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Theorem serle 11117
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 2804 . . . . . 6  |-  k  e. 
_V
3 fveq2 5541 . . . . . . . 8  |-  ( x  =  k  ->  ( G `  x )  =  ( G `  k ) )
4 fveq2 5541 . . . . . . . 8  |-  ( x  =  k  ->  ( F `  x )  =  ( F `  k ) )
53, 4oveq12d 5892 . . . . . . 7  |-  ( x  =  k  ->  (
( G `  x
)  -  ( F `
 x ) )  =  ( ( G `
 k )  -  ( F `  k ) ) )
6 eqid 2296 . . . . . . 7  |-  ( x  e.  _V  |->  ( ( G `  x )  -  ( F `  x ) ) )  =  ( x  e. 
_V  |->  ( ( G `
 x )  -  ( F `  x ) ) )
7 ovex 5899 . . . . . . 7  |-  ( ( G `  k )  -  ( F `  k ) )  e. 
_V
85, 6, 7fvmpt 5618 . . . . . 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 9227 . . . . 5  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( ( G `  k )  -  ( F `  k ) )  e.  RR )
139, 12syl5eqel 2380 . . . 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 9378 . . . . . 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 4079 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  0  <_  ( ( x  e.  _V  |->  ( ( G `  x )  -  ( F `  x )
) ) `  k
) )
181, 13, 17serge0 11116 . . 3  |-  ( ph  ->  0  <_  (  seq  M (  +  ,  ( x  e.  _V  |->  ( ( G `  x
)  -  ( F `
 x ) ) ) ) `  N
) )
1910recnd 8877 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( G `  k )  e.  CC )
2011recnd 8877 . . . 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 11105 . . 3  |-  ( ph  ->  (  seq  M (  +  ,  ( x  e.  _V  |->  ( ( G `  x )  -  ( F `  x ) ) ) ) `  N )  =  ( (  seq 
M (  +  ,  G ) `  N
)  -  (  seq 
M (  +  ,  F ) `  N
) ) )
2318, 22breqtrd 4063 . 2  |-  ( ph  ->  0  <_  ( (  seq  M (  +  ,  G ) `  N
)  -  (  seq 
M (  +  ,  F ) `  N
) ) )
24 readdcl 8836 . . . . 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 11082 . . 3  |-  ( ph  ->  (  seq  M (  +  ,  G ) `
 N )  e.  RR )
271, 11, 25seqcl 11082 . . 3  |-  ( ph  ->  (  seq  M (  +  ,  F ) `
 N )  e.  RR )
2826, 27subge0d 9378 . 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 1632    e. wcel 1696   _Vcvv 2801   class class class wbr 4039    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753    + caddc 8756    <_ cle 8884    - cmin 9053   ZZ>=cuz 10246   ...cfz 10798    seq cseq 11062
This theorem is referenced by:  iserle  12149  cvgcmpub  12291  ioombl1lem4  18934  stirlinglem10  27935
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063
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