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

Theorem cvgratnnlembern 11544
Description: Lemma for cvgratnn 11552. Upper bound for a geometric progression of positive ratio less than one. (Contributed by Jim Kingdon, 24-Nov-2022.)
Hypotheses
Ref Expression
cvgratnnlembern.3  |-  ( ph  ->  A  e.  RR )
cvgratnnlembern.4  |-  ( ph  ->  A  <  1 )
cvgratnnlembern.gt0  |-  ( ph  ->  0  <  A )
cvgratnnlembern.m  |-  ( ph  ->  M  e.  NN )
Assertion
Ref Expression
cvgratnnlembern  |-  ( ph  ->  ( A ^ M
)  <  ( (
1  /  ( ( 1  /  A )  -  1 ) )  /  M ) )

Proof of Theorem cvgratnnlembern
StepHypRef Expression
1 cvgratnnlembern.3 . . . . . . . . 9  |-  ( ph  ->  A  e.  RR )
2 cvgratnnlembern.gt0 . . . . . . . . . 10  |-  ( ph  ->  0  <  A )
31, 2gt0ap0d 8599 . . . . . . . . 9  |-  ( ph  ->  A #  0 )
41, 3rerecclapd 8804 . . . . . . . 8  |-  ( ph  ->  ( 1  /  A
)  e.  RR )
5 1red 7985 . . . . . . . 8  |-  ( ph  ->  1  e.  RR )
64, 5resubcld 8351 . . . . . . 7  |-  ( ph  ->  ( ( 1  /  A )  -  1 )  e.  RR )
7 cvgratnnlembern.m . . . . . . . 8  |-  ( ph  ->  M  e.  NN )
87nnred 8945 . . . . . . 7  |-  ( ph  ->  M  e.  RR )
96, 8remulcld 8001 . . . . . 6  |-  ( ph  ->  ( ( ( 1  /  A )  - 
1 )  x.  M
)  e.  RR )
109recnd 7999 . . . . 5  |-  ( ph  ->  ( ( ( 1  /  A )  - 
1 )  x.  M
)  e.  CC )
11 cvgratnnlembern.4 . . . . . . . . . 10  |-  ( ph  ->  A  <  1 )
121, 2elrpd 9706 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  RR+ )
1312reclt1d 9723 . . . . . . . . . 10  |-  ( ph  ->  ( A  <  1  <->  1  <  ( 1  /  A ) ) )
1411, 13mpbid 147 . . . . . . . . 9  |-  ( ph  ->  1  <  ( 1  /  A ) )
155, 4posdifd 8502 . . . . . . . . 9  |-  ( ph  ->  ( 1  <  (
1  /  A )  <->  0  <  ( ( 1  /  A )  -  1 ) ) )
1614, 15mpbid 147 . . . . . . . 8  |-  ( ph  ->  0  <  ( ( 1  /  A )  -  1 ) )
176, 16elrpd 9706 . . . . . . 7  |-  ( ph  ->  ( ( 1  /  A )  -  1 )  e.  RR+ )
187nnrpd 9707 . . . . . . 7  |-  ( ph  ->  M  e.  RR+ )
1917, 18rpmulcld 9726 . . . . . 6  |-  ( ph  ->  ( ( ( 1  /  A )  - 
1 )  x.  M
)  e.  RR+ )
2019rpap0d 9715 . . . . 5  |-  ( ph  ->  ( ( ( 1  /  A )  - 
1 )  x.  M
) #  0 )
2110, 20recrecapd 8755 . . . 4  |-  ( ph  ->  ( 1  /  (
1  /  ( ( ( 1  /  A
)  -  1 )  x.  M ) ) )  =  ( ( ( 1  /  A
)  -  1 )  x.  M ) )
229, 5readdcld 8000 . . . . 5  |-  ( ph  ->  ( ( ( ( 1  /  A )  -  1 )  x.  M )  +  1 )  e.  RR )
237nnnn0d 9242 . . . . . . 7  |-  ( ph  ->  M  e.  NN0 )
241, 23reexpcld 10684 . . . . . 6  |-  ( ph  ->  ( A ^ M
)  e.  RR )
251recnd 7999 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
267nnzd 9387 . . . . . . 7  |-  ( ph  ->  M  e.  ZZ )
2725, 3, 26expap0d 10673 . . . . . 6  |-  ( ph  ->  ( A ^ M
) #  0 )
2824, 27rerecclapd 8804 . . . . 5  |-  ( ph  ->  ( 1  /  ( A ^ M ) )  e.  RR )
299ltp1d 8900 . . . . 5  |-  ( ph  ->  ( ( ( 1  /  A )  - 
1 )  x.  M
)  <  ( (
( ( 1  /  A )  -  1 )  x.  M )  +  1 ) )
30 0le1 8451 . . . . . . . . 9  |-  0  <_  1
3130a1i 9 . . . . . . . 8  |-  ( ph  ->  0  <_  1 )
325, 12, 31divge0d 9750 . . . . . . 7  |-  ( ph  ->  0  <_  ( 1  /  A ) )
33 bernneq2 10655 . . . . . . 7  |-  ( ( ( 1  /  A
)  e.  RR  /\  M  e.  NN0  /\  0  <_  ( 1  /  A
) )  ->  (
( ( ( 1  /  A )  - 
1 )  x.  M
)  +  1 )  <_  ( ( 1  /  A ) ^ M ) )
344, 23, 32, 33syl3anc 1248 . . . . . 6  |-  ( ph  ->  ( ( ( ( 1  /  A )  -  1 )  x.  M )  +  1 )  <_  ( (
1  /  A ) ^ M ) )
3525, 3, 26exprecapd 10675 . . . . . 6  |-  ( ph  ->  ( ( 1  /  A ) ^ M
)  =  ( 1  /  ( A ^ M ) ) )
3634, 35breqtrd 4041 . . . . 5  |-  ( ph  ->  ( ( ( ( 1  /  A )  -  1 )  x.  M )  +  1 )  <_  ( 1  /  ( A ^ M ) ) )
379, 22, 28, 29, 36ltletrd 8393 . . . 4  |-  ( ph  ->  ( ( ( 1  /  A )  - 
1 )  x.  M
)  <  ( 1  /  ( A ^ M ) ) )
3821, 37eqbrtrd 4037 . . 3  |-  ( ph  ->  ( 1  /  (
1  /  ( ( ( 1  /  A
)  -  1 )  x.  M ) ) )  <  ( 1  /  ( A ^ M ) ) )
3912, 26rpexpcld 10691 . . . 4  |-  ( ph  ->  ( A ^ M
)  e.  RR+ )
4019rpreccld 9720 . . . 4  |-  ( ph  ->  ( 1  /  (
( ( 1  /  A )  -  1 )  x.  M ) )  e.  RR+ )
4139, 40ltrecd 9728 . . 3  |-  ( ph  ->  ( ( A ^ M )  <  (
1  /  ( ( ( 1  /  A
)  -  1 )  x.  M ) )  <-> 
( 1  /  (
1  /  ( ( ( 1  /  A
)  -  1 )  x.  M ) ) )  <  ( 1  /  ( A ^ M ) ) ) )
4238, 41mpbird 167 . 2  |-  ( ph  ->  ( A ^ M
)  <  ( 1  /  ( ( ( 1  /  A )  -  1 )  x.  M ) ) )
436recnd 7999 . . 3  |-  ( ph  ->  ( ( 1  /  A )  -  1 )  e.  CC )
447nncnd 8946 . . 3  |-  ( ph  ->  M  e.  CC )
4517rpap0d 9715 . . 3  |-  ( ph  ->  ( ( 1  /  A )  -  1 ) #  0 )
4618rpap0d 9715 . . 3  |-  ( ph  ->  M #  0 )
4743, 44, 45, 46recdivap2d 8778 . 2  |-  ( ph  ->  ( ( 1  / 
( ( 1  /  A )  -  1 ) )  /  M
)  =  ( 1  /  ( ( ( 1  /  A )  -  1 )  x.  M ) ) )
4842, 47breqtrrd 4043 1  |-  ( ph  ->  ( A ^ M
)  <  ( (
1  /  ( ( 1  /  A )  -  1 ) )  /  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2158   class class class wbr 4015  (class class class)co 5888   RRcr 7823   0cc0 7824   1c1 7825    + caddc 7827    x. cmul 7829    < clt 8005    <_ cle 8006    - cmin 8141    / cdiv 8642   NNcn 8932   NN0cn0 9189   ^cexp 10532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-mulrcl 7923  ax-addcom 7924  ax-mulcom 7925  ax-addass 7926  ax-mulass 7927  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-1rid 7931  ax-0id 7932  ax-rnegex 7933  ax-precex 7934  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-apti 7939  ax-pre-ltadd 7940  ax-pre-mulgt0 7941  ax-pre-mulext 7942
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-frec 6405  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-reap 8545  df-ap 8552  df-div 8643  df-inn 8933  df-n0 9190  df-z 9267  df-uz 9542  df-rp 9667  df-seqfrec 10459  df-exp 10533
This theorem is referenced by:  cvgratnnlemfm  11550
  Copyright terms: Public domain W3C validator