Theorem cvgratnnlembern 12034

Description: Lemma for cvgratnn 12042. 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 8776	. . . . . . . . 9 |- ( ph -> A # 0 )
4	1, 3	rerecclapd 8981	. . . . . . . 8 |- ( ph -> ( 1 / A ) e. RR )
5		1red 8161	. . . . . . . 8 |- ( ph -> 1 e. RR )
6	4, 5	resubcld 8527	. . . . . . 7 |- ( ph -> ( ( 1 / A ) - 1 ) e. RR )
7		cvgratnnlembern.m	. . . . . . . 8 |- ( ph -> M e. NN )
8	7	nnred 9123	. . . . . . 7 |- ( ph -> M e. RR )
9	6, 8	remulcld 8177	. . . . . 6 |- ( ph -> ( ( ( 1 / A ) - 1 ) x. M ) e. RR )
10	9	recnd 8175	. . . . 5 |- ( ph -> ( ( ( 1 / A ) - 1 ) x. M ) e. CC )
11		cvgratnnlembern.4	. . . . . . . . . 10 |- ( ph -> A < 1 )
12	1, 2	elrpd 9889	. . . . . . . . . . 11 |- ( ph -> A e. RR+ )
13	12	reclt1d 9906	. . . . . . . . . 10 |- ( ph -> ( A < 1 <-> 1 < ( 1 / A ) ) )
14	11, 13	mpbid 147	. . . . . . . . 9 |- ( ph -> 1 < ( 1 / A ) )
15	5, 4	posdifd 8679	. . . . . . . . 9 |- ( ph -> ( 1 < ( 1 / A ) <-> 0 < ( ( 1 / A ) - 1 ) ) )
16	14, 15	mpbid 147	. . . . . . . 8 |- ( ph -> 0 < ( ( 1 / A ) - 1 ) )
17	6, 16	elrpd 9889	. . . . . . 7 |- ( ph -> ( ( 1 / A ) - 1 ) e. RR+ )
18	7	nnrpd 9890	. . . . . . 7 |- ( ph -> M e. RR+ )
19	17, 18	rpmulcld 9909	. . . . . 6 |- ( ph -> ( ( ( 1 / A ) - 1 ) x. M ) e. RR+ )
20	19	rpap0d 9898	. . . . 5 |- ( ph -> ( ( ( 1 / A ) - 1 ) x. M ) # 0 )
21	10, 20	recrecapd 8932	. . . 4 |- ( ph -> ( 1 / ( 1 / ( ( ( 1 / A ) - 1 ) x. M ) ) ) = ( ( ( 1 / A ) - 1 ) x. M ) )
22	9, 5	readdcld 8176	. . . . 5 |- ( ph -> ( ( ( ( 1 / A ) - 1 ) x. M ) + 1 ) e. RR )
23	7	nnnn0d 9422	. . . . . . 7 |- ( ph -> M e. NN0 )
24	1, 23	reexpcld 10912	. . . . . 6 |- ( ph -> ( A ^ M ) e. RR )
25	1	recnd 8175	. . . . . . 7 |- ( ph -> A e. CC )
26	7	nnzd 9568	. . . . . . 7 |- ( ph -> M e. ZZ )
27	25, 3, 26	expap0d 10901	. . . . . 6 |- ( ph -> ( A ^ M ) # 0 )
28	24, 27	rerecclapd 8981	. . . . 5 |- ( ph -> ( 1 / ( A ^ M ) ) e. RR )
29	9	ltp1d 9077	. . . . 5 |- ( ph -> ( ( ( 1 / A ) - 1 ) x. M ) < ( ( ( ( 1 / A ) - 1 ) x. M ) + 1 ) )
30		0le1 8628	. . . . . . . . 9 |- 0 <_ 1
31	30	a1i 9	. . . . . . . 8 |- ( ph -> 0 <_ 1 )
32	5, 12, 31	divge0d 9933	. . . . . . 7 |- ( ph -> 0 <_ ( 1 / A ) )
33		bernneq2 10883	. . . . . . 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 1271	. . . . . 6 |- ( ph -> ( ( ( ( 1 / A ) - 1 ) x. M ) + 1 ) <_ ( ( 1 / A ) ^ M ) )
35	25, 3, 26	exprecapd 10903	. . . . . 6 |- ( ph -> ( ( 1 / A ) ^ M ) = ( 1 / ( A ^ M ) ) )
36	34, 35	breqtrd 4109	. . . . 5 |- ( ph -> ( ( ( ( 1 / A ) - 1 ) x. M ) + 1 ) <_ ( 1 / ( A ^ M ) ) )
37	9, 22, 28, 29, 36	ltletrd 8570	. . . 4 |- ( ph -> ( ( ( 1 / A ) - 1 ) x. M ) < ( 1 / ( A ^ M ) ) )
38	21, 37	eqbrtrd 4105	. . 3 |- ( ph -> ( 1 / ( 1 / ( ( ( 1 / A ) - 1 ) x. M ) ) ) < ( 1 / ( A ^ M ) ) )
39	12, 26	rpexpcld 10919	. . . 4 |- ( ph -> ( A ^ M ) e. RR+ )
40	19	rpreccld 9903	. . . 4 |- ( ph -> ( 1 / ( ( ( 1 / A ) - 1 ) x. M ) ) e. RR+ )
41	39, 40	ltrecd 9911	. . 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 8175	. . 3 |- ( ph -> ( ( 1 / A ) - 1 ) e. CC )
44	7	nncnd 9124	. . 3 |- ( ph -> M e. CC )
45	17	rpap0d 9898	. . 3 |- ( ph -> ( ( 1 / A ) - 1 ) # 0 )
46	18	rpap0d 9898	. . 3 |- ( ph -> M # 0 )
47	43, 44, 45, 46	recdivap2d 8955	. 2 |- ( ph -> ( ( 1 / ( ( 1 / A ) - 1 ) ) / M ) = ( 1 / ( ( ( 1 / A ) - 1 ) x. M ) ) )
48	42, 47	breqtrrd 4111	1 |- ( ph -> ( A ^ M ) < ( ( 1 / ( ( 1 / A ) - 1 ) ) / M ) )

Syntax hints:   -> wi 4   e. wcel 2200   class class class wbr 4083  (class class class)co 6001   RRcr 7998   0cc0 7999   1c1 8000   + caddc 8002   x. cmul 8004   < clt 8181   <_ cle 8182   - cmin 8317   / cdiv 8819   NNcn 9110   NN0cn0 9369   ^cexp 10760

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-n0 9370  df-z 9447  df-uz 9723  df-rp 9850  df-seqfrec 10670  df-exp 10761

This theorem is referenced by:  cvgratnnlemfm  12040