Theorem cxplim 24698

Description: A power to a negative exponent goes to zero as the base becomes large. (Contributed by Mario Carneiro, 15-Sep-2014.) (Revised by Mario Carneiro, 18-May-2016.)

Assertion

Ref	Expression
cxplim	|- ( A e. RR+ -> ( n e. RR+ |-> ( 1 / ( n ^c A ) ) ) ~~> r 0 )

Distinct variable group:   A, n

Proof of Theorem cxplim

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rpre 11839	. . . . . 6 |- ( x e. RR+ -> x e. RR )
2	1	adantl 482	. . . . 5 |- ( ( A e. RR+ /\ x e. RR+ ) -> x e. RR )
3		rpge0 11845	. . . . . 6 |- ( x e. RR+ -> 0 <_ x )
4	3	adantl 482	. . . . 5 |- ( ( A e. RR+ /\ x e. RR+ ) -> 0 <_ x )
5		rpre 11839	. . . . . . . 8 |- ( A e. RR+ -> A e. RR )
6	5	renegcld 10457	. . . . . . 7 |- ( A e. RR+ -> -u A e. RR )
7	6	adantr 481	. . . . . 6 |- ( ( A e. RR+ /\ x e. RR+ ) -> -u A e. RR )
8		rpcn 11841	. . . . . . . 8 |- ( A e. RR+ -> A e. CC )
9		rpne0 11848	. . . . . . . 8 |- ( A e. RR+ -> A =/= 0 )
10	8, 9	negne0d 10390	. . . . . . 7 |- ( A e. RR+ -> -u A =/= 0 )
11	10	adantr 481	. . . . . 6 |- ( ( A e. RR+ /\ x e. RR+ ) -> -u A =/= 0 )
12	7, 11	rereccld 10852	. . . . 5 |- ( ( A e. RR+ /\ x e. RR+ ) -> ( 1 / -u A ) e. RR )
13	2, 4, 12	recxpcld 24469	. . . 4 |- ( ( A e. RR+ /\ x e. RR+ ) -> ( x ^c ( 1 / -u A ) ) e. RR )
14		simprl 794	. . . . . . . . . . 11 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> n e. RR+ )
15	5	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> A e. RR )
16	14, 15	rpcxpcld 24476	. . . . . . . . . 10 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( n ^c A ) e. RR+ )
17	16	rpreccld 11882	. . . . . . . . 9 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( 1 / ( n ^c A ) ) e. RR+ )
18	17	rprege0d 11879	. . . . . . . 8 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( ( 1 / ( n ^c A ) ) e. RR /\ 0 <_ ( 1 / ( n ^c A ) ) ) )
19		absid 14036	. . . . . . . 8 |- ( ( ( 1 / ( n ^c A ) ) e. RR /\ 0 <_ ( 1 / ( n ^c A ) ) ) -> ( abs ` ( 1 / ( n ^c A ) ) ) = ( 1 / ( n ^c A ) ) )
20	18, 19	syl 17	. . . . . . 7 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( abs ` ( 1 / ( n ^c A ) ) ) = ( 1 / ( n ^c A ) ) )
21		simplr 792	. . . . . . . 8 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> x e. RR+ )
22		simprr 796	. . . . . . . . . 10 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( x ^c ( 1 / -u A ) ) < n )
23		rpreccl 11857	. . . . . . . . . . . . . 14 |- ( A e. RR+ -> ( 1 / A ) e. RR+ )
24	23	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( 1 / A ) e. RR+ )
25	24	rpcnd 11874	. . . . . . . . . . . 12 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( 1 / A ) e. CC )
26	21, 25	cxprecd 24475	. . . . . . . . . . 11 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( ( 1 / x ) ^c ( 1 / A ) ) = ( 1 / ( x ^c ( 1 / A ) ) ) )
27		rpcn 11841	. . . . . . . . . . . . 13 |- ( x e. RR+ -> x e. CC )
28	27	ad2antlr 763	. . . . . . . . . . . 12 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> x e. CC )
29		rpne0 11848	. . . . . . . . . . . . 13 |- ( x e. RR+ -> x =/= 0 )
30	29	ad2antlr 763	. . . . . . . . . . . 12 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> x =/= 0 )
31	28, 30, 25	cxpnegd 24461	. . . . . . . . . . 11 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( x ^c -u ( 1 / A ) ) = ( 1 / ( x ^c ( 1 / A ) ) ) )
32		1cnd 10056	. . . . . . . . . . . . 13 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> 1 e. CC )
33	8	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> A e. CC )
34	9	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> A =/= 0 )
35	32, 33, 34	divneg2d 10815	. . . . . . . . . . . 12 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> -u ( 1 / A ) = ( 1 / -u A ) )
36	35	oveq2d 6666	. . . . . . . . . . 11 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( x ^c -u ( 1 / A ) ) = ( x ^c ( 1 / -u A ) ) )
37	26, 31, 36	3eqtr2d 2662	. . . . . . . . . 10 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( ( 1 / x ) ^c ( 1 / A ) ) = ( x ^c ( 1 / -u A ) ) )
38	33, 34	recidd 10796	. . . . . . . . . . . 12 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( A x. ( 1 / A ) ) = 1 )
39	38	oveq2d 6666	. . . . . . . . . . 11 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( n ^c ( A x. ( 1 / A ) ) ) = ( n ^c 1 ) )
40	14, 15, 25	cxpmuld 24480	. . . . . . . . . . 11 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( n ^c ( A x. ( 1 / A ) ) ) = ( ( n ^c A ) ^c ( 1 / A ) ) )
41	14	rpcnd 11874	. . . . . . . . . . . 12 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> n e. CC )
42	41	cxp1d 24452	. . . . . . . . . . 11 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( n ^c 1 ) = n )
43	39, 40, 42	3eqtr3d 2664	. . . . . . . . . 10 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( ( n ^c A ) ^c ( 1 / A ) ) = n )
44	22, 37, 43	3brtr4d 4685	. . . . . . . . 9 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( ( 1 / x ) ^c ( 1 / A ) ) < ( ( n ^c A ) ^c ( 1 / A ) ) )
45		rpreccl 11857	. . . . . . . . . . . 12 |- ( x e. RR+ -> ( 1 / x ) e. RR+ )
46	45	ad2antlr 763	. . . . . . . . . . 11 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( 1 / x ) e. RR+ )
47	46	rpred 11872	. . . . . . . . . 10 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( 1 / x ) e. RR )
48	46	rpge0d 11876	. . . . . . . . . 10 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> 0 <_ ( 1 / x ) )
49	16	rpred 11872	. . . . . . . . . 10 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( n ^c A ) e. RR )
50	16	rpge0d 11876	. . . . . . . . . 10 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> 0 <_ ( n ^c A ) )
51	47, 48, 49, 50, 24	cxplt2d 24472	. . . . . . . . 9 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( ( 1 / x ) < ( n ^c A ) <-> ( ( 1 / x ) ^c ( 1 / A ) ) < ( ( n ^c A ) ^c ( 1 / A ) ) ) )
52	44, 51	mpbird 247	. . . . . . . 8 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( 1 / x ) < ( n ^c A ) )
53	21, 16, 52	ltrec1d 11892	. . . . . . 7 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( 1 / ( n ^c A ) ) < x )
54	20, 53	eqbrtrd 4675	. . . . . 6 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ ( n e. RR+ /\ ( x ^c ( 1 / -u A ) ) < n ) ) -> ( abs ` ( 1 / ( n ^c A ) ) ) < x )
55	54	expr 643	. . . . 5 |- ( ( ( A e. RR+ /\ x e. RR+ ) /\ n e. RR+ ) -> ( ( x ^c ( 1 / -u A ) ) < n -> ( abs ` ( 1 / ( n ^c A ) ) ) < x ) )
56	55	ralrimiva 2966	. . . 4 |- ( ( A e. RR+ /\ x e. RR+ ) -> A. n e. RR+ ( ( x ^c ( 1 / -u A ) ) < n -> ( abs ` ( 1 / ( n ^c A ) ) ) < x ) )
57		breq1 4656	. . . . . . 7 |- ( y = ( x ^c ( 1 / -u A ) ) -> ( y < n <-> ( x ^c ( 1 / -u A ) ) < n ) )
58	57	imbi1d 331	. . . . . 6 |- ( y = ( x ^c ( 1 / -u A ) ) -> ( ( y < n -> ( abs ` ( 1 / ( n ^c A ) ) ) < x ) <-> ( ( x ^c ( 1 / -u A ) ) < n -> ( abs ` ( 1 / ( n ^c A ) ) ) < x ) ) )
59	58	ralbidv 2986	. . . . 5 |- ( y = ( x ^c ( 1 / -u A ) ) -> ( A. n e. RR+ ( y < n -> ( abs ` ( 1 / ( n ^c A ) ) ) < x ) <-> A. n e. RR+ ( ( x ^c ( 1 / -u A ) ) < n -> ( abs ` ( 1 / ( n ^c A ) ) ) < x ) ) )
60	59	rspcev 3309	. . . 4 |- ( ( ( x ^c ( 1 / -u A ) ) e. RR /\ A. n e. RR+ ( ( x ^c ( 1 / -u A ) ) < n -> ( abs ` ( 1 / ( n ^c A ) ) ) < x ) ) -> E. y e. RR A. n e. RR+ ( y < n -> ( abs ` ( 1 / ( n ^c A ) ) ) < x ) )
61	13, 56, 60	syl2anc 693	. . 3 |- ( ( A e. RR+ /\ x e. RR+ ) -> E. y e. RR A. n e. RR+ ( y < n -> ( abs ` ( 1 / ( n ^c A ) ) ) < x ) )
62	61	ralrimiva 2966	. 2 |- ( A e. RR+ -> A. x e. RR+ E. y e. RR A. n e. RR+ ( y < n -> ( abs ` ( 1 / ( n ^c A ) ) ) < x ) )
63		id 22	. . . . . . 7 |- ( n e. RR+ -> n e. RR+ )
64		rpcxpcl 24422	. . . . . . 7 |- ( ( n e. RR+ /\ A e. RR ) -> ( n ^c A ) e. RR+ )
65	63, 5, 64	syl2anr 495	. . . . . 6 |- ( ( A e. RR+ /\ n e. RR+ ) -> ( n ^c A ) e. RR+ )
66	65	rpreccld 11882	. . . . 5 |- ( ( A e. RR+ /\ n e. RR+ ) -> ( 1 / ( n ^c A ) ) e. RR+ )
67	66	rpcnd 11874	. . . 4 |- ( ( A e. RR+ /\ n e. RR+ ) -> ( 1 / ( n ^c A ) ) e. CC )
68	67	ralrimiva 2966	. . 3 |- ( A e. RR+ -> A. n e. RR+ ( 1 / ( n ^c A ) ) e. CC )
69		rpssre 11843	. . . 4 |- RR+ C_ RR
70	69	a1i 11	. . 3 |- ( A e. RR+ -> RR+ C_ RR )
71	68, 70	rlim0lt 14240	. 2 |- ( A e. RR+ -> ( ( n e. RR+ |-> ( 1 / ( n ^c A ) ) ) ~~> r 0 <-> A. x e. RR+ E. y e. RR A. n e. RR+ ( y < n -> ( abs ` ( 1 / ( n ^c A ) ) ) < x ) ) )
72	62, 71	mpbird 247	1 |- ( A e. RR+ -> ( n e. RR+ |-> ( 1 / ( n ^c A ) ) ) ~~> r 0 )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   < clt 10074   <_ cle 10075   -ucneg 10267   / cdiv 10684   RR+crp 11832   abscabs 13974   ~~> r crli 14216   ^c ccxp 24302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303  df-cxp 24304

This theorem is referenced by:  sqrtlim  24699  signsplypnf  30627