, 40, 43fsum3ser 11437 . . . . 5  |-  ( (
ph  /\  n  e.  Z )  ->  sum_ k  e.  ( M ... n
) ( F `  k )  =  (  seq M (  +  ,  F ) `  n ) )
4544fveq2d 5538 . . . 4  |-  ( (
ph  /\  n  e.  Z )  ->  ( abs `  sum_ k  e.  ( M ... n ) ( F `  k
) )  =  ( abs `  (  seq M (  +  ,  F ) `  n
) ) )
4622adantlr 477 . . . . . 6  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  Z )  ->  ( G `  k )  =  ( abs `  ( F `  k )
) )
4741, 46sylan2br 288 . . . . 5  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( G `  k )  =  ( abs `  ( F `
 k ) ) )
4823adantlr 477 . . . . . . 7  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  Z )  ->  ( abs `  ( F `  k ) )  e.  RR )
4941, 48sylan2br 288 . . . . . 6  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( abs `  ( F `  k
) )  e.  RR )
5049recnd 8016 . . . . 5  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( abs `  ( F `  k
) )  e.  CC )
5147, 40, 50fsum3ser 11437 . . . 4  |-  ( (
ph  /\  n  e.  Z )  ->  sum_ k  e.  ( M ... n
) ( abs `  ( F `  k )
)  =  (  seq M (  +  ,  G ) `  n
) )
5236, 45, 513brtr3d 4049 . . 3  |-  ( (
ph  /\  n  e.  Z )  ->  ( abs `  (  seq M
(  +  ,  F
) `  n )
)  <_  (  seq M (  +  ,  G ) `  n
) )
5318, 52eqbrtrd 4040 . 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 11374 1  |-  ( ph  ->  ( abs `  A
)  <_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160   _Vcvv 2752   class class class wbr 4018    |-> cmpt 4079   ` cfv 5235  (class class class)co 5896   CCcc 7839   RRcr 7840    + caddc 7844    <_ cle 8023   ZZcz 9283   ZZ>=cuz 9558   ...cfz 10038    seqcseq 10476   abscabs 11038    ~~> cli 11318   sum_csu 11393
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-mulrcl 7940  ax-addcom 7941  ax-mulcom 7942  ax-addass 7943  ax-mulass 7944  ax-distr 7945  ax-i2m1 7946  ax-0lt1 7947  ax-1rid 7948  ax-0id 7949  ax-rnegex 7950  ax-precex 7951  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-ltwlin 7954  ax-pre-lttrn 7955  ax-pre-apti 7956  ax-pre-ltadd 7957  ax-pre-mulgt0 7958  ax-pre-mulext 7959  ax-arch 7960  ax-caucvg 7961
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-isom 5244  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-irdg 6395  df-frec 6416  df-1o 6441  df-oadd 6445  df-er 6559  df-en 6767  df-dom 6768  df-fin 6769  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028  df-sub 8160  df-neg 8161  df-reap 8562  df-ap 8569  df-div 8660  df-inn 8950  df-2 9008  df-3 9009  df-4 9010  df-n0 9207  df-z 9284  df-uz 9559  df-q 9650  df-rp 9684  df-fz 10039  df-fzo 10173  df-seqfrec 10477  df-exp 10551  df-ihash 10788  df-cj 10883  df-re 10884  df-im 10885  df-rsqrt 11039  df-abs 11040  df-clim 11319  df-sumdc 11394
This theorem is referenced by:  eftlub  11730
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