Theorem expcnvre 11668

Description: A sequence of powers of a nonnegative real number less than one converges to zero. (Contributed by Jim Kingdon, 28-Oct-2022.)

Hypotheses

Ref	Expression
expcnvre.ar	|- ( ph -> A e. RR )
expcnvre.a1	|- ( ph -> A < 1 )
expcnvre.a0	|- ( ph -> 0 <_ A )

Assertion

Ref	Expression
expcnvre	|- ( ph -> ( n e. NN0 |-> ( A ^ n ) ) ~~> 0 )

Distinct variable group:   A, n

Allowed substitution hint:   ph( n)

Proof of Theorem expcnvre

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

Step	Hyp	Ref	Expression
1		expcnvre.ar	. . 3 |- ( ph -> A e. RR )
2		1red 8041	. . 3 |- ( ph -> 1 e. RR )
3		expcnvre.a1	. . 3 |- ( ph -> A < 1 )
4		qbtwnre 10346	. . 3 |- ( ( A e. RR /\ 1 e. RR /\ A < 1 ) -> E. x e. QQ ( A < x /\ x < 1 ) )
5	1, 2, 3, 4	syl3anc 1249	. 2 |- ( ph -> E. x e. QQ ( A < x /\ x < 1 ) )
6		nn0uz 9636	. . 3 |- NN0 = ( ZZ>= ` 0 )
7		0zd 9338	. . 3 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> 0 e. ZZ )
8		qre 9699	. . . . . 6 |- ( x e. QQ -> x e. RR )
9	8	ad2antrl 490	. . . . 5 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> x e. RR )
10	9	recnd 8055	. . . 4 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> x e. CC )
11		0red 8027	. . . . . . 7 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> 0 e. RR )
12	1	adantr 276	. . . . . . . 8 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> A e. RR )
13		expcnvre.a0	. . . . . . . . 9 |- ( ph -> 0 <_ A )
14	13	adantr 276	. . . . . . . 8 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> 0 <_ A )
15		simprrl 539	. . . . . . . 8 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> A < x )
16	11, 12, 9, 14, 15	lelttrd 8151	. . . . . . 7 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> 0 < x )
17	11, 9, 16	ltled 8145	. . . . . 6 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> 0 <_ x )
18	9, 17	absidd 11332	. . . . 5 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> ( abs ` x ) = x )
19		simprrr 540	. . . . 5 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> x < 1 )
20	18, 19	eqbrtrd 4055	. . . 4 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> ( abs ` x ) < 1 )
21	9, 16	gt0ap0d 8656	. . . 4 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> x # 0 )
22	10, 20, 21	expcnvap0 11667	. . 3 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> ( n e. NN0 |-> ( x ^ n ) ) ~~> 0 )
23		nn0ex 9255	. . . . 5 |- NN0 e. _V
24	23	mptex 5788	. . . 4 |- ( n e. NN0 |-> ( A ^ n ) ) e. _V
25	24	a1i 9	. . 3 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> ( n e. NN0 |-> ( A ^ n ) ) e. _V )
26		simpr 110	. . . . 5 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> k e. NN0 )
27	9	adantr 276	. . . . . 6 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> x e. RR )
28	27, 26	reexpcld 10782	. . . . 5 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> ( x ^ k ) e. RR )
29		oveq2 5930	. . . . . 6 |- ( n = k -> ( x ^ n ) = ( x ^ k ) )
30		eqid 2196	. . . . . 6 |- ( n e. NN0 |-> ( x ^ n ) ) = ( n e. NN0 |-> ( x ^ n ) )
31	29, 30	fvmptg 5637	. . . . 5 |- ( ( k e. NN0 /\ ( x ^ k ) e. RR ) -> ( ( n e. NN0 |-> ( x ^ n ) ) ` k ) = ( x ^ k ) )
32	26, 28, 31	syl2anc 411	. . . 4 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( x ^ n ) ) ` k ) = ( x ^ k ) )
33	32, 28	eqeltrd 2273	. . 3 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( x ^ n ) ) ` k ) e. RR )
34	12	adantr 276	. . . . . 6 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> A e. RR )
35	34, 26	reexpcld 10782	. . . . 5 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> ( A ^ k ) e. RR )
36		oveq2 5930	. . . . . 6 |- ( n = k -> ( A ^ n ) = ( A ^ k ) )
37		eqid 2196	. . . . . 6 |- ( n e. NN0 |-> ( A ^ n ) ) = ( n e. NN0 |-> ( A ^ n ) )
38	36, 37	fvmptg 5637	. . . . 5 |- ( ( k e. NN0 /\ ( A ^ k ) e. RR ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) = ( A ^ k ) )
39	26, 35, 38	syl2anc 411	. . . 4 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) = ( A ^ k ) )
40	39, 35	eqeltrd 2273	. . 3 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) e. RR )
41	14	adantr 276	. . . . 5 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> 0 <_ A )
42	15	adantr 276	. . . . . 6 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> A < x )
43	34, 27, 42	ltled 8145	. . . . 5 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> A <_ x )
44		leexp1a 10686	. . . . 5 |- ( ( ( A e. RR /\ x e. RR /\ k e. NN0 ) /\ ( 0 <_ A /\ A <_ x ) ) -> ( A ^ k ) <_ ( x ^ k ) )
45	34, 27, 26, 41, 43, 44	syl32anc 1257	. . . 4 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> ( A ^ k ) <_ ( x ^ k ) )
46	45, 39, 32	3brtr4d 4065	. . 3 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) <_ ( ( n e. NN0 |-> ( x ^ n ) ) ` k ) )
47	34, 26, 41	expge0d 10783	. . . 4 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> 0 <_ ( A ^ k ) )
48	47, 39	breqtrrd 4061	. . 3 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> 0 <_ ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) )
49	6, 7, 22, 25, 33, 40, 46, 48	climsqz2 11501	. 2 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> ( n e. NN0 |-> ( A ^ n ) ) ~~> 0 )
50	5, 49	rexlimddv 2619	1 |- ( ph -> ( n e. NN0 |-> ( A ^ n ) ) ~~> 0 )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   E.wrex 2476   _Vcvv 2763   class class class wbr 4033   |-> cmpt 4094   ` cfv 5258  (class class class)co 5922   RRcr 7878   0cc0 7879   1c1 7880   < clt 8061   <_ cle 8062   NN0cn0 9249   QQcq 9693   ^cexp 10630   abscabs 11162   ~~> cli 11443

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-seqfrec 10540  df-exp 10631  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-clim 11444

This theorem is referenced by:  expcnv  11669