Theorem cvgratnnlembern 12234

Description: Lemma for cvgratnn 12242. 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

Step	Hyp	Ref	Expression
1		cvgratnnlembern.3	. . . . . . . . 9 |- ( ph -> A e. RR )
2		cvgratnnlembern.gt0	. . . . . . . . . 10 |- ( ph -> 0 < A )
3	1, 2	gt0ap0d 8920	. . . . . . . . 9 |- ( ph -> A # 0 )
4	1, 3	rerecclapd 9125	. . . . . . . 8 |- ( ph -> ( 1 / A ) e. RR )
5		1red 8305	. . . . . . . 8 |- ( ph -> 1 e. RR )
6	4, 5	resubcld 8671	. . . . . . 7 |- ( ph -> ( ( 1 / A ) - 1 ) e. RR )
7		cvgratnnlembern.m	. . . . . . . 8 |- ( ph -> M e. NN )
8	7	nnred 9267	. . . . . . 7 |- ( ph -> M e. RR )
9	6, 8	remulcld 8320	. . . . . 6 |- ( ph -> ( ( ( 1 / A ) - 1 ) x. M ) e. RR )
10	9	recnd 8318	. . . . 5 |- ( ph -> ( ( ( 1 / A ) - 1 ) x. M ) e. CC )
11		cvgratnnlembern.4	. . . . . . . . . 10 |- ( ph -> A < 1 )
12	1, 2	elrpd 10044	. . . . . . . . . . 11 |- ( ph -> A e. RR+ )
13	12	reclt1d 10061	. . . . . . . . . 10 |- ( ph -> ( A < 1 <-> 1 < ( 1 / A ) ) )
14	11, 13	mpbid 147	. . . . . . . . 9 |- ( ph -> 1 < ( 1 / A ) )
15	5, 4	posdifd 8823	. . . . . . . . 9 |- ( ph -> ( 1 < ( 1 / A ) <-> 0 < ( ( 1 / A ) - 1 ) ) )
16	14, 15	mpbid 147	. . . . . . . 8 |- ( ph -> 0 < ( ( 1 / A ) - 1 ) )
17	6, 16	elrpd 10044	. . . . . . 7 |- ( ph -> ( ( 1 / A ) - 1 ) e. RR+ )
18	7	nnrpd 10045	. . . . . . 7 |- ( ph -> M e. RR+ )
19	17, 18	rpmulcld 10064	. . . . . 6 |- ( ph -> ( ( ( 1 / A ) - 1 ) x. M ) e. RR+ )
20	19	rpap0d 10053	. . . . 5 |- ( ph -> ( ( ( 1 / A ) - 1 ) x. M ) # 0 )
21	10, 20	recrecapd 9076	. . . 4 |- ( ph -> ( 1 / ( 1 / ( ( ( 1 / A ) - 1 ) x. M ) ) ) = ( ( ( 1 / A ) - 1 ) x. M ) )
22	9, 5	readdcld 8319	. . . . 5 |- ( ph -> ( ( ( ( 1 / A ) - 1 ) x. M ) + 1 ) e. RR )
23	7	nnnn0d 9570	. . . . . . 7 |- ( ph -> M e. NN0 )
24	1, 23	reexpcld 11077	. . . . . 6 |- ( ph -> ( A ^ M ) e. RR )
25	1	recnd 8318	. . . . . . 7 |- ( ph -> A e. CC )
26	7	nnzd 9717	. . . . . . 7 |- ( ph -> M e. ZZ )
27	25, 3, 26	expap0d 11066	. . . . . 6 |- ( ph -> ( A ^ M ) # 0 )
28	24, 27	rerecclapd 9125	. . . . 5 |- ( ph -> ( 1 / ( A ^ M ) ) e. RR )
29	9	ltp1d 9221	. . . . 5 |- ( ph -> ( ( ( 1 / A ) - 1 ) x. M ) < ( ( ( ( 1 / A ) - 1 ) x. M ) + 1 ) )
30		0le1 8772	. . . . . . . . 9 |- 0 <_ 1
31	30	a1i 9	. . . . . . . 8 |- ( ph -> 0 <_ 1 )
32	5, 12, 31	divge0d 10088	. . . . . . 7 |- ( ph -> 0 <_ ( 1 / A ) )
33		bernneq2 11048	. . . . . . 7 |- ( ( ( 1 / A ) e. RR /\ M e. NN0 /\ 0 <_ ( 1 / A ) ) -> ( ( ( ( 1 / A ) - 1 ) x. M ) + 1 ) <_ ( ( 1 / A ) ^ M ) )
34	4, 23, 32, 33	syl3anc 1274	. . . . . 6 |- ( ph -> ( ( ( ( 1 / A ) - 1 ) x. M ) + 1 ) <_ ( ( 1 / A ) ^ M ) )
35	25, 3, 26	exprecapd 11068	. . . . . 6 |- ( ph -> ( ( 1 / A ) ^ M ) = ( 1 / ( A ^ M ) ) )
36	34, 35	breqtrd 4140	. . . . 5 |- ( ph -> ( ( ( ( 1 / A ) - 1 ) x. M ) + 1 ) <_ ( 1 / ( A ^ M ) ) )
37	9, 22, 28, 29, 36	ltletrd 8714	. . . 4 |- ( ph -> ( ( ( 1 / A ) - 1 ) x. M ) < ( 1 / ( A ^ M ) ) )
38	21, 37	eqbrtrd 4136	. . 3 |- ( ph -> ( 1 / ( 1 / ( ( ( 1 / A ) - 1 ) x. M ) ) ) < ( 1 / ( A ^ M ) ) )
39	12, 26	rpexpcld 11084	. . . 4 |- ( ph -> ( A ^ M ) e. RR+ )
40	19	rpreccld 10058	. . . 4 |- ( ph -> ( 1 / ( ( ( 1 / A ) - 1 ) x. M ) ) e. RR+ )
41	39, 40	ltrecd 10066	. . 3 |- ( ph -> ( ( A ^ M ) < ( 1 / ( ( ( 1 / A ) - 1 ) x. M ) ) <-> ( 1 / ( 1 / ( ( ( 1 / A ) - 1 ) x. M ) ) ) < ( 1 / ( A ^ M ) ) ) )
42	38, 41	mpbird 167	. 2 |- ( ph -> ( A ^ M ) < ( 1 / ( ( ( 1 / A ) - 1 ) x. M ) ) )
43	6	recnd 8318	. . 3 |- ( ph -> ( ( 1 / A ) - 1 ) e. CC )
44	7	nncnd 9268	. . 3 |- ( ph -> M e. CC )
45	17	rpap0d 10053	. . 3 |- ( ph -> ( ( 1 / A ) - 1 ) # 0 )
46	18	rpap0d 10053	. . 3 |- ( ph -> M # 0 )
47	43, 44, 45, 46	recdivap2d 9099	. 2 |- ( ph -> ( ( 1 / ( ( 1 / A ) - 1 ) ) / M ) = ( 1 / ( ( ( 1 / A ) - 1 ) x. M ) ) )
48	42, 47	breqtrrd 4142	1 |- ( ph -> ( A ^ M ) < ( ( 1 / ( ( 1 / A ) - 1 ) ) / M ) )

Syntax hints:   -> wi 4   e. wcel 2205   class class class wbr 4114  (class class class)co 6058   RRcr 8142   0cc0 8143   1c1 8144   + caddc 8146   x. cmul 8148   < clt 8324   <_ cle 8325   - cmin 8460   / cdiv 8963   NNcn 9254   NN0cn0 9513   ^cexp 10924

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-rp 10005  df-seqfrec 10834  df-exp 10925

This theorem is referenced by:  cvgratnnlemfm  12240