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Theorem cvgratnnlembern 10917
Description: Lemma for cvgratnn 10925. 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 8105 . . . . . . . . 9  |-  ( ph  ->  A #  0 )
41, 3rerecclapd 8300 . . . . . . . 8  |-  ( ph  ->  ( 1  /  A
)  e.  RR )
5 1red 7503 . . . . . . . 8  |-  ( ph  ->  1  e.  RR )
64, 5resubcld 7859 . . . . . . 7  |-  ( ph  ->  ( ( 1  /  A )  -  1 )  e.  RR )
7 cvgratnnlembern.m . . . . . . . 8  |-  ( ph  ->  M  e.  NN )
87nnred 8435 . . . . . . 7  |-  ( ph  ->  M  e.  RR )
96, 8remulcld 7518 . . . . . 6  |-  ( ph  ->  ( ( ( 1  /  A )  - 
1 )  x.  M
)  e.  RR )
109recnd 7516 . . . . 5  |-  ( ph  ->  ( ( ( 1  /  A )  - 
1 )  x.  M
)  e.  CC )
11 cvgratnnlembern.4 . . . . . . . . . 10  |-  ( ph  ->  A  <  1 )
121, 2elrpd 9171 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  RR+ )
1312reclt1d 9187 . . . . . . . . . 10  |-  ( ph  ->  ( A  <  1  <->  1  <  ( 1  /  A ) ) )
1411, 13mpbid 145 . . . . . . . . 9  |-  ( ph  ->  1  <  ( 1  /  A ) )
155, 4posdifd 8009 . . . . . . . . 9  |-  ( ph  ->  ( 1  <  (
1  /  A )  <->  0  <  ( ( 1  /  A )  -  1 ) ) )
1614, 15mpbid 145 . . . . . . . 8  |-  ( ph  ->  0  <  ( ( 1  /  A )  -  1 ) )
176, 16elrpd 9171 . . . . . . 7  |-  ( ph  ->  ( ( 1  /  A )  -  1 )  e.  RR+ )
187nnrpd 9172 . . . . . . 7  |-  ( ph  ->  M  e.  RR+ )
1917, 18rpmulcld 9190 . . . . . 6  |-  ( ph  ->  ( ( ( 1  /  A )  - 
1 )  x.  M
)  e.  RR+ )
2019rpap0d 9179 . . . . 5  |-  ( ph  ->  ( ( ( 1  /  A )  - 
1 )  x.  M
) #  0 )
2110, 20recrecapd 8252 . . . 4  |-  ( ph  ->  ( 1  /  (
1  /  ( ( ( 1  /  A
)  -  1 )  x.  M ) ) )  =  ( ( ( 1  /  A
)  -  1 )  x.  M ) )
229, 5readdcld 7517 . . . . 5  |-  ( ph  ->  ( ( ( ( 1  /  A )  -  1 )  x.  M )  +  1 )  e.  RR )
237nnnn0d 8726 . . . . . . 7  |-  ( ph  ->  M  e.  NN0 )
241, 23reexpcld 10103 . . . . . 6  |-  ( ph  ->  ( A ^ M
)  e.  RR )
251recnd 7516 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
267nnzd 8867 . . . . . . 7  |-  ( ph  ->  M  e.  ZZ )
2725, 3, 26expap0d 10092 . . . . . 6  |-  ( ph  ->  ( A ^ M
) #  0 )
2824, 27rerecclapd 8300 . . . . 5  |-  ( ph  ->  ( 1  /  ( A ^ M ) )  e.  RR )
299ltp1d 8391 . . . . 5  |-  ( ph  ->  ( ( ( 1  /  A )  - 
1 )  x.  M
)  <  ( (
( ( 1  /  A )  -  1 )  x.  M )  +  1 ) )
30 0le1 7959 . . . . . . . . 9  |-  0  <_  1
3130a1i 9 . . . . . . . 8  |-  ( ph  ->  0  <_  1 )
325, 12, 31divge0d 9214 . . . . . . 7  |-  ( ph  ->  0  <_  ( 1  /  A ) )
33 bernneq2 10075 . . . . . . 7  |-  ( ( ( 1  /  A
)  e.  RR  /\  M  e.  NN0  /\  0  <_  ( 1  /  A
) )  ->  (
( ( ( 1  /  A )  - 
1 )  x.  M
)  +  1 )  <_  ( ( 1  /  A ) ^ M ) )
344, 23, 32, 33syl3anc 1174 . . . . . 6  |-  ( ph  ->  ( ( ( ( 1  /  A )  -  1 )  x.  M )  +  1 )  <_  ( (
1  /  A ) ^ M ) )
3525, 3, 26exprecapd 10094 . . . . . 6  |-  ( ph  ->  ( ( 1  /  A ) ^ M
)  =  ( 1  /  ( A ^ M ) ) )
3634, 35breqtrd 3869 . . . . 5  |-  ( ph  ->  ( ( ( ( 1  /  A )  -  1 )  x.  M )  +  1 )  <_  ( 1  /  ( A ^ M ) ) )
379, 22, 28, 29, 36ltletrd 7901 . . . 4  |-  ( ph  ->  ( ( ( 1  /  A )  - 
1 )  x.  M
)  <  ( 1  /  ( A ^ M ) ) )
3821, 37eqbrtrd 3865 . . 3  |-  ( ph  ->  ( 1  /  (
1  /  ( ( ( 1  /  A
)  -  1 )  x.  M ) ) )  <  ( 1  /  ( A ^ M ) ) )
3912, 26rpexpcld 10110 . . . 4  |-  ( ph  ->  ( A ^ M
)  e.  RR+ )
4019rpreccld 9184 . . . 4  |-  ( ph  ->  ( 1  /  (
( ( 1  /  A )  -  1 )  x.  M ) )  e.  RR+ )
4139, 40ltrecd 9192 . . 3  |-  ( ph  ->  ( ( A ^ M )  <  (
1  /  ( ( ( 1  /  A
)  -  1 )  x.  M ) )  <-> 
( 1  /  (
1  /  ( ( ( 1  /  A
)  -  1 )  x.  M ) ) )  <  ( 1  /  ( A ^ M ) ) ) )
4238, 41mpbird 165 . 2  |-  ( ph  ->  ( A ^ M
)  <  ( 1  /  ( ( ( 1  /  A )  -  1 )  x.  M ) ) )
436recnd 7516 . . 3  |-  ( ph  ->  ( ( 1  /  A )  -  1 )  e.  CC )
447nncnd 8436 . . 3  |-  ( ph  ->  M  e.  CC )
4517rpap0d 9179 . . 3  |-  ( ph  ->  ( ( 1  /  A )  -  1 ) #  0 )
4618rpap0d 9179 . . 3  |-  ( ph  ->  M #  0 )
4743, 44, 45, 46recdivap2d 8275 . 2  |-  ( ph  ->  ( ( 1  / 
( ( 1  /  A )  -  1 ) )  /  M
)  =  ( 1  /  ( ( ( 1  /  A )  -  1 )  x.  M ) ) )
4842, 47breqtrrd 3871 1  |-  ( ph  ->  ( A ^ M
)  <  ( (
1  /  ( ( 1  /  A )  -  1 ) )  /  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1438   class class class wbr 3845  (class class class)co 5652   RRcr 7349   0cc0 7350   1c1 7351    + caddc 7353    x. cmul 7355    < clt 7522    <_ cle 7523    - cmin 7653    / cdiv 8139   NNcn 8422   NN0cn0 8673   ^cexp 9954
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-mulrcl 7444  ax-addcom 7445  ax-mulcom 7446  ax-addass 7447  ax-mulass 7448  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-1rid 7452  ax-0id 7453  ax-rnegex 7454  ax-precex 7455  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-apti 7460  ax-pre-ltadd 7461  ax-pre-mulgt0 7462  ax-pre-mulext 7463
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-frec 6156  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-reap 8052  df-ap 8059  df-div 8140  df-inn 8423  df-n0 8674  df-z 8751  df-uz 9020  df-rp 9135  df-iseq 9853  df-seq3 9854  df-exp 9955
This theorem is referenced by:  cvgratnnlemfm  10923
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