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Theorem expcnvre 11543
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 8002 . . 3  |-  ( ph  ->  1  e.  RR )
3 expcnvre.a1 . . 3  |-  ( ph  ->  A  <  1 )
4 qbtwnre 10287 . . 3  |-  ( ( A  e.  RR  /\  1  e.  RR  /\  A  <  1 )  ->  E. x  e.  QQ  ( A  < 
x  /\  x  <  1 ) )
51, 2, 3, 4syl3anc 1249 . 2  |-  ( ph  ->  E. x  e.  QQ  ( A  <  x  /\  x  <  1 ) )
6 nn0uz 9592 . . 3  |-  NN0  =  ( ZZ>= `  0 )
7 0zd 9295 . . 3  |-  ( (
ph  /\  ( x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  ->  0  e.  ZZ )
8 qre 9655 . . . . . 6  |-  ( x  e.  QQ  ->  x  e.  RR )
98ad2antrl 490 . . . . 5  |-  ( (
ph  /\  ( x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  ->  x  e.  RR )
109recnd 8016 . . . 4  |-  ( (
ph  /\  ( x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  ->  x  e.  CC )
11 0red 7988 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  ->  0  e.  RR )
121adantr 276 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  ->  A  e.  RR )
13 expcnvre.a0 . . . . . . . . 9  |-  ( ph  ->  0  <_  A )
1413adantr 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
)
1611, 12, 9, 14, 15lelttrd 8112 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  ->  0  <  x
)
1711, 9, 16ltled 8106 . . . . . 6  |-  ( (
ph  /\  ( x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  ->  0  <_  x
)
189, 17absidd 11208 . . . . 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
)
2018, 19eqbrtrd 4040 . . . 4  |-  ( (
ph  /\  ( x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  ->  ( abs `  x
)  <  1 )
219, 16gt0ap0d 8616 . . . 4  |-  ( (
ph  /\  ( x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  ->  x #  0 )
2210, 20, 21expcnvap0 11542 . . 3  |-  ( (
ph  /\  ( x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  ->  ( n  e. 
NN0  |->  ( x ^
n ) )  ~~>  0 )
23 nn0ex 9212 . . . . 5  |-  NN0  e.  _V
2423mptex 5763 . . . 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 110 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  /\  k  e. 
NN0 )  ->  k  e.  NN0 )
279adantr 276 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  /\  k  e. 
NN0 )  ->  x  e.  RR )
2827, 26reexpcld 10702 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  /\  k  e. 
NN0 )  ->  (
x ^ k )  e.  RR )
29 oveq2 5904 . . . . . 6  |-  ( n  =  k  ->  (
x ^ n )  =  ( x ^
k ) )
30 eqid 2189 . . . . . 6  |-  ( n  e.  NN0  |->  ( x ^ n ) )  =  ( n  e. 
NN0  |->  ( x ^
n ) )
3129, 30fvmptg 5613 . . . . 5  |-  ( ( k  e.  NN0  /\  ( x ^ k
)  e.  RR )  ->  ( ( n  e.  NN0  |->  ( x ^ n ) ) `
 k )  =  ( x ^ k
) )
3226, 28, 31syl2anc 411 . . . 4  |-  ( ( ( ph  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  /\  k  e. 
NN0 )  ->  (
( n  e.  NN0  |->  ( x ^ n
) ) `  k
)  =  ( x ^ k ) )
3332, 28eqeltrd 2266 . . 3  |-  ( ( ( ph  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  /\  k  e. 
NN0 )  ->  (
( n  e.  NN0  |->  ( x ^ n
) ) `  k
)  e.  RR )
3412adantr 276 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  /\  k  e. 
NN0 )  ->  A  e.  RR )
3534, 26reexpcld 10702 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  /\  k  e. 
NN0 )  ->  ( A ^ k )  e.  RR )
36 oveq2 5904 . . . . . 6  |-  ( n  =  k  ->  ( A ^ n )  =  ( A ^ k
) )
37 eqid 2189 . . . . . 6  |-  ( n  e.  NN0  |->  ( A ^ n ) )  =  ( n  e. 
NN0  |->  ( A ^
n ) )
3836, 37fvmptg 5613 . . . . 5  |-  ( ( k  e.  NN0  /\  ( A ^ k )  e.  RR )  -> 
( ( n  e. 
NN0  |->  ( A ^
n ) ) `  k )  =  ( A ^ k ) )
3926, 35, 38syl2anc 411 . . . 4  |-  ( ( ( ph  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  /\  k  e. 
NN0 )  ->  (
( n  e.  NN0  |->  ( A ^ n ) ) `  k )  =  ( A ^
k ) )
4039, 35eqeltrd 2266 . . 3  |-  ( ( ( ph  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  /\  k  e. 
NN0 )  ->  (
( n  e.  NN0  |->  ( A ^ n ) ) `  k )  e.  RR )
4114adantr 276 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  /\  k  e. 
NN0 )  ->  0  <_  A )
4215adantr 276 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  /\  k  e. 
NN0 )  ->  A  <  x )
4334, 27, 42ltled 8106 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  /\  k  e. 
NN0 )  ->  A  <_  x )
44 leexp1a 10606 . . . . 5  |-  ( ( ( A  e.  RR  /\  x  e.  RR  /\  k  e.  NN0 )  /\  ( 0  <_  A  /\  A  <_  x ) )  ->  ( A ^ k )  <_ 
( x ^ k
) )
4534, 27, 26, 41, 43, 44syl32anc 1257 . . . 4  |-  ( ( ( ph  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  /\  k  e. 
NN0 )  ->  ( A ^ k )  <_ 
( x ^ k
) )
4645, 39, 323brtr4d 4050 . . 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 10703 . . . 4  |-  ( ( ( ph  /\  (
x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  /\  k  e. 
NN0 )  ->  0  <_  ( A ^ k
) )
4847, 39breqtrrd 4046 . . 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 11376 . 2  |-  ( (
ph  /\  ( x  e.  QQ  /\  ( A  <  x  /\  x  <  1 ) ) )  ->  ( n  e. 
NN0  |->  ( A ^
n ) )  ~~>  0 )
505, 49rexlimddv 2612 1  |-  ( ph  ->  ( n  e.  NN0  |->  ( A ^ n ) )  ~~>  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160   E.wrex 2469   _Vcvv 2752   class class class wbr 4018    |-> cmpt 4079   ` cfv 5235  (class class class)co 5896   RRcr 7840   0cc0 7841   1c1 7842    < clt 8022    <_ cle 8023   NN0cn0 9206   QQcq 9649   ^cexp 10550   abscabs 11038    ~~> cli 11318
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-mulrcl 7940  ax-addcom 7941  ax-mulcom 7942  ax-addass 7943  ax-mulass 7944  ax-distr 7945  ax-i2m1 7946  ax-0lt1 7947  ax-1rid 7948  ax-0id 7949  ax-rnegex 7950  ax-precex 7951  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-ltwlin 7954  ax-pre-lttrn 7955  ax-pre-apti 7956  ax-pre-ltadd 7957  ax-pre-mulgt0 7958  ax-pre-mulext 7959  ax-arch 7960  ax-caucvg 7961
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-frec 6416  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028  df-sub 8160  df-neg 8161  df-reap 8562  df-ap 8569  df-div 8660  df-inn 8950  df-2 9008  df-3 9009  df-4 9010  df-n0 9207  df-z 9284  df-uz 9559  df-q 9650  df-rp 9684  df-seqfrec 10477  df-exp 10551  df-cj 10883  df-re 10884  df-im 10885  df-rsqrt 11039  df-abs 11040  df-clim 11319
This theorem is referenced by:  expcnv  11544
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