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Theorem cvgratnnlembern 11431
Description: Lemma for cvgratnn 11439. 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 8508 . . . . . . . . 9  |-  ( ph  ->  A #  0 )
41, 3rerecclapd 8711 . . . . . . . 8  |-  ( ph  ->  ( 1  /  A
)  e.  RR )
5 1red 7895 . . . . . . . 8  |-  ( ph  ->  1  e.  RR )
64, 5resubcld 8260 . . . . . . 7  |-  ( ph  ->  ( ( 1  /  A )  -  1 )  e.  RR )
7 cvgratnnlembern.m . . . . . . . 8  |-  ( ph  ->  M  e.  NN )
87nnred 8851 . . . . . . 7  |-  ( ph  ->  M  e.  RR )
96, 8remulcld 7910 . . . . . 6  |-  ( ph  ->  ( ( ( 1  /  A )  - 
1 )  x.  M
)  e.  RR )
109recnd 7908 . . . . 5  |-  ( ph  ->  ( ( ( 1  /  A )  - 
1 )  x.  M
)  e.  CC )
11 cvgratnnlembern.4 . . . . . . . . . 10  |-  ( ph  ->  A  <  1 )
121, 2elrpd 9606 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  RR+ )
1312reclt1d 9623 . . . . . . . . . 10  |-  ( ph  ->  ( A  <  1  <->  1  <  ( 1  /  A ) ) )
1411, 13mpbid 146 . . . . . . . . 9  |-  ( ph  ->  1  <  ( 1  /  A ) )
155, 4posdifd 8411 . . . . . . . . 9  |-  ( ph  ->  ( 1  <  (
1  /  A )  <->  0  <  ( ( 1  /  A )  -  1 ) ) )
1614, 15mpbid 146 . . . . . . . 8  |-  ( ph  ->  0  <  ( ( 1  /  A )  -  1 ) )
176, 16elrpd 9606 . . . . . . 7  |-  ( ph  ->  ( ( 1  /  A )  -  1 )  e.  RR+ )
187nnrpd 9607 . . . . . . 7  |-  ( ph  ->  M  e.  RR+ )
1917, 18rpmulcld 9626 . . . . . 6  |-  ( ph  ->  ( ( ( 1  /  A )  - 
1 )  x.  M
)  e.  RR+ )
2019rpap0d 9615 . . . . 5  |-  ( ph  ->  ( ( ( 1  /  A )  - 
1 )  x.  M
) #  0 )
2110, 20recrecapd 8662 . . . 4  |-  ( ph  ->  ( 1  /  (
1  /  ( ( ( 1  /  A
)  -  1 )  x.  M ) ) )  =  ( ( ( 1  /  A
)  -  1 )  x.  M ) )
229, 5readdcld 7909 . . . . 5  |-  ( ph  ->  ( ( ( ( 1  /  A )  -  1 )  x.  M )  +  1 )  e.  RR )
237nnnn0d 9148 . . . . . . 7  |-  ( ph  ->  M  e.  NN0 )
241, 23reexpcld 10577 . . . . . 6  |-  ( ph  ->  ( A ^ M
)  e.  RR )
251recnd 7908 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
267nnzd 9290 . . . . . . 7  |-  ( ph  ->  M  e.  ZZ )
2725, 3, 26expap0d 10566 . . . . . 6  |-  ( ph  ->  ( A ^ M
) #  0 )
2824, 27rerecclapd 8711 . . . . 5  |-  ( ph  ->  ( 1  /  ( A ^ M ) )  e.  RR )
299ltp1d 8806 . . . . 5  |-  ( ph  ->  ( ( ( 1  /  A )  - 
1 )  x.  M
)  <  ( (
( ( 1  /  A )  -  1 )  x.  M )  +  1 ) )
30 0le1 8360 . . . . . . . . 9  |-  0  <_  1
3130a1i 9 . . . . . . . 8  |-  ( ph  ->  0  <_  1 )
325, 12, 31divge0d 9650 . . . . . . 7  |-  ( ph  ->  0  <_  ( 1  /  A ) )
33 bernneq2 10548 . . . . . . 7  |-  ( ( ( 1  /  A
)  e.  RR  /\  M  e.  NN0  /\  0  <_  ( 1  /  A
) )  ->  (
( ( ( 1  /  A )  - 
1 )  x.  M
)  +  1 )  <_  ( ( 1  /  A ) ^ M ) )
344, 23, 32, 33syl3anc 1220 . . . . . 6  |-  ( ph  ->  ( ( ( ( 1  /  A )  -  1 )  x.  M )  +  1 )  <_  ( (
1  /  A ) ^ M ) )
3525, 3, 26exprecapd 10568 . . . . . 6  |-  ( ph  ->  ( ( 1  /  A ) ^ M
)  =  ( 1  /  ( A ^ M ) ) )
3634, 35breqtrd 3992 . . . . 5  |-  ( ph  ->  ( ( ( ( 1  /  A )  -  1 )  x.  M )  +  1 )  <_  ( 1  /  ( A ^ M ) ) )
379, 22, 28, 29, 36ltletrd 8302 . . . 4  |-  ( ph  ->  ( ( ( 1  /  A )  - 
1 )  x.  M
)  <  ( 1  /  ( A ^ M ) ) )
3821, 37eqbrtrd 3988 . . 3  |-  ( ph  ->  ( 1  /  (
1  /  ( ( ( 1  /  A
)  -  1 )  x.  M ) ) )  <  ( 1  /  ( A ^ M ) ) )
3912, 26rpexpcld 10584 . . . 4  |-  ( ph  ->  ( A ^ M
)  e.  RR+ )
4019rpreccld 9620 . . . 4  |-  ( ph  ->  ( 1  /  (
( ( 1  /  A )  -  1 )  x.  M ) )  e.  RR+ )
4139, 40ltrecd 9628 . . 3  |-  ( ph  ->  ( ( A ^ M )  <  (
1  /  ( ( ( 1  /  A
)  -  1 )  x.  M ) )  <-> 
( 1  /  (
1  /  ( ( ( 1  /  A
)  -  1 )  x.  M ) ) )  <  ( 1  /  ( A ^ M ) ) ) )
4238, 41mpbird 166 . 2  |-  ( ph  ->  ( A ^ M
)  <  ( 1  /  ( ( ( 1  /  A )  -  1 )  x.  M ) ) )
436recnd 7908 . . 3  |-  ( ph  ->  ( ( 1  /  A )  -  1 )  e.  CC )
447nncnd 8852 . . 3  |-  ( ph  ->  M  e.  CC )
4517rpap0d 9615 . . 3  |-  ( ph  ->  ( ( 1  /  A )  -  1 ) #  0 )
4618rpap0d 9615 . . 3  |-  ( ph  ->  M #  0 )
4743, 44, 45, 46recdivap2d 8685 . 2  |-  ( ph  ->  ( ( 1  / 
( ( 1  /  A )  -  1 ) )  /  M
)  =  ( 1  /  ( ( ( 1  /  A )  -  1 )  x.  M ) ) )
4842, 47breqtrrd 3994 1  |-  ( ph  ->  ( A ^ M
)  <  ( (
1  /  ( ( 1  /  A )  -  1 ) )  /  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2128   class class class wbr 3967  (class class class)co 5826   RRcr 7733   0cc0 7734   1c1 7735    + caddc 7737    x. cmul 7739    < clt 7914    <_ cle 7915    - cmin 8050    / cdiv 8549   NNcn 8838   NN0cn0 9095   ^cexp 10427
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4081  ax-sep 4084  ax-nul 4092  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498  ax-iinf 4549  ax-cnex 7825  ax-resscn 7826  ax-1cn 7827  ax-1re 7828  ax-icn 7829  ax-addcl 7830  ax-addrcl 7831  ax-mulcl 7832  ax-mulrcl 7833  ax-addcom 7834  ax-mulcom 7835  ax-addass 7836  ax-mulass 7837  ax-distr 7838  ax-i2m1 7839  ax-0lt1 7840  ax-1rid 7841  ax-0id 7842  ax-rnegex 7843  ax-precex 7844  ax-cnre 7845  ax-pre-ltirr 7846  ax-pre-ltwlin 7847  ax-pre-lttrn 7848  ax-pre-apti 7849  ax-pre-ltadd 7850  ax-pre-mulgt0 7851  ax-pre-mulext 7852
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-if 3507  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-tr 4065  df-id 4255  df-po 4258  df-iso 4259  df-iord 4328  df-on 4330  df-ilim 4331  df-suc 4333  df-iom 4552  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180  df-riota 5782  df-ov 5829  df-oprab 5830  df-mpo 5831  df-1st 6090  df-2nd 6091  df-recs 6254  df-frec 6340  df-pnf 7916  df-mnf 7917  df-xr 7918  df-ltxr 7919  df-le 7920  df-sub 8052  df-neg 8053  df-reap 8454  df-ap 8461  df-div 8550  df-inn 8839  df-n0 9096  df-z 9173  df-uz 9445  df-rp 9567  df-seqfrec 10354  df-exp 10428
This theorem is referenced by:  cvgratnnlemfm  11437
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