Theorem expcnvre 12029

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 8172	. . 3 |- ( ph -> 1 e. RR )
3		expcnvre.a1	. . 3 |- ( ph -> A < 1 )
4		qbtwnre 10488	. . 3 |- ( ( A e. RR /\ 1 e. RR /\ A < 1 ) -> E. x e. QQ ( A < x /\ x < 1 ) )
5	1, 2, 3, 4	syl3anc 1271	. 2 |- ( ph -> E. x e. QQ ( A < x /\ x < 1 ) )
6		nn0uz 9769	. . 3 |- NN0 = ( ZZ>= ` 0 )
7		0zd 9469	. . 3 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> 0 e. ZZ )
8		qre 9832	. . . . . 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 8186	. . . 4 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> x e. CC )
11		0red 8158	. . . . . . 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 8282	. . . . . . 7 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> 0 < x )
17	11, 9, 16	ltled 8276	. . . . . 6 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> 0 <_ x )
18	9, 17	absidd 11693	. . . . 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 4105	. . . 4 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> ( abs ` x ) < 1 )
21	9, 16	gt0ap0d 8787	. . . 4 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> x # 0 )
22	10, 20, 21	expcnvap0 12028	. . 3 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> ( n e. NN0 |-> ( x ^ n ) ) ~~> 0 )
23		nn0ex 9386	. . . . 5 |- NN0 e. _V
24	23	mptex 5869	. . . 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 10924	. . . . 5 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> ( x ^ k ) e. RR )
29		oveq2 6015	. . . . . 6 |- ( n = k -> ( x ^ n ) = ( x ^ k ) )
30		eqid 2229	. . . . . 6 |- ( n e. NN0 |-> ( x ^ n ) ) = ( n e. NN0 |-> ( x ^ n ) )
31	29, 30	fvmptg 5712	. . . . 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 2306	. . 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 10924	. . . . 5 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> ( A ^ k ) e. RR )
36		oveq2 6015	. . . . . 6 |- ( n = k -> ( A ^ n ) = ( A ^ k ) )
37		eqid 2229	. . . . . 6 |- ( n e. NN0 |-> ( A ^ n ) ) = ( n e. NN0 |-> ( A ^ n ) )
38	36, 37	fvmptg 5712	. . . . 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 2306	. . 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 8276	. . . . 5 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> A <_ x )
44		leexp1a 10828	. . . . 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 1279	. . . 4 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> ( A ^ k ) <_ ( x ^ k ) )
46	45, 39, 32	3brtr4d 4115	. . 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 10925	. . . 4 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> 0 <_ ( A ^ k ) )
48	47, 39	breqtrrd 4111	. . 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 11862	. 2 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> ( n e. NN0 |-> ( A ^ n ) ) ~~> 0 )
50	5, 49	rexlimddv 2653	1 |- ( ph -> ( n e. NN0 |-> ( A ^ n ) ) ~~> 0 )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   E.wrex 2509   _Vcvv 2799   class class class wbr 4083   |-> cmpt 4145   ` cfv 5318  (class class class)co 6007   RRcr 8009   0cc0 8010   1c1 8011   < clt 8192   <_ cle 8193   NN0cn0 9380   QQcq 9826   ^cexp 10772   abscabs 11523   ~~> cli 11804

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 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128  ax-arch 8129  ax-caucvg 8130

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 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-n0 9381  df-z 9458  df-uz 9734  df-q 9827  df-rp 9862  df-seqfrec 10682  df-exp 10773  df-cj 11368  df-re 11369  df-im 11370  df-rsqrt 11524  df-abs 11525  df-clim 11805

This theorem is referenced by:  expcnv  12030