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Theorem iserabs 11425
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 9208 . . . . . . 7  |-  ZZ  e.  _V
5 uzssz 9493 . . . . . . 7  |-  ( ZZ>= `  M )  C_  ZZ
64, 5ssexi 4125 . . . . . 6  |-  ( ZZ>= `  M )  e.  _V
71, 6eqeltri 2243 . . . . 5  |-  Z  e. 
_V
87mptex 5719 . . . 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 10417 . . . 4  |-  ( ph  ->  seq M (  +  ,  F ) : Z --> CC )
1211ffvelrnda 5628 . . 3  |-  ( (
ph  /\  n  e.  Z )  ->  (  seq M (  +  ,  F ) `  n
)  e.  CC )
13 simpr 109 . . . 4  |-  ( (
ph  /\  n  e.  Z )  ->  n  e.  Z )
1412abscld 11132 . . . 4  |-  ( (
ph  /\  n  e.  Z )  ->  ( abs `  (  seq M
(  +  ,  F
) `  n )
)  e.  RR )
15 2fveq3 5499 . . . . 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 5570 . . . 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 11270 . 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 11132 . . . . 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 10418 . . 3  |-  ( ph  ->  seq M (  +  ,  G ) : Z --> RR )
2625ffvelrnda 5628 . 2  |-  ( (
ph  /\  n  e.  Z )  ->  (  seq M (  +  ,  G ) `  n
)  e.  RR )
272adantr 274 . . . . . 6  |-  ( (
ph  /\  n  e.  Z )  ->  M  e.  ZZ )
28 eluzelz 9483 . . . . . . . 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 10374 . . . . 5  |-  ( (
ph  /\  n  e.  Z )  ->  ( M ... n )  e. 
Fin )
32 elfzuz 9964 . . . . . . . 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 11415 . . . 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 11347 . . . . 5  |-  ( (
ph  /\  n  e.  Z )  ->  sum_ k  e.  ( M ... n
) ( F `  k )  =  (  seq M (  +  ,  F ) `  n ) )
4544fveq2d 5498 . . . 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 7935 . . . . 5  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( abs `  ( F `  k
) )  e.  CC )
5147, 40, 50fsum3ser 11347 . . . 4  |-  ( (
ph  /\  n  e.  Z )  ->  sum_ k  e.  ( M ... n
) ( abs `  ( F `  k )
)  =  (  seq M (  +  ,  G ) `  n
) )
5236, 45, 513brtr3d 4018 . . 3  |-  ( (
ph  /\  n  e.  Z )  ->  ( abs `  (  seq M
(  +  ,  F
) `  n )
)  <_  (  seq M (  +  ,  G ) `  n
) )
5318, 52eqbrtrd 4009 . 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 11284 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 3987    |-> cmpt 4048   ` cfv 5196  (class class class)co 5850   CCcc 7759   RRcr 7760    + caddc 7764    <_ cle 7942   ZZcz 9199   ZZ>=cuz 9474   ...cfz 9952    seqcseq 10388   abscabs 10948    ~~> cli 11228   sum_csu 11303
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 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-mulrcl 7860  ax-addcom 7861  ax-mulcom 7862  ax-addass 7863  ax-mulass 7864  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-1rid 7868  ax-0id 7869  ax-rnegex 7870  ax-precex 7871  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-apti 7876  ax-pre-ltadd 7877  ax-pre-mulgt0 7878  ax-pre-mulext 7879  ax-arch 7880  ax-caucvg 7881
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 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-isom 5205  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-irdg 6346  df-frec 6367  df-1o 6392  df-oadd 6396  df-er 6509  df-en 6715  df-dom 6716  df-fin 6717  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-reap 8481  df-ap 8488  df-div 8577  df-inn 8866  df-2 8924  df-3 8925  df-4 8926  df-n0 9123  df-z 9200  df-uz 9475  df-q 9566  df-rp 9598  df-fz 9953  df-fzo 10086  df-seqfrec 10389  df-exp 10463  df-ihash 10697  df-cj 10793  df-re 10794  df-im 10795  df-rsqrt 10949  df-abs 10950  df-clim 11229  df-sumdc 11304
This theorem is referenced by:  eftlub  11640
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