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Theorem serile 9571
Description: Comparison of partial sums of two infinite series of reals. (Contributed by Jim Kingdon, 22-Aug-2021.)
Hypotheses
Ref Expression
serige0.1  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
serige0.2  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  RR )
serile.3  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( G `  k )  e.  RR )
serile.4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  <_  ( G `  k )
)
Assertion
Ref Expression
serile  |-  ( ph  ->  (  seq M (  +  ,  F ,  CC ) `  N )  <_  (  seq M
(  +  ,  G ,  CC ) `  N
) )
Distinct variable groups:    k, F    k, G    k, M    k, N    ph, k

Proof of Theorem serile
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 serige0.1 . . . 4  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
2 vex 2605 . . . . . 6  |-  k  e. 
_V
3 serile.3 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( G `  k )  e.  RR )
4 serige0.2 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  RR )
53, 4resubcld 7552 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( ( G `  k )  -  ( F `  k ) )  e.  RR )
6 fveq2 5209 . . . . . . . 8  |-  ( x  =  k  ->  ( G `  x )  =  ( G `  k ) )
7 fveq2 5209 . . . . . . . 8  |-  ( x  =  k  ->  ( F `  x )  =  ( F `  k ) )
86, 7oveq12d 5561 . . . . . . 7  |-  ( x  =  k  ->  (
( G `  x
)  -  ( F `
 x ) )  =  ( ( G `
 k )  -  ( F `  k ) ) )
9 eqid 2082 . . . . . . 7  |-  ( x  e.  _V  |->  ( ( G `  x )  -  ( F `  x ) ) )  =  ( x  e. 
_V  |->  ( ( G `
 x )  -  ( F `  x ) ) )
108, 9fvmptg 5280 . . . . . 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 405 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (
x  e.  _V  |->  ( ( G `  x
)  -  ( F `
 x ) ) ) `  k )  =  ( ( G `
 k )  -  ( F `  k ) ) )
1211, 5eqeltrd 2156 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (
x  e.  _V  |->  ( ( G `  x
)  -  ( F `
 x ) ) ) `  k )  e.  RR )
13 serile.4 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  <_  ( G `  k )
)
143, 4subge0d 7702 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( 0  <_  ( ( G `
 k )  -  ( F `  k ) )  <->  ( F `  k )  <_  ( G `  k )
) )
1513, 14mpbird 165 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  0  <_  ( ( G `  k
)  -  ( F `
 k ) ) )
1615, 11breqtrrd 3819 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  0  <_  ( ( x  e.  _V  |->  ( ( G `  x )  -  ( F `  x )
) ) `  k
) )
171, 12, 16serige0 9570 . . 3  |-  ( ph  ->  0  <_  (  seq M (  +  , 
( x  e.  _V  |->  ( ( G `  x )  -  ( F `  x )
) ) ,  CC ) `  N )
)
183recnd 7209 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( G `  k )  e.  CC )
194recnd 7209 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
201, 18, 19, 11isersub 9560 . . 3  |-  ( ph  ->  (  seq M (  +  ,  ( x  e.  _V  |->  ( ( G `  x )  -  ( F `  x ) ) ) ,  CC ) `  N )  =  ( (  seq M (  +  ,  G ,  CC ) `  N )  -  (  seq M
(  +  ,  F ,  CC ) `  N
) ) )
2117, 20breqtrd 3817 . 2  |-  ( ph  ->  0  <_  ( (  seq M (  +  ,  G ,  CC ) `  N )  -  (  seq M (  +  ,  F ,  CC ) `  N ) ) )
22 eluzel2 8705 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
231, 22syl 14 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
24 cnex 7159 . . . . . . 7  |-  CC  e.  _V
2524a1i 9 . . . . . 6  |-  ( ph  ->  CC  e.  _V )
26 ax-resscn 7130 . . . . . . 7  |-  RR  C_  CC
2726a1i 9 . . . . . 6  |-  ( ph  ->  RR  C_  CC )
28 readdcl 7161 . . . . . . 7  |-  ( ( k  e.  RR  /\  x  e.  RR )  ->  ( k  +  x
)  e.  RR )
2928adantl 271 . . . . . 6  |-  ( (
ph  /\  ( k  e.  RR  /\  x  e.  RR ) )  -> 
( k  +  x
)  e.  RR )
30 addcl 7160 . . . . . . 7  |-  ( ( k  e.  CC  /\  x  e.  CC )  ->  ( k  +  x
)  e.  CC )
3130adantl 271 . . . . . 6  |-  ( (
ph  /\  ( k  e.  CC  /\  x  e.  CC ) )  -> 
( k  +  x
)  e.  CC )
3223, 25, 27, 3, 29, 31iseqss 9541 . . . . 5  |-  ( ph  ->  seq M (  +  ,  G ,  RR )  =  seq M (  +  ,  G ,  CC ) )
3332fveq1d 5211 . . . 4  |-  ( ph  ->  (  seq M (  +  ,  G ,  RR ) `  N )  =  (  seq M
(  +  ,  G ,  CC ) `  N
) )
341, 3, 29iseqcl 9537 . . . 4  |-  ( ph  ->  (  seq M (  +  ,  G ,  RR ) `  N )  e.  RR )
3533, 34eqeltrrd 2157 . . 3  |-  ( ph  ->  (  seq M (  +  ,  G ,  CC ) `  N )  e.  RR )
3623, 25, 27, 4, 29, 31iseqss 9541 . . . . 5  |-  ( ph  ->  seq M (  +  ,  F ,  RR )  =  seq M (  +  ,  F ,  CC ) )
3736fveq1d 5211 . . . 4  |-  ( ph  ->  (  seq M (  +  ,  F ,  RR ) `  N )  =  (  seq M
(  +  ,  F ,  CC ) `  N
) )
381, 4, 29iseqcl 9537 . . . 4  |-  ( ph  ->  (  seq M (  +  ,  F ,  RR ) `  N )  e.  RR )
3937, 38eqeltrrd 2157 . . 3  |-  ( ph  ->  (  seq M (  +  ,  F ,  CC ) `  N )  e.  RR )
4035, 39subge0d 7702 . 2  |-  ( ph  ->  ( 0  <_  (
(  seq M (  +  ,  G ,  CC ) `  N )  -  (  seq M (  +  ,  F ,  CC ) `  N ) )  <->  (  seq M
(  +  ,  F ,  CC ) `  N
)  <_  (  seq M (  +  ,  G ,  CC ) `  N ) ) )
4121, 40mpbid 145 1  |-  ( ph  ->  (  seq M (  +  ,  F ,  CC ) `  N )  <_  (  seq M
(  +  ,  G ,  CC ) `  N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   _Vcvv 2602    C_ wss 2974   class class class wbr 3793    |-> cmpt 3847   ` cfv 4932  (class class class)co 5543   CCcc 7041   RRcr 7042   0cc0 7043    + caddc 7046    <_ cle 7216    - cmin 7346   ZZcz 8432   ZZ>=cuz 8700    seqcseq 9521
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-addass 7140  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-ltadd 7154
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-iord 4129  df-on 4131  df-ilim 4132  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-frec 6040  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-inn 8107  df-n0 8356  df-z 8433  df-uz 8701  df-fz 9106  df-fzo 9230  df-iseq 9522
This theorem is referenced by:  iserile  10318
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