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Theorem serle 12088
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 3054 . . . . . 6  |-  k  e. 
_V
3 fveq2 5791 . . . . . . . 8  |-  ( x  =  k  ->  ( G `  x )  =  ( G `  k ) )
4 fveq2 5791 . . . . . . . 8  |-  ( x  =  k  ->  ( F `  x )  =  ( F `  k ) )
53, 4oveq12d 6236 . . . . . . 7  |-  ( x  =  k  ->  (
( G `  x
)  -  ( F `
 x ) )  =  ( ( G `
 k )  -  ( F `  k ) ) )
6 eqid 2396 . . . . . . 7  |-  ( x  e.  _V  |->  ( ( G `  x )  -  ( F `  x ) ) )  =  ( x  e. 
_V  |->  ( ( G `
 x )  -  ( F `  x ) ) )
7 ovex 6246 . . . . . . 7  |-  ( ( G `  k )  -  ( F `  k ) )  e. 
_V
85, 6, 7fvmpt 5874 . . . . . 6  |-  ( k  e.  _V  ->  (
( x  e.  _V  |->  ( ( G `  x )  -  ( F `  x )
) ) `  k
)  =  ( ( G `  k )  -  ( F `  k ) ) )
92, 8ax-mp 5 . . . . 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 9927 . . . . 5  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( ( G `  k )  -  ( F `  k ) )  e.  RR )
139, 12syl5eqel 2488 . . . 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 10081 . . . . . 6  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( 0  <_  ( ( G `
 k )  -  ( F `  k ) )  <->  ( F `  k )  <_  ( G `  k )
) )
1614, 15mpbird 232 . . . . 5  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  0  <_  ( ( G `  k
)  -  ( F `
 k ) ) )
1716, 9syl6breqr 4424 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  0  <_  ( ( x  e.  _V  |->  ( ( G `  x )  -  ( F `  x )
) ) `  k
) )
181, 13, 17serge0 12087 . . 3  |-  ( ph  ->  0  <_  (  seq M (  +  , 
( x  e.  _V  |->  ( ( G `  x )  -  ( F `  x )
) ) ) `  N ) )
1910recnd 9555 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( G `  k )  e.  CC )
2011recnd 9555 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( F `  k )  e.  CC )
219a1i 11 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( (
x  e.  _V  |->  ( ( G `  x
)  -  ( F `
 x ) ) ) `  k )  =  ( ( G `
 k )  -  ( F `  k ) ) )
221, 19, 20, 21sersub 12076 . . 3  |-  ( ph  ->  (  seq M (  +  ,  ( x  e.  _V  |->  ( ( G `  x )  -  ( F `  x ) ) ) ) `  N )  =  ( (  seq M (  +  ,  G ) `  N
)  -  (  seq M (  +  ,  F ) `  N
) ) )
2318, 22breqtrd 4408 . 2  |-  ( ph  ->  0  <_  ( (  seq M (  +  ,  G ) `  N
)  -  (  seq M (  +  ,  F ) `  N
) ) )
24 readdcl 9508 . . . . 5  |-  ( ( k  e.  RR  /\  x  e.  RR )  ->  ( k  +  x
)  e.  RR )
2524adantl 464 . . . 4  |-  ( (
ph  /\  ( k  e.  RR  /\  x  e.  RR ) )  -> 
( k  +  x
)  e.  RR )
261, 10, 25seqcl 12053 . . 3  |-  ( ph  ->  (  seq M (  +  ,  G ) `
 N )  e.  RR )
271, 11, 25seqcl 12053 . . 3  |-  ( ph  ->  (  seq M (  +  ,  F ) `
 N )  e.  RR )
2826, 27subge0d 10081 . 2  |-  ( ph  ->  ( 0  <_  (
(  seq M (  +  ,  G ) `  N )  -  (  seq M (  +  ,  F ) `  N
) )  <->  (  seq M (  +  ,  F ) `  N
)  <_  (  seq M (  +  ,  G ) `  N
) ) )
2923, 28mpbid 210 1  |-  ( ph  ->  (  seq M (  +  ,  F ) `
 N )  <_ 
(  seq M (  +  ,  G ) `  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1836   _Vcvv 3051   class class class wbr 4384    |-> cmpt 4442   ` cfv 5513  (class class class)co 6218   RRcr 9424   0cc0 9425    + caddc 9428    <_ cle 9562    - cmin 9740   ZZ>=cuz 11023   ...cfz 11615    seqcseq 12033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513  ax-cnex 9481  ax-resscn 9482  ax-1cn 9483  ax-icn 9484  ax-addcl 9485  ax-addrcl 9486  ax-mulcl 9487  ax-mulrcl 9488  ax-mulcom 9489  ax-addass 9490  ax-mulass 9491  ax-distr 9492  ax-i2m1 9493  ax-1ne0 9494  ax-1rid 9495  ax-rnegex 9496  ax-rrecex 9497  ax-cnre 9498  ax-pre-lttri 9499  ax-pre-lttrn 9500  ax-pre-ltadd 9501  ax-pre-mulgt0 9502
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-nel 2594  df-ral 2751  df-rex 2752  df-reu 2753  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4181  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-tr 4478  df-eprel 4722  df-id 4726  df-po 4731  df-so 4732  df-fr 4769  df-we 4771  df-ord 4812  df-on 4813  df-lim 4814  df-suc 4815  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6180  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-om 6622  df-1st 6721  df-2nd 6722  df-recs 6982  df-rdg 7016  df-er 7251  df-en 7458  df-dom 7459  df-sdom 7460  df-pnf 9563  df-mnf 9564  df-xr 9565  df-ltxr 9566  df-le 9567  df-sub 9742  df-neg 9743  df-nn 10475  df-n0 10735  df-z 10804  df-uz 11024  df-fz 11616  df-fzo 11740  df-seq 12034
This theorem is referenced by:  iserle  13507  cvgcmpub  13656  ioombl1lem4  22079  stirlinglem10  32070
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