Theorem expcnvre 11503

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 7968	. . 3 |- ( ph -> 1 e. RR )
3		expcnvre.a1	. . 3 |- ( ph -> A < 1 )
4		qbtwnre 10251	. . 3 |- ( ( A e. RR /\ 1 e. RR /\ A < 1 ) -> E. x e. QQ ( A < x /\ x < 1 ) )
5	1, 2, 3, 4	syl3anc 1238	. 2 |- ( ph -> E. x e. QQ ( A < x /\ x < 1 ) )
6		nn0uz 9557	. . 3 |- NN0 = ( ZZ>= ` 0 )
7		0zd 9260	. . 3 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> 0 e. ZZ )
8		qre 9620	. . . . . 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 7981	. . . 4 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> x e. CC )
11		0red 7954	. . . . . . 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 8077	. . . . . . 7 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> 0 < x )
17	11, 9, 16	ltled 8071	. . . . . 6 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> 0 <_ x )
18	9, 17	absidd 11168	. . . . 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 4024	. . . 4 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> ( abs ` x ) < 1 )
21	9, 16	gt0ap0d 8581	. . . 4 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> x # 0 )
22	10, 20, 21	expcnvap0 11502	. . 3 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> ( n e. NN0 |-> ( x ^ n ) ) ~~> 0 )
23		nn0ex 9177	. . . . 5 |- NN0 e. _V
24	23	mptex 5740	. . . 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 10664	. . . . 5 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> ( x ^ k ) e. RR )
29		oveq2 5879	. . . . . 6 |- ( n = k -> ( x ^ n ) = ( x ^ k ) )
30		eqid 2177	. . . . . 6 |- ( n e. NN0 |-> ( x ^ n ) ) = ( n e. NN0 |-> ( x ^ n ) )
31	29, 30	fvmptg 5590	. . . . 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 2254	. . 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 10664	. . . . 5 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> ( A ^ k ) e. RR )
36		oveq2 5879	. . . . . 6 |- ( n = k -> ( A ^ n ) = ( A ^ k ) )
37		eqid 2177	. . . . . 6 |- ( n e. NN0 |-> ( A ^ n ) ) = ( n e. NN0 |-> ( A ^ n ) )
38	36, 37	fvmptg 5590	. . . . 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 2254	. . 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 8071	. . . . 5 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> A <_ x )
44		leexp1a 10569	. . . . 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 1246	. . . 4 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> ( A ^ k ) <_ ( x ^ k ) )
46	45, 39, 32	3brtr4d 4034	. . 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 10665	. . . 4 |- ( ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) /\ k e. NN0 ) -> 0 <_ ( A ^ k ) )
48	47, 39	breqtrrd 4030	. . 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 11336	. 2 |- ( ( ph /\ ( x e. QQ /\ ( A < x /\ x < 1 ) ) ) -> ( n e. NN0 |-> ( A ^ n ) ) ~~> 0 )
50	5, 49	rexlimddv 2599	1 |- ( ph -> ( n e. NN0 |-> ( A ^ n ) ) ~~> 0 )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   E.wrex 2456   _Vcvv 2737   class class class wbr 4002   |-> cmpt 4063   ` cfv 5214  (class class class)co 5871   RRcr 7806   0cc0 7807   1c1 7808   < clt 7987   <_ cle 7988   NN0cn0 9171   QQcq 9614   ^cexp 10513   abscabs 10998   ~~> cli 11278

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-iinf 4586  ax-cnex 7898  ax-resscn 7899  ax-1cn 7900  ax-1re 7901  ax-icn 7902  ax-addcl 7903  ax-addrcl 7904  ax-mulcl 7905  ax-mulrcl 7906  ax-addcom 7907  ax-mulcom 7908  ax-addass 7909  ax-mulass 7910  ax-distr 7911  ax-i2m1 7912  ax-0lt1 7913  ax-1rid 7914  ax-0id 7915  ax-rnegex 7916  ax-precex 7917  ax-cnre 7918  ax-pre-ltirr 7919  ax-pre-ltwlin 7920  ax-pre-lttrn 7921  ax-pre-apti 7922  ax-pre-ltadd 7923  ax-pre-mulgt0 7924  ax-pre-mulext 7925  ax-arch 7926  ax-caucvg 7927

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-tr 4101  df-id 4292  df-po 4295  df-iso 4296  df-iord 4365  df-on 4367  df-ilim 4368  df-suc 4370  df-iom 4589  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5176  df-fun 5216  df-fn 5217  df-f 5218  df-f1 5219  df-fo 5220  df-f1o 5221  df-fv 5222  df-riota 5827  df-ov 5874  df-oprab 5875  df-mpo 5876  df-1st 6137  df-2nd 6138  df-recs 6302  df-frec 6388  df-pnf 7989  df-mnf 7990  df-xr 7991  df-ltxr 7992  df-le 7993  df-sub 8125  df-neg 8126  df-reap 8527  df-ap 8534  df-div 8625  df-inn 8915  df-2 8973  df-3 8974  df-4 8975  df-n0 9172  df-z 9249  df-uz 9524  df-q 9615  df-rp 9649  df-seqfrec 10440  df-exp 10514  df-cj 10843  df-re 10844  df-im 10845  df-rsqrt 10999  df-abs 11000  df-clim 11279

This theorem is referenced by:  expcnv  11504