Theorem expcnvre 12270

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 8341	. . 3 |- ( ph -> 1 e. RR )
3		expcnvre.a1	. . 3 |- ( ph -> A < 1 )
4		qbtwnre 10691	. . 3 |- ( ( A e. RR /\ 1 e. RR /\ A < 1 ) -> E. x e. QQ ( A < x /\ x < 1 ) )
5	1, 2, 3, 4	syl3anc 1278	. 2 |- ( ph -> E. x e. QQ ( A < x /\ x < 1 ) )
6		nn0uz 9957	. . 3 |- NN0 = ( ZZ>= ` 0 )
7		0zd 9656	. . 3 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> 0 e. ZZ )
8		qre 10025	. . . . . 6 |- ( x e. QQ -> x e. RR )
9	8	ad2antrl 494	. . . . 5 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> x e. RR )
10	9	recnd 8354	. . . 4 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> x e. CC )
11		0red 8327	. . . . . . 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 545	. . . . . . . 8 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> A < x )
16	11, 12, 9, 14, 15	lelttrd 8451	. . . . . . 7 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> 0 < x )
17	11, 9, 16	ltled 8445	. . . . . 6 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> 0 <_ x )
18	9, 17	absidd 11933	. . . . 5 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> ( abs ` x ) = x )
19		simprrr 546	. . . . 5 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> x < 1 )
20	18, 19	eqbrtrd 4152	. . . 4 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> ( abs ` x ) < 1 )
21	9, 16	gt0ap0d 8957	. . . 4 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> x # 0 )
22	10, 20, 21	expcnvap0 12269	. . 3 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> ( n e. NN0 |-> ( x ^ n ) ) ~~> 0 )
23		nn0ex 9569	. . . . 5 |- NN0 e. _V
24	23	mptex 5943	. . . 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 11128	. . . . 5 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> ( x ^ k ) e. RR )
29		oveq2 6093	. . . . . 6 |- ( n = k -> ( x ^ n ) = ( x ^ k ) )
30		eqid 2238	. . . . . 6 |- ( n e. NN0 |-> ( x ^ n ) ) = ( n e. NN0 |-> ( x ^ n ) )
31	29, 30	fvmptg 5781	. . . . 5 |- ( ( k e. NN0 /\ ( x ^ k ) e. RR ) -> ( ( n e. NN0 |-> ( x ^ n ) ) ` k ) = ( x ^ k ) )
32	26, 28, 31	syl2anc 415	. . . 4 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( x ^ n ) ) ` k ) = ( x ^ k ) )
33	32, 28	eqeltrd 2315	. . 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 11128	. . . . 5 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> ( A ^ k ) e. RR )
36		oveq2 6093	. . . . . 6 |- ( n = k -> ( A ^ n ) = ( A ^ k ) )
37		eqid 2238	. . . . . 6 |- ( n e. NN0 |-> ( A ^ n ) ) = ( n e. NN0 |-> ( A ^ n ) )
38	36, 37	fvmptg 5781	. . . . 5 |- ( ( k e. NN0 /\ ( A ^ k ) e. RR ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) = ( A ^ k ) )
39	26, 35, 38	syl2anc 415	. . . 4 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> ( ( n e. NN0 |-> ( A ^ n ) ) ` k ) = ( A ^ k ) )
40	39, 35	eqeltrd 2315	. . 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 8445	. . . . 5 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> A <_ x )
44		leexp1a 11031	. . . . 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 1286	. . . 4 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> ( A ^ k ) <_ ( x ^ k ) )
46	45, 39, 32	3brtr4d 4162	. . 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 11129	. . . 4 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> 0 <_ ( A ^ k ) )
48	47, 39	breqtrrd 4158	. . 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 12102	. 2 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> ( n e. NN0 |-> ( A ^ n ) ) ~~> 0 )
50	5, 49	rexlimddv 2673	1 |- ( ph -> ( n e. NN0 |-> ( A ^ n ) ) ~~> 0 )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   E.wrex 2529   _Vcvv 2821   class class class wbr 4130   |-> cmpt 4192   ` cfv 5377  (class class class)co 6085   RRcr 8178   0cc0 8179   1c1 8180   < clt 8360   <_ cle 8361   NN0cn0 9563   QQcq 10019   ^cexp 10975   abscabs 11763   ~~> cli 12044

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-seqfrec 10885  df-exp 10976  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-clim 12045

This theorem is used by:  expcnv  12271