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

Step	Hyp	Ref	Expression
1		cvgratnnlembern.3	. . . . . . . . 9 |- ( ph -> A e. RR )
2		cvgratnnlembern.gt0	. . . . . . . . . 10 |- ( ph -> 0 < A )
3	1, 2	gt0ap0d 8508	. . . . . . . . 9 |- ( ph -> A # 0 )
4	1, 3	rerecclapd 8711	. . . . . . . 8 |- ( ph -> ( 1 / A ) e. RR )
5		1red 7895	. . . . . . . 8 |- ( ph -> 1 e. RR )
6	4, 5	resubcld 8260	. . . . . . 7 |- ( ph -> ( ( 1 / A ) - 1 ) e. RR )
7		cvgratnnlembern.m	. . . . . . . 8 |- ( ph -> M e. NN )
8	7	nnred 8851	. . . . . . 7 |- ( ph -> M e. RR )
9	6, 8	remulcld 7910	. . . . . 6 |- ( ph -> ( ( ( 1 / A ) - 1 ) x. M ) e. RR )
10	9	recnd 7908	. . . . 5 |- ( ph -> ( ( ( 1 / A ) - 1 ) x. M ) e. CC )
11		cvgratnnlembern.4	. . . . . . . . . 10 |- ( ph -> A < 1 )
12	1, 2	elrpd 9606	. . . . . . . . . . 11 |- ( ph -> A e. RR+ )
13	12	reclt1d 9623	. . . . . . . . . 10 |- ( ph -> ( A < 1 <-> 1 < ( 1 / A ) ) )
14	11, 13	mpbid 146	. . . . . . . . 9 |- ( ph -> 1 < ( 1 / A ) )
15	5, 4	posdifd 8411	. . . . . . . . 9 |- ( ph -> ( 1 < ( 1 / A ) <-> 0 < ( ( 1 / A ) - 1 ) ) )
16	14, 15	mpbid 146	. . . . . . . 8 |- ( ph -> 0 < ( ( 1 / A ) - 1 ) )
17	6, 16	elrpd 9606	. . . . . . 7 |- ( ph -> ( ( 1 / A ) - 1 ) e. RR+ )
18	7	nnrpd 9607	. . . . . . 7 |- ( ph -> M e. RR+ )
19	17, 18	rpmulcld 9626	. . . . . 6 |- ( ph -> ( ( ( 1 / A ) - 1 ) x. M ) e. RR+ )
20	19	rpap0d 9615	. . . . 5 |- ( ph -> ( ( ( 1 / A ) - 1 ) x. M ) # 0 )
21	10, 20	recrecapd 8662	. . . 4 |- ( ph -> ( 1 / ( 1 / ( ( ( 1 / A ) - 1 ) x. M ) ) ) = ( ( ( 1 / A ) - 1 ) x. M ) )
22	9, 5	readdcld 7909	. . . . 5 |- ( ph -> ( ( ( ( 1 / A ) - 1 ) x. M ) + 1 ) e. RR )
23	7	nnnn0d 9148	. . . . . . 7 |- ( ph -> M e. NN0 )
24	1, 23	reexpcld 10577	. . . . . 6 |- ( ph -> ( A ^ M ) e. RR )
25	1	recnd 7908	. . . . . . 7 |- ( ph -> A e. CC )
26	7	nnzd 9290	. . . . . . 7 |- ( ph -> M e. ZZ )
27	25, 3, 26	expap0d 10566	. . . . . 6 |- ( ph -> ( A ^ M ) # 0 )
28	24, 27	rerecclapd 8711	. . . . 5 |- ( ph -> ( 1 / ( A ^ M ) ) e. RR )
29	9	ltp1d 8806	. . . . 5 |- ( ph -> ( ( ( 1 / A ) - 1 ) x. M ) < ( ( ( ( 1 / A ) - 1 ) x. M ) + 1 ) )
30		0le1 8360	. . . . . . . . 9 |- 0 <_ 1
31	30	a1i 9	. . . . . . . 8 |- ( ph -> 0 <_ 1 )
32	5, 12, 31	divge0d 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 ) )
34	4, 23, 32, 33	syl3anc 1220	. . . . . 6 |- ( ph -> ( ( ( ( 1 / A ) - 1 ) x. M ) + 1 ) <_ ( ( 1 / A ) ^ M ) )
35	25, 3, 26	exprecapd 10568	. . . . . 6 |- ( ph -> ( ( 1 / A ) ^ M ) = ( 1 / ( A ^ M ) ) )
36	34, 35	breqtrd 3992	. . . . 5 |- ( ph -> ( ( ( ( 1 / A ) - 1 ) x. M ) + 1 ) <_ ( 1 / ( A ^ M ) ) )
37	9, 22, 28, 29, 36	ltletrd 8302	. . . 4 |- ( ph -> ( ( ( 1 / A ) - 1 ) x. M ) < ( 1 / ( A ^ M ) ) )
38	21, 37	eqbrtrd 3988	. . 3 |- ( ph -> ( 1 / ( 1 / ( ( ( 1 / A ) - 1 ) x. M ) ) ) < ( 1 / ( A ^ M ) ) )
39	12, 26	rpexpcld 10584	. . . 4 |- ( ph -> ( A ^ M ) e. RR+ )
40	19	rpreccld 9620	. . . 4 |- ( ph -> ( 1 / ( ( ( 1 / A ) - 1 ) x. M ) ) e. RR+ )
41	39, 40	ltrecd 9628	. . 3 |- ( ph -> ( ( A ^ M ) < ( 1 / ( ( ( 1 / A ) - 1 ) x. M ) ) <-> ( 1 / ( 1 / ( ( ( 1 / A ) - 1 ) x. M ) ) ) < ( 1 / ( A ^ M ) ) ) )
42	38, 41	mpbird 166	. 2 |- ( ph -> ( A ^ M ) < ( 1 / ( ( ( 1 / A ) - 1 ) x. M ) ) )
43	6	recnd 7908	. . 3 |- ( ph -> ( ( 1 / A ) - 1 ) e. CC )
44	7	nncnd 8852	. . 3 |- ( ph -> M e. CC )
45	17	rpap0d 9615	. . 3 |- ( ph -> ( ( 1 / A ) - 1 ) # 0 )
46	18	rpap0d 9615	. . 3 |- ( ph -> M # 0 )
47	43, 44, 45, 46	recdivap2d 8685	. 2 |- ( ph -> ( ( 1 / ( ( 1 / A ) - 1 ) ) / M ) = ( 1 / ( ( ( 1 / A ) - 1 ) x. M ) ) )
48	42, 47	breqtrrd 3994	1 |- ( ph -> ( A ^ M ) < ( ( 1 / ( ( 1 / A ) - 1 ) ) / M ) )

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