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Theorem rlimcxp 22347
Description: Any power to a positive exponent of a converging sequence also converges. (Contributed by Mario Carneiro, 18-Sep-2014.)
Hypotheses
Ref Expression
rlimcxp.1  |-  ( (
ph  /\  n  e.  A )  ->  B  e.  V )
rlimcxp.2  |-  ( ph  ->  ( n  e.  A  |->  B )  ~~> r  0 )
rlimcxp.3  |-  ( ph  ->  C  e.  RR+ )
Assertion
Ref Expression
rlimcxp  |-  ( ph  ->  ( n  e.  A  |->  ( B  ^c  C ) )  ~~> r  0 )
Distinct variable groups:    A, n    C, n    ph, n
Allowed substitution hints:    B( n)    V( n)

Proof of Theorem rlimcxp
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rlimcxp.2 . . . . . . . . 9  |-  ( ph  ->  ( n  e.  A  |->  B )  ~~> r  0 )
2 rlimf 12971 . . . . . . . . 9  |-  ( ( n  e.  A  |->  B )  ~~> r  0  -> 
( n  e.  A  |->  B ) : dom  ( n  e.  A  |->  B ) --> CC )
31, 2syl 16 . . . . . . . 8  |-  ( ph  ->  ( n  e.  A  |->  B ) : dom  ( n  e.  A  |->  B ) --> CC )
4 rlimcxp.1 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  A )  ->  B  e.  V )
54ralrimiva 2794 . . . . . . . . . 10  |-  ( ph  ->  A. n  e.  A  B  e.  V )
6 dmmptg 5330 . . . . . . . . . 10  |-  ( A. n  e.  A  B  e.  V  ->  dom  (
n  e.  A  |->  B )  =  A )
75, 6syl 16 . . . . . . . . 9  |-  ( ph  ->  dom  ( n  e.  A  |->  B )  =  A )
87feq2d 5542 . . . . . . . 8  |-  ( ph  ->  ( ( n  e.  A  |->  B ) : dom  ( n  e.  A  |->  B ) --> CC  <->  ( n  e.  A  |->  B ) : A --> CC ) )
93, 8mpbid 210 . . . . . . 7  |-  ( ph  ->  ( n  e.  A  |->  B ) : A --> CC )
10 eqid 2438 . . . . . . . 8  |-  ( n  e.  A  |->  B )  =  ( n  e.  A  |->  B )
1110fmpt 5859 . . . . . . 7  |-  ( A. n  e.  A  B  e.  CC  <->  ( n  e.  A  |->  B ) : A --> CC )
129, 11sylibr 212 . . . . . 6  |-  ( ph  ->  A. n  e.  A  B  e.  CC )
1312adantr 465 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  A. n  e.  A  B  e.  CC )
14 simpr 461 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  x  e.  RR+ )
15 rlimcxp.3 . . . . . . . 8  |-  ( ph  ->  C  e.  RR+ )
1615adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR+ )  ->  C  e.  RR+ )
1716rprecred 11030 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 1  /  C )  e.  RR )
1814, 17rpcxpcld 22155 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( x  ^c  ( 1  /  C ) )  e.  RR+ )
191adantr 465 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( n  e.  A  |->  B )  ~~> r  0 )
2013, 18, 19rlimi 12983 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  E. y  e.  RR  A. n  e.  A  ( y  <_  n  ->  ( abs `  ( B  -  0 ) )  <  ( x  ^c  ( 1  /  C ) ) ) )
214, 1rlimmptrcl 13077 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  A )  ->  B  e.  CC )
2221adantlr 714 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  B  e.  CC )
2322abscld 12914 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  ( abs `  B )  e.  RR )
2422absge0d 12922 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  0  <_  ( abs `  B
) )
2518adantr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
x  ^c  ( 1  /  C ) )  e.  RR+ )
2625rpred 11019 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
x  ^c  ( 1  /  C ) )  e.  RR )
2725rpge0d 11023 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  0  <_  ( x  ^c 
( 1  /  C
) ) )
2815ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  C  e.  RR+ )
2923, 24, 26, 27, 28cxplt2d 22151 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
( abs `  B
)  <  ( x  ^c  ( 1  /  C ) )  <-> 
( ( abs `  B
)  ^c  C )  <  ( ( x  ^c  ( 1  /  C ) )  ^c  C ) ) )
3022subid1d 9700 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  ( B  -  0 )  =  B )
3130fveq2d 5690 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  ( abs `  ( B  - 
0 ) )  =  ( abs `  B
) )
3231breq1d 4297 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
( abs `  ( B  -  0 ) )  <  ( x  ^c  ( 1  /  C ) )  <-> 
( abs `  B
)  <  ( x  ^c  ( 1  /  C ) ) ) )
3328rpred 11019 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  C  e.  RR )
34 abscxp2 22118 . . . . . . . . . . 11  |-  ( ( B  e.  CC  /\  C  e.  RR )  ->  ( abs `  ( B  ^c  C ) )  =  ( ( abs `  B )  ^c  C ) )
3522, 33, 34syl2anc 661 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  ( abs `  ( B  ^c  C ) )  =  ( ( abs `  B
)  ^c  C ) )
3628rpcnd 11021 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  C  e.  CC )
3728rpne0d 11024 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  C  =/=  0 )
3836, 37recid2d 10095 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
( 1  /  C
)  x.  C )  =  1 )
3938oveq2d 6102 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
x  ^c  ( ( 1  /  C
)  x.  C ) )  =  ( x  ^c  1 ) )
40 simplr 754 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  x  e.  RR+ )
4117adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
1  /  C )  e.  RR )
4240, 41, 36cxpmuld 22159 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
x  ^c  ( ( 1  /  C
)  x.  C ) )  =  ( ( x  ^c  ( 1  /  C ) )  ^c  C ) )
4340rpcnd 11021 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  x  e.  CC )
4443cxp1d 22131 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
x  ^c  1 )  =  x )
4539, 42, 443eqtr3rd 2479 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  x  =  ( ( x  ^c  ( 1  /  C ) )  ^c  C ) )
4635, 45breq12d 4300 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
( abs `  ( B  ^c  C ) )  <  x  <->  ( ( abs `  B )  ^c  C )  <  (
( x  ^c 
( 1  /  C
) )  ^c  C ) ) )
4729, 32, 463bitr4d 285 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
( abs `  ( B  -  0 ) )  <  ( x  ^c  ( 1  /  C ) )  <-> 
( abs `  ( B  ^c  C ) )  <  x ) )
4847biimpd 207 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
( abs `  ( B  -  0 ) )  <  ( x  ^c  ( 1  /  C ) )  ->  ( abs `  ( B  ^c  C ) )  <  x ) )
4948imim2d 52 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
( y  <_  n  ->  ( abs `  ( B  -  0 ) )  <  ( x  ^c  ( 1  /  C ) ) )  ->  ( y  <_  n  ->  ( abs `  ( B  ^c  C ) )  < 
x ) ) )
5049ralimdva 2789 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( A. n  e.  A  (
y  <_  n  ->  ( abs `  ( B  -  0 ) )  <  ( x  ^c  ( 1  /  C ) ) )  ->  A. n  e.  A  ( y  <_  n  ->  ( abs `  ( B  ^c  C ) )  <  x ) ) )
5150reximdv 2822 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( E. y  e.  RR  A. n  e.  A  ( y  <_  n  ->  ( abs `  ( B  -  0 ) )  <  (
x  ^c  ( 1  /  C ) ) )  ->  E. y  e.  RR  A. n  e.  A  ( y  <_  n  ->  ( abs `  ( B  ^c  C ) )  <  x ) ) )
5220, 51mpd 15 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  E. y  e.  RR  A. n  e.  A  ( y  <_  n  ->  ( abs `  ( B  ^c  C ) )  <  x ) )
5352ralrimiva 2794 . 2  |-  ( ph  ->  A. x  e.  RR+  E. y  e.  RR  A. n  e.  A  (
y  <_  n  ->  ( abs `  ( B  ^c  C ) )  <  x ) )
5415rpcnd 11021 . . . . . 6  |-  ( ph  ->  C  e.  CC )
5554adantr 465 . . . . 5  |-  ( (
ph  /\  n  e.  A )  ->  C  e.  CC )
5621, 55cxpcld 22133 . . . 4  |-  ( (
ph  /\  n  e.  A )  ->  ( B  ^c  C )  e.  CC )
5756ralrimiva 2794 . . 3  |-  ( ph  ->  A. n  e.  A  ( B  ^c  C )  e.  CC )
58 rlimss 12972 . . . . 5  |-  ( ( n  e.  A  |->  B )  ~~> r  0  ->  dom  ( n  e.  A  |->  B )  C_  RR )
591, 58syl 16 . . . 4  |-  ( ph  ->  dom  ( n  e.  A  |->  B )  C_  RR )
607, 59eqsstr3d 3386 . . 3  |-  ( ph  ->  A  C_  RR )
6157, 60rlim0 12978 . 2  |-  ( ph  ->  ( ( n  e.  A  |->  ( B  ^c  C ) )  ~~> r  0  <->  A. x  e.  RR+  E. y  e.  RR  A. n  e.  A  ( y  <_  n  ->  ( abs `  ( B  ^c  C ) )  <  x ) ) )
6253, 61mpbird 232 1  |-  ( ph  ->  ( n  e.  A  |->  ( B  ^c  C ) )  ~~> r  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2710   E.wrex 2711    C_ wss 3323   class class class wbr 4287    e. cmpt 4345   dom cdm 4835   -->wf 5409   ` cfv 5413  (class class class)co 6086   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275    x. cmul 9279    < clt 9410    <_ cle 9411    - cmin 9587    / cdiv 9985   RR+crp 10983   abscabs 12715    ~~> r crli 12955    ^c ccxp 21987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ioc 11297  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-mod 11701  df-seq 11799  df-exp 11858  df-fac 12044  df-bc 12071  df-hash 12096  df-shft 12548  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-limsup 12941  df-clim 12958  df-rlim 12959  df-sum 13156  df-ef 13345  df-sin 13347  df-cos 13348  df-pi 13350  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17789  df-xmet 17790  df-met 17791  df-bl 17792  df-mopn 17793  df-fbas 17794  df-fg 17795  df-cnfld 17799  df-top 18483  df-bases 18485  df-topon 18486  df-topsp 18487  df-cld 18603  df-ntr 18604  df-cls 18605  df-nei 18682  df-lp 18720  df-perf 18721  df-cn 18811  df-cnp 18812  df-haus 18899  df-tx 19115  df-hmeo 19308  df-fil 19399  df-fm 19491  df-flim 19492  df-flf 19493  df-xms 19875  df-ms 19876  df-tms 19877  df-cncf 20434  df-limc 21321  df-dv 21322  df-log 21988  df-cxp 21989
This theorem is referenced by:  cxp2lim  22350  cxploglim2  22352
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