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Theorem iserabs 11438
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 9221 . . . . . . 7  |-  ZZ  e.  _V
5 uzssz 9506 . . . . . . 7  |-  ( ZZ>= `  M )  C_  ZZ
64, 5ssexi 4127 . . . . . 6  |-  ( ZZ>= `  M )  e.  _V
71, 6eqeltri 2243 . . . . 5  |-  Z  e. 
_V
87mptex 5722 . . . 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 10430 . . . 4  |-  ( ph  ->  seq M (  +  ,  F ) : Z --> CC )
1211ffvelrnda 5631 . . 3  |-  ( (
ph  /\  n  e.  Z )  ->  (  seq M (  +  ,  F ) `  n
)  e.  CC )
13 simpr 109 . . . 4  |-  ( (
ph  /\  n  e.  Z )  ->  n  e.  Z )
1412abscld 11145 . . . 4  |-  ( (
ph  /\  n  e.  Z )  ->  ( abs `  (  seq M
(  +  ,  F
) `  n )
)  e.  RR )
15 2fveq3 5501 . . . . 5  |-  ( m  =  n  ->  ( abs `  (  seq M
(  +  ,  F
) `  m )
)  =  ( abs `  (  seq M (  +  ,  F ) `
 n ) ) )
16 eqid 2170 . . . . 5  |-  ( m  e.  Z  |->  ( abs `  (  seq M (  +  ,  F ) `
 m ) ) )  =  ( m  e.  Z  |->  ( abs `  (  seq M (  +  ,  F ) `
 m ) ) )
1715, 16fvmptg 5572 . . . 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 409 . . 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 11283 . 2  |-  ( ph  ->  ( m  e.  Z  |->  ( abs `  (  seq M (  +  ,  F ) `  m
) ) )  ~~>  ( abs `  A ) )
20 iserabs.3 . 2  |-  ( ph  ->  seq M (  +  ,  G )  ~~>  B )
2118, 14eqeltrd 2247 . 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 11145 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( F `  k ) )  e.  RR )
2422, 23eqeltrd 2247 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  RR )
251, 2, 24serfre 10431 . . 3  |-  ( ph  ->  seq M (  +  ,  G ) : Z --> RR )
2625ffvelrnda 5631 . 2  |-  ( (
ph  /\  n  e.  Z )  ->  (  seq M (  +  ,  G ) `  n
)  e.  RR )
272adantr 274 . . . . . 6  |-  ( (
ph  /\  n  e.  Z )  ->  M  e.  ZZ )
28 eluzelz 9496 . . . . . . . 8  |-  ( n  e.  ( ZZ>= `  M
)  ->  n  e.  ZZ )
2928, 1eleq2s 2265 . . . . . . 7  |-  ( n  e.  Z  ->  n  e.  ZZ )
3029adantl 275 . . . . . 6  |-  ( (
ph  /\  n  e.  Z )  ->  n  e.  ZZ )
3127, 30fzfigd 10387 . . . . 5  |-  ( (
ph  /\  n  e.  Z )  ->  ( M ... n )  e. 
Fin )
32 elfzuz 9977 . . . . . . . 8  |-  ( k  e.  ( M ... n )  ->  k  e.  ( ZZ>= `  M )
)
3332, 1eleqtrrdi 2264 . . . . . . 7  |-  ( k  e.  ( M ... n )  ->  k  e.  Z )
3433, 10sylan2 284 . . . . . 6  |-  ( (
ph  /\  k  e.  ( M ... n ) )  ->  ( F `  k )  e.  CC )
3534adantlr 474 . . . . 5  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  ( M ... n
) )  ->  ( F `  k )  e.  CC )
3631, 35fsumabs 11428 . . . 4  |-  ( (
ph  /\  n  e.  Z )  ->  ( abs `  sum_ k  e.  ( M ... n ) ( F `  k
) )  <_  sum_ k  e.  ( M ... n
) ( abs `  ( F `  k )
) )
37 eqidd 2171 . . . . . 6  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  =  ( F `  k ) )
381eleq2i 2237 . . . . . . . 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 2237 . . . . . . . 8  |-  ( k  e.  Z  <->  k  e.  ( ZZ>= `  M )
)
4241, 10sylan2br 286 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
4342adantlr 474 . . . . . 6  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
4437, 40, 43fsum3ser 11360 . . . . 5  |-  ( (
ph  /\  n  e.  Z )  ->  sum_ k  e.  ( M ... n
) ( F `  k )  =  (  seq M (  +  ,  F ) `  n ) )
4544fveq2d 5500 . . . 4  |-  ( (
ph  /\  n  e.  Z )  ->  ( abs `  sum_ k  e.  ( M ... n ) ( F `  k
) )  =  ( abs `  (  seq M (  +  ,  F ) `  n
) ) )
4622adantlr 474 . . . . . 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 474 . . . . . . 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 7948 . . . . 5  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( abs `  ( F `  k
) )  e.  CC )
5147, 40, 50fsum3ser 11360 . . . 4  |-  ( (
ph  /\  n  e.  Z )  ->  sum_ k  e.  ( M ... n
) ( abs `  ( F `  k )
)  =  (  seq M (  +  ,  G ) `  n
) )
5236, 45, 513brtr3d 4020 . . 3  |-  ( (
ph  /\  n  e.  Z )  ->  ( abs `  (  seq M
(  +  ,  F
) `  n )
)  <_  (  seq M (  +  ,  G ) `  n
) )
5318, 52eqbrtrd 4011 . 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 11297 1  |-  ( ph  ->  ( abs `  A
)  <_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1348    e. wcel 2141   _Vcvv 2730   class class class wbr 3989    |-> cmpt 4050   ` cfv 5198  (class class class)co 5853   CCcc 7772   RRcr 7773    + caddc 7777    <_ cle 7955   ZZcz 9212   ZZ>=cuz 9487   ...cfz 9965    seqcseq 10401   abscabs 10961    ~~> cli 11241   sum_csu 11316
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-oadd 6399  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-seqfrec 10402  df-exp 10476  df-ihash 10710  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-sumdc 11317
This theorem is referenced by:  eftlub  11653
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