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Theorem ser3le 10417
Description: Comparison of partial sums of two infinite series of reals. (Contributed by NM, 27-Dec-2005.) (Revised by Jim Kingdon, 23-Apr-2023.)
Hypotheses
Ref Expression
ser3ge0.1  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
ser3ge0.2  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  RR )
ser3le.3  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( G `  k )  e.  RR )
serle.4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  <_  ( G `  k )
)
Assertion
Ref Expression
ser3le  |-  ( 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 ser3le
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ser3ge0.1 . . . 4  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
2 vex 2715 . . . . . 6  |-  k  e. 
_V
3 ser3le.3 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( G `  k )  e.  RR )
4 ser3ge0.2 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  RR )
53, 4resubcld 8256 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( ( G `  k )  -  ( F `  k ) )  e.  RR )
6 fveq2 5468 . . . . . . . 8  |-  ( x  =  k  ->  ( G `  x )  =  ( G `  k ) )
7 fveq2 5468 . . . . . . . 8  |-  ( x  =  k  ->  ( F `  x )  =  ( F `  k ) )
86, 7oveq12d 5842 . . . . . . 7  |-  ( x  =  k  ->  (
( G `  x
)  -  ( F `
 x ) )  =  ( ( G `
 k )  -  ( F `  k ) ) )
9 eqid 2157 . . . . . . 7  |-  ( x  e.  _V  |->  ( ( G `  x )  -  ( F `  x ) ) )  =  ( x  e. 
_V  |->  ( ( G `
 x )  -  ( F `  x ) ) )
108, 9fvmptg 5544 . . . . . 6  |-  ( ( k  e.  _V  /\  ( ( G `  k )  -  ( F `  k )
)  e.  RR )  ->  ( ( x  e.  _V  |->  ( ( G `  x )  -  ( F `  x ) ) ) `
 k )  =  ( ( G `  k )  -  ( F `  k )
) )
112, 5, 10sylancr 411 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (
x  e.  _V  |->  ( ( G `  x
)  -  ( F `
 x ) ) ) `  k )  =  ( ( G `
 k )  -  ( F `  k ) ) )
1211, 5eqeltrd 2234 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (
x  e.  _V  |->  ( ( G `  x
)  -  ( F `
 x ) ) ) `  k )  e.  RR )
13 elfzuz 9924 . . . . . . 7  |-  ( k  e.  ( M ... N )  ->  k  e.  ( ZZ>= `  M )
)
14 serle.4 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  <_  ( G `  k )
)
1513, 14sylan2 284 . . . . . 6  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( F `  k )  <_  ( G `  k )
)
163, 4subge0d 8410 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( 0  <_  ( ( G `
 k )  -  ( F `  k ) )  <->  ( F `  k )  <_  ( G `  k )
) )
1713, 16sylan2 284 . . . . . 6  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( 0  <_  ( ( G `
 k )  -  ( F `  k ) )  <->  ( F `  k )  <_  ( G `  k )
) )
1815, 17mpbird 166 . . . . 5  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  0  <_  ( ( G `  k
)  -  ( F `
 k ) ) )
1913, 11sylan2 284 . . . . 5  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( (
x  e.  _V  |->  ( ( G `  x
)  -  ( F `
 x ) ) ) `  k )  =  ( ( G `
 k )  -  ( F `  k ) ) )
2018, 19breqtrrd 3992 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  0  <_  ( ( x  e.  _V  |->  ( ( G `  x )  -  ( F `  x )
) ) `  k
) )
211, 12, 20ser3ge0 10416 . . 3  |-  ( ph  ->  0  <_  (  seq M (  +  , 
( x  e.  _V  |->  ( ( G `  x )  -  ( F `  x )
) ) ) `  N ) )
223recnd 7906 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( G `  k )  e.  CC )
234recnd 7906 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
241, 22, 23, 11ser3sub 10405 . . 3  |-  ( ph  ->  (  seq M (  +  ,  ( x  e.  _V  |->  ( ( G `  x )  -  ( F `  x ) ) ) ) `  N )  =  ( (  seq M (  +  ,  G ) `  N
)  -  (  seq M (  +  ,  F ) `  N
) ) )
2521, 24breqtrd 3990 . 2  |-  ( ph  ->  0  <_  ( (  seq M (  +  ,  G ) `  N
)  -  (  seq M (  +  ,  F ) `  N
) ) )
26 eqid 2157 . . . . 5  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
27 eluzel2 9444 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
281, 27syl 14 . . . . 5  |-  ( ph  ->  M  e.  ZZ )
2926, 28, 3serfre 10374 . . . 4  |-  ( ph  ->  seq M (  +  ,  G ) : ( ZZ>= `  M ) --> RR )
3029, 1ffvelrnd 5603 . . 3  |-  ( ph  ->  (  seq M (  +  ,  G ) `
 N )  e.  RR )
3126, 28, 4serfre 10374 . . . 4  |-  ( ph  ->  seq M (  +  ,  F ) : ( ZZ>= `  M ) --> RR )
3231, 1ffvelrnd 5603 . . 3  |-  ( ph  ->  (  seq M (  +  ,  F ) `
 N )  e.  RR )
3330, 32subge0d 8410 . 2  |-  ( ph  ->  ( 0  <_  (
(  seq M (  +  ,  G ) `  N )  -  (  seq M (  +  ,  F ) `  N
) )  <->  (  seq M (  +  ,  F ) `  N
)  <_  (  seq M (  +  ,  G ) `  N
) ) )
3425, 33mpbid 146 1  |-  ( ph  ->  (  seq M (  +  ,  F ) `
 N )  <_ 
(  seq M (  +  ,  G ) `  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1335    e. wcel 2128   _Vcvv 2712   class class class wbr 3965    |-> cmpt 4025   ` cfv 5170  (class class class)co 5824   RRcr 7731   0cc0 7732    + caddc 7735    <_ cle 7913    - cmin 8046   ZZcz 9167   ZZ>=cuz 9439   ...cfz 9912    seqcseq 10344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-addcom 7832  ax-addass 7834  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-0id 7840  ax-rnegex 7841  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-ltadd 7848
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-frec 6338  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-inn 8834  df-n0 9091  df-z 9168  df-uz 9440  df-fz 9913  df-fzo 10042  df-seqfrec 10345
This theorem is referenced by:  iserle  11239  cvgcmpub  11373
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