ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  iserile Unicode version

Theorem iserile 10318
Description: Comparison of the limits of two infinite series. (Contributed by Jim Kingdon, 22-Aug-2021.)
Hypotheses
Ref Expression
clim2ser.1  |-  Z  =  ( ZZ>= `  M )
iserile.2  |-  ( ph  ->  M  e.  ZZ )
iserile.4  |-  ( ph  ->  seq M (  +  ,  F ,  CC ) 
~~>  A )
iserile.5  |-  ( ph  ->  seq M (  +  ,  G ,  CC ) 
~~>  B )
iserile.6  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  RR )
iserile.7  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  RR )
iserile.8  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  <_  ( G `  k
) )
Assertion
Ref Expression
iserile  |-  ( ph  ->  A  <_  B )
Distinct variable groups:    A, k    B, k    k, F    k, M    k, G    ph, k    k, Z

Proof of Theorem iserile
Dummy variables  j  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 clim2ser.1 . 2  |-  Z  =  ( ZZ>= `  M )
2 iserile.2 . 2  |-  ( ph  ->  M  e.  ZZ )
3 iserile.4 . 2  |-  ( ph  ->  seq M (  +  ,  F ,  CC ) 
~~>  A )
4 iserile.5 . 2  |-  ( ph  ->  seq M (  +  ,  G ,  CC ) 
~~>  B )
5 cnex 7159 . . . . . . 7  |-  CC  e.  _V
65a1i 9 . . . . . 6  |-  ( ph  ->  CC  e.  _V )
7 ax-resscn 7130 . . . . . . 7  |-  RR  C_  CC
87a1i 9 . . . . . 6  |-  ( ph  ->  RR  C_  CC )
91eleq2i 2146 . . . . . . 7  |-  ( k  e.  Z  <->  k  e.  ( ZZ>= `  M )
)
10 iserile.6 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  RR )
119, 10sylan2br 282 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  RR )
12 readdcl 7161 . . . . . . 7  |-  ( ( k  e.  RR  /\  x  e.  RR )  ->  ( k  +  x
)  e.  RR )
1312adantl 271 . . . . . 6  |-  ( (
ph  /\  ( k  e.  RR  /\  x  e.  RR ) )  -> 
( k  +  x
)  e.  RR )
14 addcl 7160 . . . . . . 7  |-  ( ( k  e.  CC  /\  x  e.  CC )  ->  ( k  +  x
)  e.  CC )
1514adantl 271 . . . . . 6  |-  ( (
ph  /\  ( k  e.  CC  /\  x  e.  CC ) )  -> 
( k  +  x
)  e.  CC )
162, 6, 8, 11, 13, 15iseqss 9541 . . . . 5  |-  ( ph  ->  seq M (  +  ,  F ,  RR )  =  seq M (  +  ,  F ,  CC ) )
1716adantr 270 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  seq M (  +  ,  F ,  RR )  =  seq M (  +  ,  F ,  CC ) )
1817fveq1d 5211 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  ,  F ,  RR ) `  j )  =  (  seq M (  +  ,  F ,  CC ) `  j )
)
191, 2, 10iserfre 9550 . . . 4  |-  ( ph  ->  seq M (  +  ,  F ,  RR ) : Z --> RR )
2019ffvelrnda 5334 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  ,  F ,  RR ) `  j )  e.  RR )
2118, 20eqeltrrd 2157 . 2  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  ,  F ,  CC ) `  j )  e.  RR )
22 iserile.7 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  RR )
239, 22sylan2br 282 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( G `  k )  e.  RR )
242, 6, 8, 23, 13, 15iseqss 9541 . . . . 5  |-  ( ph  ->  seq M (  +  ,  G ,  RR )  =  seq M (  +  ,  G ,  CC ) )
2524adantr 270 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  seq M (  +  ,  G ,  RR )  =  seq M (  +  ,  G ,  CC ) )
2625fveq1d 5211 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  ,  G ,  RR ) `  j )  =  (  seq M (  +  ,  G ,  CC ) `  j )
)
271, 2, 22iserfre 9550 . . . 4  |-  ( ph  ->  seq M (  +  ,  G ,  RR ) : Z --> RR )
2827ffvelrnda 5334 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  ,  G ,  RR ) `  j )  e.  RR )
2926, 28eqeltrrd 2157 . 2  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  ,  G ,  CC ) `  j )  e.  RR )
30 simpr 108 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  Z )
3130, 1syl6eleq 2172 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  ( ZZ>= `  M )
)
3211adantlr 461 . . 3  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  RR )
3323adantlr 461 . . 3  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( G `  k )  e.  RR )
34 simpll 496 . . . 4  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ph )
359biimpri 131 . . . . 5  |-  ( k  e.  ( ZZ>= `  M
)  ->  k  e.  Z )
3635adantl 271 . . . 4  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  k  e.  Z )
37 iserile.8 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  <_  ( G `  k
) )
3834, 36, 37syl2anc 403 . . 3  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  <_  ( G `  k )
)
3931, 32, 33, 38serile 9571 . 2  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  ,  F ,  CC ) `  j )  <_  (  seq M (  +  ,  G ,  CC ) `  j ) )
401, 2, 3, 4, 21, 29, 39climle 10310 1  |-  ( ph  ->  A  <_  B )
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   ` cfv 4932  (class class class)co 5543   CCcc 7041   RRcr 7042    + caddc 7046    <_ cle 7216   ZZcz 8432   ZZ>=cuz 8700    seqcseq 9521    ~~> cli 10255
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-mulrcl 7137  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-1rid 7145  ax-0id 7146  ax-rnegex 7147  ax-precex 7148  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-apti 7153  ax-pre-ltadd 7154  ax-pre-mulgt0 7155  ax-pre-mulext 7156  ax-arch 7157  ax-caucvg 7158
This theorem depends on definitions:  df-bi 115  df-dc 777  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-rmo 2357  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-if 3360  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-po 4059  df-iso 4060  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-reap 7742  df-ap 7749  df-div 7828  df-inn 8107  df-2 8165  df-3 8166  df-4 8167  df-n0 8356  df-z 8433  df-uz 8701  df-rp 8816  df-fz 9106  df-fzo 9230  df-iseq 9522  df-iexp 9573  df-cj 9867  df-re 9868  df-im 9869  df-rsqrt 10022  df-abs 10023  df-clim 10256
This theorem is referenced by:  iserige0  10319
  Copyright terms: Public domain W3C validator