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Theorem iserabs 12273
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 Mario Carneiro, 27-May-2014.)
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 fvex 5539 . . . . . 6  |-  ( ZZ>= `  M )  e.  _V
51, 4eqeltri 2353 . . . . 5  |-  Z  e. 
_V
65mptex 5746 . . . 4  |-  ( m  e.  Z  |->  ( abs `  (  seq  M (  +  ,  F ) `
 m ) ) )  e.  _V
76a1i 10 . . 3  |-  ( ph  ->  ( m  e.  Z  |->  ( abs `  (  seq  M (  +  ,  F ) `  m
) ) )  e. 
_V )
8 iserabs.6 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
91, 2, 8serf 11074 . . . 4  |-  ( ph  ->  seq  M (  +  ,  F ) : Z --> CC )
10 ffvelrn 5663 . . . 4  |-  ( (  seq  M (  +  ,  F ) : Z --> CC  /\  n  e.  Z )  ->  (  seq  M (  +  ,  F ) `  n
)  e.  CC )
119, 10sylan 457 . . 3  |-  ( (
ph  /\  n  e.  Z )  ->  (  seq  M (  +  ,  F ) `  n
)  e.  CC )
12 fveq2 5525 . . . . . 6  |-  ( m  =  n  ->  (  seq  M (  +  ,  F ) `  m
)  =  (  seq 
M (  +  ,  F ) `  n
) )
1312fveq2d 5529 . . . . 5  |-  ( m  =  n  ->  ( abs `  (  seq  M
(  +  ,  F
) `  m )
)  =  ( abs `  (  seq  M (  +  ,  F ) `
 n ) ) )
14 eqid 2283 . . . . 5  |-  ( m  e.  Z  |->  ( abs `  (  seq  M (  +  ,  F ) `
 m ) ) )  =  ( m  e.  Z  |->  ( abs `  (  seq  M (  +  ,  F ) `
 m ) ) )
15 fvex 5539 . . . . 5  |-  ( abs `  (  seq  M (  +  ,  F ) `
 n ) )  e.  _V
1613, 14, 15fvmpt 5602 . . . 4  |-  ( n  e.  Z  ->  (
( m  e.  Z  |->  ( abs `  (  seq  M (  +  ,  F ) `  m
) ) ) `  n )  =  ( abs `  (  seq 
M (  +  ,  F ) `  n
) ) )
1716adantl 452 . . 3  |-  ( (
ph  /\  n  e.  Z )  ->  (
( m  e.  Z  |->  ( abs `  (  seq  M (  +  ,  F ) `  m
) ) ) `  n )  =  ( abs `  (  seq 
M (  +  ,  F ) `  n
) ) )
181, 3, 7, 2, 11, 17climabs 12077 . 2  |-  ( ph  ->  ( m  e.  Z  |->  ( abs `  (  seq  M (  +  ,  F ) `  m
) ) )  ~~>  ( abs `  A ) )
19 iserabs.3 . 2  |-  ( ph  ->  seq  M (  +  ,  G )  ~~>  B )
2011abscld 11918 . . 3  |-  ( (
ph  /\  n  e.  Z )  ->  ( abs `  (  seq  M
(  +  ,  F
) `  n )
)  e.  RR )
2117, 20eqeltrd 2357 . 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 )
) )
238abscld 11918 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( F `  k ) )  e.  RR )
2422, 23eqeltrd 2357 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  RR )
251, 2, 24serfre 11075 . . 3  |-  ( ph  ->  seq  M (  +  ,  G ) : Z --> RR )
26 ffvelrn 5663 . . 3  |-  ( (  seq  M (  +  ,  G ) : Z --> RR  /\  n  e.  Z )  ->  (  seq  M (  +  ,  G ) `  n
)  e.  RR )
2725, 26sylan 457 . 2  |-  ( (
ph  /\  n  e.  Z )  ->  (  seq  M (  +  ,  G ) `  n
)  e.  RR )
28 simpr 447 . . . . 5  |-  ( (
ph  /\  n  e.  Z )  ->  n  e.  Z )
2928, 1syl6eleq 2373 . . . 4  |-  ( (
ph  /\  n  e.  Z )  ->  n  e.  ( ZZ>= `  M )
)
30 elfzuz 10794 . . . . . . 7  |-  ( k  e.  ( M ... n )  ->  k  e.  ( ZZ>= `  M )
)
3130, 1syl6eleqr 2374 . . . . . 6  |-  ( k  e.  ( M ... n )  ->  k  e.  Z )
3231, 8sylan2 460 . . . . 5  |-  ( (
ph  /\  k  e.  ( M ... n ) )  ->  ( F `  k )  e.  CC )
3332adantlr 695 . . . 4  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  ( M ... n
) )  ->  ( F `  k )  e.  CC )
3431, 22sylan2 460 . . . . 5  |-  ( (
ph  /\  k  e.  ( M ... n ) )  ->  ( G `  k )  =  ( abs `  ( F `
 k ) ) )
3534adantlr 695 . . . 4  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  ( M ... n
) )  ->  ( G `  k )  =  ( abs `  ( F `  k )
) )
3629, 33, 35seqabs 12272 . . 3  |-  ( (
ph  /\  n  e.  Z )  ->  ( abs `  (  seq  M
(  +  ,  F
) `  n )
)  <_  (  seq  M (  +  ,  G
) `  n )
)
3717, 36eqbrtrd 4043 . 2  |-  ( (
ph  /\  n  e.  Z )  ->  (
( m  e.  Z  |->  ( abs `  (  seq  M (  +  ,  F ) `  m
) ) ) `  n )  <_  (  seq  M (  +  ,  G ) `  n
) )
381, 2, 18, 19, 21, 27, 37climle 12113 1  |-  ( ph  ->  ( abs `  A
)  <_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   class class class wbr 4023    e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736    + caddc 8740    <_ cle 8868   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782    seq cseq 11046   abscabs 11719    ~~> cli 11958
This theorem is referenced by:  eftlub  12389  abelthlem7  19814
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159
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