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

Step	Hyp	Ref	Expression
1		cvgratnnlembern.3	. . . . . . . . 9 |- ( ph -> A e. RR )
2		cvgratnnlembern.gt0	. . . . . . . . . 10 |- ( ph -> 0 < A )
3	1, 2	gt0ap0d 8548	. . . . . . . . 9 |- ( ph -> A # 0 )
4	1, 3	rerecclapd 8751	. . . . . . . 8 |- ( ph -> ( 1 / A ) e. RR )
5		1red 7935	. . . . . . . 8 |- ( ph -> 1 e. RR )
6	4, 5	resubcld 8300	. . . . . . 7 |- ( ph -> ( ( 1 / A ) - 1 ) e. RR )
7		cvgratnnlembern.m	. . . . . . . 8 |- ( ph -> M e. NN )
8	7	nnred 8891	. . . . . . 7 |- ( ph -> M e. RR )
9	6, 8	remulcld 7950	. . . . . 6 |- ( ph -> ( ( ( 1 / A ) - 1 ) x. M ) e. RR )
10	9	recnd 7948	. . . . 5 |- ( ph -> ( ( ( 1 / A ) - 1 ) x. M ) e. CC )
11		cvgratnnlembern.4	. . . . . . . . . 10 |- ( ph -> A < 1 )
12	1, 2	elrpd 9650	. . . . . . . . . . 11 |- ( ph -> A e. RR+ )
13	12	reclt1d 9667	. . . . . . . . . 10 |- ( ph -> ( A < 1 <-> 1 < ( 1 / A ) ) )
14	11, 13	mpbid 146	. . . . . . . . 9 |- ( ph -> 1 < ( 1 / A ) )
15	5, 4	posdifd 8451	. . . . . . . . 9 |- ( ph -> ( 1 < ( 1 / A ) <-> 0 < ( ( 1 / A ) - 1 ) ) )
16	14, 15	mpbid 146	. . . . . . . 8 |- ( ph -> 0 < ( ( 1 / A ) - 1 ) )
17	6, 16	elrpd 9650	. . . . . . 7 |- ( ph -> ( ( 1 / A ) - 1 ) e. RR+ )
18	7	nnrpd 9651	. . . . . . 7 |- ( ph -> M e. RR+ )
19	17, 18	rpmulcld 9670	. . . . . 6 |- ( ph -> ( ( ( 1 / A ) - 1 ) x. M ) e. RR+ )
20	19	rpap0d 9659	. . . . 5 |- ( ph -> ( ( ( 1 / A ) - 1 ) x. M ) # 0 )
21	10, 20	recrecapd 8702	. . . 4 |- ( ph -> ( 1 / ( 1 / ( ( ( 1 / A ) - 1 ) x. M ) ) ) = ( ( ( 1 / A ) - 1 ) x. M ) )
22	9, 5	readdcld 7949	. . . . 5 |- ( ph -> ( ( ( ( 1 / A ) - 1 ) x. M ) + 1 ) e. RR )
23	7	nnnn0d 9188	. . . . . . 7 |- ( ph -> M e. NN0 )
24	1, 23	reexpcld 10626	. . . . . 6 |- ( ph -> ( A ^ M ) e. RR )
25	1	recnd 7948	. . . . . . 7 |- ( ph -> A e. CC )
26	7	nnzd 9333	. . . . . . 7 |- ( ph -> M e. ZZ )
27	25, 3, 26	expap0d 10615	. . . . . 6 |- ( ph -> ( A ^ M ) # 0 )
28	24, 27	rerecclapd 8751	. . . . 5 |- ( ph -> ( 1 / ( A ^ M ) ) e. RR )
29	9	ltp1d 8846	. . . . 5 |- ( ph -> ( ( ( 1 / A ) - 1 ) x. M ) < ( ( ( ( 1 / A ) - 1 ) x. M ) + 1 ) )
30		0le1 8400	. . . . . . . . 9 |- 0 <_ 1
31	30	a1i 9	. . . . . . . 8 |- ( ph -> 0 <_ 1 )
32	5, 12, 31	divge0d 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 ) )
34	4, 23, 32, 33	syl3anc 1233	. . . . . 6 |- ( ph -> ( ( ( ( 1 / A ) - 1 ) x. M ) + 1 ) <_ ( ( 1 / A ) ^ M ) )
35	25, 3, 26	exprecapd 10617	. . . . . 6 |- ( ph -> ( ( 1 / A ) ^ M ) = ( 1 / ( A ^ M ) ) )
36	34, 35	breqtrd 4015	. . . . 5 |- ( ph -> ( ( ( ( 1 / A ) - 1 ) x. M ) + 1 ) <_ ( 1 / ( A ^ M ) ) )
37	9, 22, 28, 29, 36	ltletrd 8342	. . . 4 |- ( ph -> ( ( ( 1 / A ) - 1 ) x. M ) < ( 1 / ( A ^ M ) ) )
38	21, 37	eqbrtrd 4011	. . 3 |- ( ph -> ( 1 / ( 1 / ( ( ( 1 / A ) - 1 ) x. M ) ) ) < ( 1 / ( A ^ M ) ) )
39	12, 26	rpexpcld 10633	. . . 4 |- ( ph -> ( A ^ M ) e. RR+ )
40	19	rpreccld 9664	. . . 4 |- ( ph -> ( 1 / ( ( ( 1 / A ) - 1 ) x. M ) ) e. RR+ )
41	39, 40	ltrecd 9672	. . 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 7948	. . 3 |- ( ph -> ( ( 1 / A ) - 1 ) e. CC )
44	7	nncnd 8892	. . 3 |- ( ph -> M e. CC )
45	17	rpap0d 9659	. . 3 |- ( ph -> ( ( 1 / A ) - 1 ) # 0 )
46	18	rpap0d 9659	. . 3 |- ( ph -> M # 0 )
47	43, 44, 45, 46	recdivap2d 8725	. 2 |- ( ph -> ( ( 1 / ( ( 1 / A ) - 1 ) ) / M ) = ( 1 / ( ( ( 1 / A ) - 1 ) x. M ) ) )
48	42, 47	breqtrrd 4017	1 |- ( ph -> ( A ^ M ) < ( ( 1 / ( ( 1 / A ) - 1 ) ) / M ) )

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