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Theorem expcnvre 10951
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.
StepHypRef Expression
1 expcnvre.ar . . 3  |-  ( ph  ->  A  e.  RR )
2 1red 7557 . . 3  |-  ( ph  ->  1  e.  RR )
3 expcnvre.a1 . . 3  |-  ( ph  ->  A  <  1 )
4 qbtwnre 9722 . . 3  |-  ( ( A  e.  RR  /\  1  e.  RR  /\  A  <  1 )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  1 ) )
51, 2, 3, 4syl3anc 1175 . 2  |-  ( ph  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  1 ) )
6 nn0uz 9107 . . 3  |-  NN0  =  ( ZZ>= `  0 )
7 0zd 8816 . . 3  |-  ( (
ph  /\  ( x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  ->  0  e.  ZZ )
8 qre 9164 . . . . . 6  |-  ( x  e.  QQ  ->  x  e.  RR )
98ad2antrl 475 . . . . 5  |-  ( (
ph  /\  ( x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  ->  x  e.  RR )
109recnd 7570 . . . 4  |-  ( (
ph  /\  ( x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  ->  x  e.  CC )
11 0red 7543 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  ->  0  e.  RR )
121adantr 271 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  ->  A  e.  RR )
13 expcnvre.a0 . . . . . . . . 9  |-  ( ph  ->  0  <_  A )
1413adantr 271 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  ->  0  <_  A
)
15 simprrl 507 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  ->  A  <  x
)
1611, 12, 9, 14, 15lelttrd 7662 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  ->  0  <  x
)
1711, 9, 16ltled 7656 . . . . . 6  |-  ( (
ph  /\  ( x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  ->  0  <_  x
)
189, 17absidd 10654 . . . . 5  |-  ( (
ph  /\  ( x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  ->  ( abs `  x
)  =  x )
19 simprrr 508 . . . . 5  |-  ( (
ph  /\  ( x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  ->  x  <  1
)
2018, 19eqbrtrd 3871 . . . 4  |-  ( (
ph  /\  ( x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  ->  ( abs `  x
)  <  1 )
219, 16gt0ap0d 8159 . . . 4  |-  ( (
ph  /\  ( x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  ->  x #  0 )
2210, 20, 21expcnvap0 10950 . . 3  |-  ( (
ph  /\  ( x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  ->  ( n  e. 
NN0  |->  ( x ^
n ) )  ~~>  0 )
23 nn0ex 8733 . . . . 5  |-  NN0  e.  _V
2423mptex 5537 . . . 4  |-  ( n  e.  NN0  |->  ( A ^ n ) )  e.  _V
2524a1i 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 )
279adantr 271 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  /\  k  e. 
NN0 )  ->  x  e.  RR )
2827, 26reexpcld 10157 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  /\  k  e. 
NN0 )  ->  (
x ^ k )  e.  RR )
29 oveq2 5674 . . . . . 6  |-  ( n  =  k  ->  (
x ^ n )  =  ( x ^
k ) )
30 eqid 2089 . . . . . 6  |-  ( n  e.  NN0  |->  ( x ^ n ) )  =  ( n  e. 
NN0  |->  ( x ^
n ) )
3129, 30fvmptg 5393 . . . . 5  |-  ( ( k  e.  NN0  /\  ( x ^ k
)  e.  RR )  ->  ( ( n  e.  NN0  |->  ( x ^ n ) ) `
 k )  =  ( x ^ k
) )
3226, 28, 31syl2anc 404 . . . 4  |-  ( ( ( ph  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  /\  k  e. 
NN0 )  ->  (
( n  e.  NN0  |->  ( x ^ n
) ) `  k
)  =  ( x ^ k ) )
3332, 28eqeltrd 2165 . . 3  |-  ( ( ( ph  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  /\  k  e. 
NN0 )  ->  (
( n  e.  NN0  |->  ( x ^ n
) ) `  k
)  e.  RR )
3412adantr 271 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  /\  k  e. 
NN0 )  ->  A  e.  RR )
3534, 26reexpcld 10157 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  /\  k  e. 
NN0 )  ->  ( A ^ k )  e.  RR )
36 oveq2 5674 . . . . . 6  |-  ( n  =  k  ->  ( A ^ n )  =  ( A ^ k
) )
37 eqid 2089 . . . . . 6  |-  ( n  e.  NN0  |->  ( A ^ n ) )  =  ( n  e. 
NN0  |->  ( A ^
n ) )
3836, 37fvmptg 5393 . . . . 5  |-  ( ( k  e.  NN0  /\  ( A ^ k )  e.  RR )  -> 
( ( n  e. 
NN0  |->  ( A ^
n ) ) `  k )  =  ( A ^ k ) )
3926, 35, 38syl2anc 404 . . . 4  |-  ( ( ( ph  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  /\  k  e. 
NN0 )  ->  (
( n  e.  NN0  |->  ( A ^ n ) ) `  k )  =  ( A ^
k ) )
4039, 35eqeltrd 2165 . . 3  |-  ( ( ( ph  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  /\  k  e. 
NN0 )  ->  (
( n  e.  NN0  |->  ( A ^ n ) ) `  k )  e.  RR )
4114adantr 271 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  /\  k  e. 
NN0 )  ->  0  <_  A )
4215adantr 271 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  /\  k  e. 
NN0 )  ->  A  <  x )
4334, 27, 42ltled 7656 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  /\  k  e. 
NN0 )  ->  A  <_  x )
44 leexp1a 10064 . . . . 5  |-  ( ( ( A  e.  RR  /\  x  e.  RR  /\  k  e.  NN0 )  /\  ( 0  <_  A  /\  A  <_  x ) )  ->  ( A ^ k )  <_ 
( x ^ k
) )
4534, 27, 26, 41, 43, 44syl32anc 1183 . . . 4  |-  ( ( ( ph  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  /\  k  e. 
NN0 )  ->  ( A ^ k )  <_ 
( x ^ k
) )
4645, 39, 323brtr4d 3881 . . 3  |-  ( ( ( ph  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  /\  k  e. 
NN0 )  ->  (
( n  e.  NN0  |->  ( A ^ n ) ) `  k )  <_  ( ( n  e.  NN0  |->  ( x ^ n ) ) `
 k ) )
4734, 26, 41expge0d 10158 . . . 4  |-  ( ( ( ph  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  /\  k  e. 
NN0 )  ->  0  <_  ( A ^ k
) )
4847, 39breqtrrd 3877 . . 3  |-  ( ( ( ph  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  /\  k  e. 
NN0 )  ->  0  <_  ( ( n  e. 
NN0  |->  ( A ^
n ) ) `  k ) )
496, 7, 22, 25, 33, 40, 46, 48climsqz2 10778 . 2  |-  ( (
ph  /\  ( x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  ->  ( n  e. 
NN0  |->  ( A ^
n ) )  ~~>  0 )
505, 49rexlimddv 2494 1  |-  ( ph  ->  ( n  e.  NN0  |->  ( A ^ n ) )  ~~>  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1290    e. wcel 1439   E.wrex 2361   _Vcvv 2620   class class class wbr 3851    |-> cmpt 3905   ` cfv 5028  (class class class)co 5666   RRcr 7403   0cc0 7404   1c1 7405    < clt 7576    <_ cle 7577   NN0cn0 8727   QQcq 9158   ^cexp 10008   abscabs 10484    ~~> cli 10720
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416  ax-cnex 7490  ax-resscn 7491  ax-1cn 7492  ax-1re 7493  ax-icn 7494  ax-addcl 7495  ax-addrcl 7496  ax-mulcl 7497  ax-mulrcl 7498  ax-addcom 7499  ax-mulcom 7500  ax-addass 7501  ax-mulass 7502  ax-distr 7503  ax-i2m1 7504  ax-0lt1 7505  ax-1rid 7506  ax-0id 7507  ax-rnegex 7508  ax-precex 7509  ax-cnre 7510  ax-pre-ltirr 7511  ax-pre-ltwlin 7512  ax-pre-lttrn 7513  ax-pre-apti 7514  ax-pre-ltadd 7515  ax-pre-mulgt0 7516  ax-pre-mulext 7517  ax-arch 7518  ax-caucvg 7519
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-if 3398  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-ilim 4205  df-suc 4207  df-iom 4419  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-frec 6170  df-pnf 7578  df-mnf 7579  df-xr 7580  df-ltxr 7581  df-le 7582  df-sub 7709  df-neg 7710  df-reap 8106  df-ap 8113  df-div 8194  df-inn 8477  df-2 8535  df-3 8536  df-4 8537  df-n0 8728  df-z 8805  df-uz 9074  df-q 9159  df-rp 9189  df-iseq 9907  df-seq3 9908  df-exp 10009  df-cj 10330  df-re 10331  df-im 10332  df-rsqrt 10485  df-abs 10486  df-clim 10721
This theorem is referenced by:  expcnv  10952
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