Theorem cvgratnnlembern 11526

Description: Lemma for cvgratnn 11534. 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 8584	. . . . . . . . 9 |- ( ph -> A # 0 )
4	1, 3	rerecclapd 8789	. . . . . . . 8 |- ( ph -> ( 1 / A ) e. RR )
5		1red 7971	. . . . . . . 8 |- ( ph -> 1 e. RR )
6	4, 5	resubcld 8336	. . . . . . 7 |- ( ph -> ( ( 1 / A ) - 1 ) e. RR )
7		cvgratnnlembern.m	. . . . . . . 8 |- ( ph -> M e. NN )
8	7	nnred 8930	. . . . . . 7 |- ( ph -> M e. RR )
9	6, 8	remulcld 7986	. . . . . 6 |- ( ph -> ( ( ( 1 / A ) - 1 ) x. M ) e. RR )
10	9	recnd 7984	. . . . 5 |- ( ph -> ( ( ( 1 / A ) - 1 ) x. M ) e. CC )
11		cvgratnnlembern.4	. . . . . . . . . 10 |- ( ph -> A < 1 )
12	1, 2	elrpd 9691	. . . . . . . . . . 11 |- ( ph -> A e. RR+ )
13	12	reclt1d 9708	. . . . . . . . . 10 |- ( ph -> ( A < 1 <-> 1 < ( 1 / A ) ) )
14	11, 13	mpbid 147	. . . . . . . . 9 |- ( ph -> 1 < ( 1 / A ) )
15	5, 4	posdifd 8487	. . . . . . . . 9 |- ( ph -> ( 1 < ( 1 / A ) <-> 0 < ( ( 1 / A ) - 1 ) ) )
16	14, 15	mpbid 147	. . . . . . . 8 |- ( ph -> 0 < ( ( 1 / A ) - 1 ) )
17	6, 16	elrpd 9691	. . . . . . 7 |- ( ph -> ( ( 1 / A ) - 1 ) e. RR+ )
18	7	nnrpd 9692	. . . . . . 7 |- ( ph -> M e. RR+ )
19	17, 18	rpmulcld 9711	. . . . . 6 |- ( ph -> ( ( ( 1 / A ) - 1 ) x. M ) e. RR+ )
20	19	rpap0d 9700	. . . . 5 |- ( ph -> ( ( ( 1 / A ) - 1 ) x. M ) # 0 )
21	10, 20	recrecapd 8740	. . . 4 |- ( ph -> ( 1 / ( 1 / ( ( ( 1 / A ) - 1 ) x. M ) ) ) = ( ( ( 1 / A ) - 1 ) x. M ) )
22	9, 5	readdcld 7985	. . . . 5 |- ( ph -> ( ( ( ( 1 / A ) - 1 ) x. M ) + 1 ) e. RR )
23	7	nnnn0d 9227	. . . . . . 7 |- ( ph -> M e. NN0 )
24	1, 23	reexpcld 10667	. . . . . 6 |- ( ph -> ( A ^ M ) e. RR )
25	1	recnd 7984	. . . . . . 7 |- ( ph -> A e. CC )
26	7	nnzd 9372	. . . . . . 7 |- ( ph -> M e. ZZ )
27	25, 3, 26	expap0d 10656	. . . . . 6 |- ( ph -> ( A ^ M ) # 0 )
28	24, 27	rerecclapd 8789	. . . . 5 |- ( ph -> ( 1 / ( A ^ M ) ) e. RR )
29	9	ltp1d 8885	. . . . 5 |- ( ph -> ( ( ( 1 / A ) - 1 ) x. M ) < ( ( ( ( 1 / A ) - 1 ) x. M ) + 1 ) )
30		0le1 8436	. . . . . . . . 9 |- 0 <_ 1
31	30	a1i 9	. . . . . . . 8 |- ( ph -> 0 <_ 1 )
32	5, 12, 31	divge0d 9735	. . . . . . 7 |- ( ph -> 0 <_ ( 1 / A ) )
33		bernneq2 10638	. . . . . . 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 1238	. . . . . 6 |- ( ph -> ( ( ( ( 1 / A ) - 1 ) x. M ) + 1 ) <_ ( ( 1 / A ) ^ M ) )
35	25, 3, 26	exprecapd 10658	. . . . . 6 |- ( ph -> ( ( 1 / A ) ^ M ) = ( 1 / ( A ^ M ) ) )
36	34, 35	breqtrd 4029	. . . . 5 |- ( ph -> ( ( ( ( 1 / A ) - 1 ) x. M ) + 1 ) <_ ( 1 / ( A ^ M ) ) )
37	9, 22, 28, 29, 36	ltletrd 8378	. . . 4 |- ( ph -> ( ( ( 1 / A ) - 1 ) x. M ) < ( 1 / ( A ^ M ) ) )
38	21, 37	eqbrtrd 4025	. . 3 |- ( ph -> ( 1 / ( 1 / ( ( ( 1 / A ) - 1 ) x. M ) ) ) < ( 1 / ( A ^ M ) ) )
39	12, 26	rpexpcld 10674	. . . 4 |- ( ph -> ( A ^ M ) e. RR+ )
40	19	rpreccld 9705	. . . 4 |- ( ph -> ( 1 / ( ( ( 1 / A ) - 1 ) x. M ) ) e. RR+ )
41	39, 40	ltrecd 9713	. . 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 7984	. . 3 |- ( ph -> ( ( 1 / A ) - 1 ) e. CC )
44	7	nncnd 8931	. . 3 |- ( ph -> M e. CC )
45	17	rpap0d 9700	. . 3 |- ( ph -> ( ( 1 / A ) - 1 ) # 0 )
46	18	rpap0d 9700	. . 3 |- ( ph -> M # 0 )
47	43, 44, 45, 46	recdivap2d 8763	. 2 |- ( ph -> ( ( 1 / ( ( 1 / A ) - 1 ) ) / M ) = ( 1 / ( ( ( 1 / A ) - 1 ) x. M ) ) )
48	42, 47	breqtrrd 4031	1 |- ( ph -> ( A ^ M ) < ( ( 1 / ( ( 1 / A ) - 1 ) ) / M ) )

Syntax hints:   -> wi 4   e. wcel 2148   class class class wbr 4003  (class class class)co 5874   RRcr 7809   0cc0 7810   1c1 7811   + caddc 7813   x. cmul 7815   < clt 7990   <_ cle 7991   - cmin 8126   / cdiv 8627   NNcn 8917   NN0cn0 9174   ^cexp 10516

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-frec 6391  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-reap 8530  df-ap 8537  df-div 8628  df-inn 8918  df-n0 9175  df-z 9252  df-uz 9527  df-rp 9652  df-seqfrec 10443  df-exp 10517

This theorem is referenced by:  cvgratnnlemfm  11532