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Theorem serle 11979
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 3081 . . . . . 6  |-  k  e. 
_V
3 fveq2 5800 . . . . . . . 8  |-  ( x  =  k  ->  ( G `  x )  =  ( G `  k ) )
4 fveq2 5800 . . . . . . . 8  |-  ( x  =  k  ->  ( F `  x )  =  ( F `  k ) )
53, 4oveq12d 6219 . . . . . . 7  |-  ( x  =  k  ->  (
( G `  x
)  -  ( F `
 x ) )  =  ( ( G `
 k )  -  ( F `  k ) ) )
6 eqid 2454 . . . . . . 7  |-  ( x  e.  _V  |->  ( ( G `  x )  -  ( F `  x ) ) )  =  ( x  e. 
_V  |->  ( ( G `
 x )  -  ( F `  x ) ) )
7 ovex 6226 . . . . . . 7  |-  ( ( G `  k )  -  ( F `  k ) )  e. 
_V
85, 6, 7fvmpt 5884 . . . . . 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 9888 . . . . 5  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( ( G `  k )  -  ( F `  k ) )  e.  RR )
139, 12syl5eqel 2546 . . . 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 10041 . . . . . 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 4441 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  0  <_  ( ( x  e.  _V  |->  ( ( G `  x )  -  ( F `  x )
) ) `  k
) )
181, 13, 17serge0 11978 . . 3  |-  ( ph  ->  0  <_  (  seq M (  +  , 
( x  e.  _V  |->  ( ( G `  x )  -  ( F `  x )
) ) ) `  N ) )
1910recnd 9524 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( G `  k )  e.  CC )
2011recnd 9524 . . . 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 11967 . . 3  |-  ( ph  ->  (  seq M (  +  ,  ( x  e.  _V  |->  ( ( G `  x )  -  ( F `  x ) ) ) ) `  N )  =  ( (  seq M (  +  ,  G ) `  N
)  -  (  seq M (  +  ,  F ) `  N
) ) )
2318, 22breqtrd 4425 . 2  |-  ( ph  ->  0  <_  ( (  seq M (  +  ,  G ) `  N
)  -  (  seq M (  +  ,  F ) `  N
) ) )
24 readdcl 9477 . . . . 5  |-  ( ( k  e.  RR  /\  x  e.  RR )  ->  ( k  +  x
)  e.  RR )
2524adantl 466 . . . 4  |-  ( (
ph  /\  ( k  e.  RR  /\  x  e.  RR ) )  -> 
( k  +  x
)  e.  RR )
261, 10, 25seqcl 11944 . . 3  |-  ( ph  ->  (  seq M (  +  ,  G ) `
 N )  e.  RR )
271, 11, 25seqcl 11944 . . 3  |-  ( ph  ->  (  seq M (  +  ,  F ) `
 N )  e.  RR )
2826, 27subge0d 10041 . 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 369    = wceq 1370    e. wcel 1758   _Vcvv 3078   class class class wbr 4401    |-> cmpt 4459   ` cfv 5527  (class class class)co 6201   RRcr 9393   0cc0 9394    + caddc 9397    <_ cle 9531    - cmin 9707   ZZ>=cuz 10973   ...cfz 11555    seqcseq 11924
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-fzo 11667  df-seq 11925
This theorem is referenced by:  iserle  13256  cvgcmpub  13399  ioombl1lem4  21176  stirlinglem10  30027
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