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Theorem ser3le 10474
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 2733 . . . . . 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 8300 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( ( G `  k )  -  ( F `  k ) )  e.  RR )
6 fveq2 5496 . . . . . . . 8  |-  ( x  =  k  ->  ( G `  x )  =  ( G `  k ) )
7 fveq2 5496 . . . . . . . 8  |-  ( x  =  k  ->  ( F `  x )  =  ( F `  k ) )
86, 7oveq12d 5871 . . . . . . 7  |-  ( x  =  k  ->  (
( G `  x
)  -  ( F `
 x ) )  =  ( ( G `
 k )  -  ( F `  k ) ) )
9 eqid 2170 . . . . . . 7  |-  ( x  e.  _V  |->  ( ( G `  x )  -  ( F `  x ) ) )  =  ( x  e. 
_V  |->  ( ( G `
 x )  -  ( F `  x ) ) )
108, 9fvmptg 5572 . . . . . 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 412 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (
x  e.  _V  |->  ( ( G `  x
)  -  ( F `
 x ) ) ) `  k )  =  ( ( G `
 k )  -  ( F `  k ) ) )
1211, 5eqeltrd 2247 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (
x  e.  _V  |->  ( ( G `  x
)  -  ( F `
 x ) ) ) `  k )  e.  RR )
13 elfzuz 9977 . . . . . . 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 8454 . . . . . . 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 4017 . . . 4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  0  <_  ( ( x  e.  _V  |->  ( ( G `  x )  -  ( F `  x )
) ) `  k
) )
211, 12, 20ser3ge0 10473 . . 3  |-  ( ph  ->  0  <_  (  seq M (  +  , 
( x  e.  _V  |->  ( ( G `  x )  -  ( F `  x )
) ) ) `  N ) )
223recnd 7948 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( G `  k )  e.  CC )
234recnd 7948 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
241, 22, 23, 11ser3sub 10462 . . 3  |-  ( ph  ->  (  seq M (  +  ,  ( x  e.  _V  |->  ( ( G `  x )  -  ( F `  x ) ) ) ) `  N )  =  ( (  seq M (  +  ,  G ) `  N
)  -  (  seq M (  +  ,  F ) `  N
) ) )
2521, 24breqtrd 4015 . 2  |-  ( ph  ->  0  <_  ( (  seq M (  +  ,  G ) `  N
)  -  (  seq M (  +  ,  F ) `  N
) ) )
26 eqid 2170 . . . . 5  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
27 eluzel2 9492 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
281, 27syl 14 . . . . 5  |-  ( ph  ->  M  e.  ZZ )
2926, 28, 3serfre 10431 . . . 4  |-  ( ph  ->  seq M (  +  ,  G ) : ( ZZ>= `  M ) --> RR )
3029, 1ffvelrnd 5632 . . 3  |-  ( ph  ->  (  seq M (  +  ,  G ) `
 N )  e.  RR )
3126, 28, 4serfre 10431 . . . 4  |-  ( ph  ->  seq M (  +  ,  F ) : ( ZZ>= `  M ) --> RR )
3231, 1ffvelrnd 5632 . . 3  |-  ( ph  ->  (  seq M (  +  ,  F ) `
 N )  e.  RR )
3330, 32subge0d 8454 . 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 1348    e. wcel 2141   _Vcvv 2730   class class class wbr 3989    |-> cmpt 4050   ` cfv 5198  (class class class)co 5853   RRcr 7773   0cc0 7774    + caddc 7777    <_ cle 7955    - cmin 8090   ZZcz 9212   ZZ>=cuz 9487   ...cfz 9965    seqcseq 10401
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-fz 9966  df-fzo 10099  df-seqfrec 10402
This theorem is referenced by:  iserle  11305  cvgcmpub  11439
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