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Theorem cvgratnnlembern 11486
Description: Lemma for cvgratnn 11494. 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 8548 . . . . . . . . 9  |-  ( ph  ->  A #  0 )
41, 3rerecclapd 8751 . . . . . . . 8  |-  ( ph  ->  ( 1  /  A
)  e.  RR )
5 1red 7935 . . . . . . . 8  |-  ( ph  ->  1  e.  RR )
64, 5resubcld 8300 . . . . . . 7  |-  ( ph  ->  ( ( 1  /  A )  -  1 )  e.  RR )
7 cvgratnnlembern.m . . . . . . . 8  |-  ( ph  ->  M  e.  NN )
87nnred 8891 . . . . . . 7  |-  ( ph  ->  M  e.  RR )
96, 8remulcld 7950 . . . . . 6  |-  ( ph  ->  ( ( ( 1  /  A )  - 
1 )  x.  M
)  e.  RR )
109recnd 7948 . . . . 5  |-  ( ph  ->  ( ( ( 1  /  A )  - 
1 )  x.  M
)  e.  CC )
11 cvgratnnlembern.4 . . . . . . . . . 10  |-  ( ph  ->  A  <  1 )
121, 2elrpd 9650 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  RR+ )
1312reclt1d 9667 . . . . . . . . . 10  |-  ( ph  ->  ( A  <  1  <->  1  <  ( 1  /  A ) ) )
1411, 13mpbid 146 . . . . . . . . 9  |-  ( ph  ->  1  <  ( 1  /  A ) )
155, 4posdifd 8451 . . . . . . . . 9  |-  ( ph  ->  ( 1  <  (
1  /  A )  <->  0  <  ( ( 1  /  A )  -  1 ) ) )
1614, 15mpbid 146 . . . . . . . 8  |-  ( ph  ->  0  <  ( ( 1  /  A )  -  1 ) )
176, 16elrpd 9650 . . . . . . 7  |-  ( ph  ->  ( ( 1  /  A )  -  1 )  e.  RR+ )
187nnrpd 9651 . . . . . . 7  |-  ( ph  ->  M  e.  RR+ )
1917, 18rpmulcld 9670 . . . . . 6  |-  ( ph  ->  ( ( ( 1  /  A )  - 
1 )  x.  M
)  e.  RR+ )
2019rpap0d 9659 . . . . 5  |-  ( ph  ->  ( ( ( 1  /  A )  - 
1 )  x.  M
) #  0 )
2110, 20recrecapd 8702 . . . 4  |-  ( ph  ->  ( 1  /  (
1  /  ( ( ( 1  /  A
)  -  1 )  x.  M ) ) )  =  ( ( ( 1  /  A
)  -  1 )  x.  M ) )
229, 5readdcld 7949 . . . . 5  |-  ( ph  ->  ( ( ( ( 1  /  A )  -  1 )  x.  M )  +  1 )  e.  RR )
237nnnn0d 9188 . . . . . . 7  |-  ( ph  ->  M  e.  NN0 )
241, 23reexpcld 10626 . . . . . 6  |-  ( ph  ->  ( A ^ M
)  e.  RR )
251recnd 7948 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
267nnzd 9333 . . . . . . 7  |-  ( ph  ->  M  e.  ZZ )
2725, 3, 26expap0d 10615 . . . . . 6  |-  ( ph  ->  ( A ^ M
) #  0 )
2824, 27rerecclapd 8751 . . . . 5  |-  ( ph  ->  ( 1  /  ( A ^ M ) )  e.  RR )
299ltp1d 8846 . . . . 5  |-  ( ph  ->  ( ( ( 1  /  A )  - 
1 )  x.  M
)  <  ( (
( ( 1  /  A )  -  1 )  x.  M )  +  1 ) )
30 0le1 8400 . . . . . . . . 9  |-  0  <_  1
3130a1i 9 . . . . . . . 8  |-  ( ph  ->  0  <_  1 )
325, 12, 31divge0d 9694 . . . . . . 7  |-  ( ph  ->  0  <_  ( 1  /  A ) )
33 bernneq2 10597 . . . . . . 7  |-  ( ( ( 1  /  A
)  e.  RR  /\  M  e.  NN0  /\  0  <_  ( 1  /  A
) )  ->  (
( ( ( 1  /  A )  - 
1 )  x.  M
)  +  1 )  <_  ( ( 1  /  A ) ^ M ) )
344, 23, 32, 33syl3anc 1233 . . . . . 6  |-  ( ph  ->  ( ( ( ( 1  /  A )  -  1 )  x.  M )  +  1 )  <_  ( (
1  /  A ) ^ M ) )
3525, 3, 26exprecapd 10617 . . . . . 6  |-  ( ph  ->  ( ( 1  /  A ) ^ M
)  =  ( 1  /  ( A ^ M ) ) )
3634, 35breqtrd 4015 . . . . 5  |-  ( ph  ->  ( ( ( ( 1  /  A )  -  1 )  x.  M )  +  1 )  <_  ( 1  /  ( A ^ M ) ) )
379, 22, 28, 29, 36ltletrd 8342 . . . 4  |-  ( ph  ->  ( ( ( 1  /  A )  - 
1 )  x.  M
)  <  ( 1  /  ( A ^ M ) ) )
3821, 37eqbrtrd 4011 . . 3  |-  ( ph  ->  ( 1  /  (
1  /  ( ( ( 1  /  A
)  -  1 )  x.  M ) ) )  <  ( 1  /  ( A ^ M ) ) )
3912, 26rpexpcld 10633 . . . 4  |-  ( ph  ->  ( A ^ M
)  e.  RR+ )
4019rpreccld 9664 . . . 4  |-  ( ph  ->  ( 1  /  (
( ( 1  /  A )  -  1 )  x.  M ) )  e.  RR+ )
4139, 40ltrecd 9672 . . 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 7948 . . 3  |-  ( ph  ->  ( ( 1  /  A )  -  1 )  e.  CC )
447nncnd 8892 . . 3  |-  ( ph  ->  M  e.  CC )
4517rpap0d 9659 . . 3  |-  ( ph  ->  ( ( 1  /  A )  -  1 ) #  0 )
4618rpap0d 9659 . . 3  |-  ( ph  ->  M #  0 )
4743, 44, 45, 46recdivap2d 8725 . 2  |-  ( ph  ->  ( ( 1  / 
( ( 1  /  A )  -  1 ) )  /  M
)  =  ( 1  /  ( ( ( 1  /  A )  -  1 )  x.  M ) ) )
4842, 47breqtrrd 4017 1  |-  ( ph  ->  ( A ^ M
)  <  ( (
1  /  ( ( 1  /  A )  -  1 ) )  /  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2141   class class class wbr 3989  (class class class)co 5853   RRcr 7773   0cc0 7774   1c1 7775    + caddc 7777    x. cmul 7779    < clt 7954    <_ cle 7955    - cmin 8090    / cdiv 8589   NNcn 8878   NN0cn0 9135   ^cexp 10475
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-rp 9611  df-seqfrec 10402  df-exp 10476
This theorem is referenced by:  cvgratnnlemfm  11492
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