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Theorem serle 11101
Description: Comparison of partial sums of two infinite series of reals. (Contributed by NM, 27-Dec-2005.) (Revised by Mario Carneiro, 27-May-2014.)
Hypotheses
Ref Expression
serge0.1  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
serge0.2  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( F `  k )  e.  RR )
serle.3  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( G `  k )  e.  RR )
serle.4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( F `  k )  <_  ( G `  k )
)
Assertion
Ref Expression
serle  |-  ( ph  ->  (  seq  M (  +  ,  F ) `
 N )  <_ 
(  seq  M (  +  ,  G ) `  N ) )
Distinct variable groups:    k, F    k, G    k, M    k, N    ph, k

Proof of Theorem serle
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 serge0.1 . . . 4  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
2 vex 2791 . . . . . 6  |-  k  e. 
_V
3 fveq2 5525 . . . . . . . 8  |-  ( x  =  k  ->  ( G `  x )  =  ( G `  k ) )
4 fveq2 5525 . . . . . . . 8  |-  ( x  =  k  ->  ( F `  x )  =  ( F `  k ) )
53, 4oveq12d 5876 . . . . . . 7  |-  ( x  =  k  ->  (
( G `  x
)  -  ( F `
 x ) )  =  ( ( G `
 k )  -  ( F `  k ) ) )
6 eqid 2283 . . . . . . 7  |-  ( x  e.  _V  |->  ( ( G `  x )  -  ( F `  x ) ) )  =  ( x  e. 
_V  |->  ( ( G `
 x )  -  ( F `  x ) ) )
7 ovex 5883 . . . . . . 7  |-  ( ( G `  k )  -  ( F `  k ) )  e. 
_V
85, 6, 7fvmpt 5602 . . . . . 6  |-  ( k  e.  _V  ->  (
( x  e.  _V  |->  ( ( G `  x )  -  ( F `  x )
) ) `  k
)  =  ( ( G `  k )  -  ( F `  k ) ) )
92, 8ax-mp 8 . . . . 5  |-  ( ( x  e.  _V  |->  ( ( G `  x
)  -  ( F `
 x ) ) ) `  k )  =  ( ( G `
 k )  -  ( F `  k ) )
10 serle.3 . . . . . 6  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( G `  k )  e.  RR )
11 serge0.2 . . . . . 6  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( F `  k )  e.  RR )
1210, 11resubcld 9211 . . . . 5  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( ( G `  k )  -  ( F `  k ) )  e.  RR )
139, 12syl5eqel 2367 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( (
x  e.  _V  |->  ( ( G `  x
)  -  ( F `
 x ) ) ) `  k )  e.  RR )
14 serle.4 . . . . . 6  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( F `  k )  <_  ( G `  k )
)
1510, 11subge0d 9362 . . . . . 6  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( 0  <_  ( ( G `
 k )  -  ( F `  k ) )  <->  ( F `  k )  <_  ( G `  k )
) )
1614, 15mpbird 223 . . . . 5  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  0  <_  ( ( G `  k
)  -  ( F `
 k ) ) )
1716, 9syl6breqr 4063 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  0  <_  ( ( x  e.  _V  |->  ( ( G `  x )  -  ( F `  x )
) ) `  k
) )
181, 13, 17serge0 11100 . . 3  |-  ( ph  ->  0  <_  (  seq  M (  +  ,  ( x  e.  _V  |->  ( ( G `  x
)  -  ( F `
 x ) ) ) ) `  N
) )
1910recnd 8861 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( G `  k )  e.  CC )
2011recnd 8861 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( F `  k )  e.  CC )
219a1i 10 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( (
x  e.  _V  |->  ( ( G `  x
)  -  ( F `
 x ) ) ) `  k )  =  ( ( G `
 k )  -  ( F `  k ) ) )
221, 19, 20, 21sersub 11089 . . 3  |-  ( ph  ->  (  seq  M (  +  ,  ( x  e.  _V  |->  ( ( G `  x )  -  ( F `  x ) ) ) ) `  N )  =  ( (  seq 
M (  +  ,  G ) `  N
)  -  (  seq 
M (  +  ,  F ) `  N
) ) )
2318, 22breqtrd 4047 . 2  |-  ( ph  ->  0  <_  ( (  seq  M (  +  ,  G ) `  N
)  -  (  seq 
M (  +  ,  F ) `  N
) ) )
24 readdcl 8820 . . . . 5  |-  ( ( k  e.  RR  /\  x  e.  RR )  ->  ( k  +  x
)  e.  RR )
2524adantl 452 . . . 4  |-  ( (
ph  /\  ( k  e.  RR  /\  x  e.  RR ) )  -> 
( k  +  x
)  e.  RR )
261, 10, 25seqcl 11066 . . 3  |-  ( ph  ->  (  seq  M (  +  ,  G ) `
 N )  e.  RR )
271, 11, 25seqcl 11066 . . 3  |-  ( ph  ->  (  seq  M (  +  ,  F ) `
 N )  e.  RR )
2826, 27subge0d 9362 . 2  |-  ( ph  ->  ( 0  <_  (
(  seq  M (  +  ,  G ) `  N )  -  (  seq  M (  +  ,  F ) `  N
) )  <->  (  seq  M (  +  ,  F
) `  N )  <_  (  seq  M (  +  ,  G ) `
 N ) ) )
2923, 28mpbid 201 1  |-  ( ph  ->  (  seq  M (  +  ,  F ) `
 N )  <_ 
(  seq  M (  +  ,  G ) `  N ) )
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   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737    + caddc 8740    <_ cle 8868    - cmin 9037   ZZ>=cuz 10230   ...cfz 10782    seq cseq 11046
This theorem is referenced by:  iserle  12133  cvgcmpub  12275  ioombl1lem4  18918  stirlinglem10  27832
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  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
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-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-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-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-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047
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