Theorem expcnvre 11304

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 7805	. . 3 |- ( ph -> 1 e. RR )
3		expcnvre.a1	. . 3 |- ( ph -> A < 1 )
4		qbtwnre 10065	. . 3 |- ( ( A e. RR /\ 1 e. RR /\ A < 1 ) -> E. x e. QQ ( A < x /\ x < 1 ) )
5	1, 2, 3, 4	syl3anc 1217	. 2 |- ( ph -> E. x e. QQ ( A < x /\ x < 1 ) )
6		nn0uz 9384	. . 3 |- NN0 = ( ZZ>= ` 0 )
7		0zd 9090	. . 3 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> 0 e. ZZ )
8		qre 9444	. . . . . 6 |- ( x e. QQ -> x e. RR )
9	8	ad2antrl 482	. . . . 5 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> x e. RR )
10	9	recnd 7818	. . . 4 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> x e. CC )
11		0red 7791	. . . . . . 7 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> 0 e. RR )
12	1	adantr 274	. . . . . . . 8 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> A e. RR )
13		expcnvre.a0	. . . . . . . . 9 |- ( ph -> 0 <_ A )
14	13	adantr 274	. . . . . . . 8 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> 0 <_ A )
15		simprrl 529	. . . . . . . 8 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> A < x )
16	11, 12, 9, 14, 15	lelttrd 7911	. . . . . . 7 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> 0 < x )
17	11, 9, 16	ltled 7905	. . . . . 6 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> 0 <_ x )
18	9, 17	absidd 10971	. . . . 5 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> ( abs ` x ) = x )
19		simprrr 530	. . . . 5 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> x < 1 )
20	18, 19	eqbrtrd 3958	. . . 4 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> ( abs ` x ) < 1 )
21	9, 16	gt0ap0d 8415	. . . 4 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> x # 0 )
22	10, 20, 21	expcnvap0 11303	. . 3 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> ( n e. NN0 |-> ( x ^ n ) ) ~~> 0 )
23		nn0ex 9007	. . . . 5 |- NN0 e. _V
24	23	mptex 5654	. . . 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 109	. . . . 5 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> k e. NN0 )
27	9	adantr 274	. . . . . 6 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> x e. RR )
28	27, 26	reexpcld 10472	. . . . 5 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> ( x ^ k ) e. RR )
29		oveq2 5790	. . . . . 6 |- ( n = k -> ( x ^ n ) = ( x ^ k ) )
30		eqid 2140	. . . . . 6 |- ( n e. NN0 |-> ( x ^ n ) ) = ( n e. NN0 |-> ( x ^ n ) )
31	29, 30	fvmptg 5505	. . . . 5 |- ( ( k e. NN0 /\ ( x ^ k ) e. RR ) -> ( ( n e. NN0 |-> ( x ^ n ) ) ` k ) = ( x ^ k ) )
32	26, 28, 31	syl2anc 409	. . . 4 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( x ^ n ) ) ` k ) = ( x ^ k ) )
33	32, 28	eqeltrd 2217	. . 3 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( x ^ n ) ) ` k ) e. RR )
34	12	adantr 274	. . . . . 6 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> A e. RR )
35	34, 26	reexpcld 10472	. . . . 5 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> ( A ^ k ) e. RR )
36		oveq2 5790	. . . . . 6 |- ( n = k -> ( A ^ n ) = ( A ^ k ) )
37		eqid 2140	. . . . . 6 |- ( n e. NN0 |-> ( A ^ n ) ) = ( n e. NN0 |-> ( A ^ n ) )
38	36, 37	fvmptg 5505	. . . . 5 |- ( ( k e. NN0 /\ ( A ^ k ) e. RR ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) = ( A ^ k ) )
39	26, 35, 38	syl2anc 409	. . . 4 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) = ( A ^ k ) )
40	39, 35	eqeltrd 2217	. . 3 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) e. RR )
41	14	adantr 274	. . . . 5 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> 0 <_ A )
42	15	adantr 274	. . . . . 6 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> A < x )
43	34, 27, 42	ltled 7905	. . . . 5 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> A <_ x )
44		leexp1a 10379	. . . . 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 1225	. . . 4 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> ( A ^ k ) <_ ( x ^ k ) )
46	45, 39, 32	3brtr4d 3968	. . 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 10473	. . . 4 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> 0 <_ ( A ^ k ) )
48	47, 39	breqtrrd 3964	. . 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 11137	. 2 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> ( n e. NN0 |-> ( A ^ n ) ) ~~> 0 )
50	5, 49	rexlimddv 2557	1 |- ( ph -> ( n e. NN0 |-> ( A ^ n ) ) ~~> 0 )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 1481   E.wrex 2418   _Vcvv 2689   class class class wbr 3937   |-> cmpt 3997   ` cfv 5131  (class class class)co 5782   RRcr 7643   0cc0 7644   1c1 7645   < clt 7824   <_ cle 7825   NN0cn0 9001   QQcq 9438   ^cexp 10323   abscabs 10801   ~~> cli 11079

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762  ax-arch 7763  ax-caucvg 7764

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-frec 6296  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-2 8803  df-3 8804  df-4 8805  df-n0 9002  df-z 9079  df-uz 9351  df-q 9439  df-rp 9471  df-seqfrec 10250  df-exp 10324  df-cj 10646  df-re 10647  df-im 10648  df-rsqrt 10802  df-abs 10803  df-clim 11080

This theorem is referenced by:  expcnv  11305