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Theorem iserabs 12521
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 5682 . . . . . 6  |-  ( ZZ>= `  M )  e.  _V
51, 4eqeltri 2457 . . . . 5  |-  Z  e. 
_V
65mptex 5905 . . . 4  |-  ( m  e.  Z  |->  ( abs `  (  seq  M (  +  ,  F ) `
 m ) ) )  e.  _V
76a1i 11 . . 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 11278 . . . 4  |-  ( ph  ->  seq  M (  +  ,  F ) : Z --> CC )
109ffvelrnda 5809 . . 3  |-  ( (
ph  /\  n  e.  Z )  ->  (  seq  M (  +  ,  F ) `  n
)  e.  CC )
11 fveq2 5668 . . . . . 6  |-  ( m  =  n  ->  (  seq  M (  +  ,  F ) `  m
)  =  (  seq 
M (  +  ,  F ) `  n
) )
1211fveq2d 5672 . . . . 5  |-  ( m  =  n  ->  ( abs `  (  seq  M
(  +  ,  F
) `  m )
)  =  ( abs `  (  seq  M (  +  ,  F ) `
 n ) ) )
13 eqid 2387 . . . . 5  |-  ( m  e.  Z  |->  ( abs `  (  seq  M (  +  ,  F ) `
 m ) ) )  =  ( m  e.  Z  |->  ( abs `  (  seq  M (  +  ,  F ) `
 m ) ) )
14 fvex 5682 . . . . 5  |-  ( abs `  (  seq  M (  +  ,  F ) `
 n ) )  e.  _V
1512, 13, 14fvmpt 5745 . . . 4  |-  ( n  e.  Z  ->  (
( m  e.  Z  |->  ( abs `  (  seq  M (  +  ,  F ) `  m
) ) ) `  n )  =  ( abs `  (  seq 
M (  +  ,  F ) `  n
) ) )
1615adantl 453 . . 3  |-  ( (
ph  /\  n  e.  Z )  ->  (
( m  e.  Z  |->  ( abs `  (  seq  M (  +  ,  F ) `  m
) ) ) `  n )  =  ( abs `  (  seq 
M (  +  ,  F ) `  n
) ) )
171, 3, 7, 2, 10, 16climabs 12324 . 2  |-  ( ph  ->  ( m  e.  Z  |->  ( abs `  (  seq  M (  +  ,  F ) `  m
) ) )  ~~>  ( abs `  A ) )
18 iserabs.3 . 2  |-  ( ph  ->  seq  M (  +  ,  G )  ~~>  B )
1910abscld 12165 . . 3  |-  ( (
ph  /\  n  e.  Z )  ->  ( abs `  (  seq  M
(  +  ,  F
) `  n )
)  e.  RR )
2016, 19eqeltrd 2461 . 2  |-  ( (
ph  /\  n  e.  Z )  ->  (
( m  e.  Z  |->  ( abs `  (  seq  M (  +  ,  F ) `  m
) ) ) `  n )  e.  RR )
21 iserabs.7 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( abs `  ( F `  k )
) )
228abscld 12165 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( abs `  ( F `  k ) )  e.  RR )
2321, 22eqeltrd 2461 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  RR )
241, 2, 23serfre 11279 . . 3  |-  ( ph  ->  seq  M (  +  ,  G ) : Z --> RR )
2524ffvelrnda 5809 . 2  |-  ( (
ph  /\  n  e.  Z )  ->  (  seq  M (  +  ,  G ) `  n
)  e.  RR )
26 simpr 448 . . . . 5  |-  ( (
ph  /\  n  e.  Z )  ->  n  e.  Z )
2726, 1syl6eleq 2477 . . . 4  |-  ( (
ph  /\  n  e.  Z )  ->  n  e.  ( ZZ>= `  M )
)
28 elfzuz 10987 . . . . . . 7  |-  ( k  e.  ( M ... n )  ->  k  e.  ( ZZ>= `  M )
)
2928, 1syl6eleqr 2478 . . . . . 6  |-  ( k  e.  ( M ... n )  ->  k  e.  Z )
3029, 8sylan2 461 . . . . 5  |-  ( (
ph  /\  k  e.  ( M ... n ) )  ->  ( F `  k )  e.  CC )
3130adantlr 696 . . . 4  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  ( M ... n
) )  ->  ( F `  k )  e.  CC )
3229, 21sylan2 461 . . . . 5  |-  ( (
ph  /\  k  e.  ( M ... n ) )  ->  ( G `  k )  =  ( abs `  ( F `
 k ) ) )
3332adantlr 696 . . . 4  |-  ( ( ( ph  /\  n  e.  Z )  /\  k  e.  ( M ... n
) )  ->  ( G `  k )  =  ( abs `  ( F `  k )
) )
3427, 31, 33seqabs 12520 . . 3  |-  ( (
ph  /\  n  e.  Z )  ->  ( abs `  (  seq  M
(  +  ,  F
) `  n )
)  <_  (  seq  M (  +  ,  G
) `  n )
)
3516, 34eqbrtrd 4173 . 2  |-  ( (
ph  /\  n  e.  Z )  ->  (
( m  e.  Z  |->  ( abs `  (  seq  M (  +  ,  F ) `  m
) ) ) `  n )  <_  (  seq  M (  +  ,  G ) `  n
) )
361, 2, 17, 18, 20, 25, 35climle 12360 1  |-  ( ph  ->  ( abs `  A
)  <_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2899   class class class wbr 4153    e. cmpt 4207   ` cfv 5394  (class class class)co 6020   CCcc 8921   RRcr 8922    + caddc 8926    <_ cle 9054   ZZcz 10214   ZZ>=cuz 10420   ...cfz 10975    seq cseq 11250   abscabs 11966    ~~> cli 12205
This theorem is referenced by:  eftlub  12637  abelthlem7  20221
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-pm 6957  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-oi 7412  df-card 7759  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-n0 10154  df-z 10215  df-uz 10421  df-rp 10545  df-fz 10976  df-fzo 11066  df-fl 11129  df-seq 11251  df-exp 11310  df-hash 11546  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-clim 12209  df-rlim 12210  df-sum 12407
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