MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rlimcxp Unicode version

Theorem rlimcxp 20679
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 12222 . . . . . . . . 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 2732 . . . . . . . . . 10  |-  ( ph  ->  A. n  e.  A  B  e.  V )
6 dmmptg 5307 . . . . . . . . . 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 5521 . . . . . . . 8  |-  ( ph  ->  ( ( n  e.  A  |->  B ) : dom  ( n  e.  A  |->  B ) --> CC  <->  ( n  e.  A  |->  B ) : A --> CC ) )
93, 8mpbid 202 . . . . . . 7  |-  ( ph  ->  ( n  e.  A  |->  B ) : A --> CC )
10 eqid 2387 . . . . . . . 8  |-  ( n  e.  A  |->  B )  =  ( n  e.  A  |->  B )
1110fmpt 5829 . . . . . . 7  |-  ( A. n  e.  A  B  e.  CC  <->  ( n  e.  A  |->  B ) : A --> CC )
129, 11sylibr 204 . . . . . 6  |-  ( ph  ->  A. n  e.  A  B  e.  CC )
1312adantr 452 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  A. n  e.  A  B  e.  CC )
14 simpr 448 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  x  e.  RR+ )
15 rlimcxp.3 . . . . . . . 8  |-  ( ph  ->  C  e.  RR+ )
1615adantr 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR+ )  ->  C  e.  RR+ )
1716rprecred 10591 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 1  /  C )  e.  RR )
1814, 17rpcxpcld 20488 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( x  ^ c  ( 1  /  C ) )  e.  RR+ )
191adantr 452 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( n  e.  A  |->  B )  ~~> r  0 )
2013, 18, 19rlimi 12234 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  E. y  e.  RR  A. n  e.  A  ( y  <_  n  ->  ( abs `  ( B  -  0 ) )  <  ( x  ^ c  ( 1  /  C ) ) ) )
214, 1rlimmptrcl 12328 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  A )  ->  B  e.  CC )
2221adantlr 696 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  B  e.  CC )
2322abscld 12165 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  ( abs `  B )  e.  RR )
2422absge0d 12173 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  0  <_  ( abs `  B
) )
2518adantr 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
x  ^ c  ( 1  /  C ) )  e.  RR+ )
2625rpred 10580 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
x  ^ c  ( 1  /  C ) )  e.  RR )
2725rpge0d 10584 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  0  <_  ( x  ^ c 
( 1  /  C
) ) )
2815ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  C  e.  RR+ )
2923, 24, 26, 27, 28cxplt2d 20484 . . . . . . . . 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 9332 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  ( B  -  0 )  =  B )
3130fveq2d 5672 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  ( abs `  ( B  - 
0 ) )  =  ( abs `  B
) )
3231breq1d 4163 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
( abs `  ( B  -  0 ) )  <  ( x  ^ c  ( 1  /  C ) )  <-> 
( abs `  B
)  <  ( x  ^ c  ( 1  /  C ) ) ) )
3328rpred 10580 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  C  e.  RR )
34 abscxp2 20451 . . . . . . . . . . 11  |-  ( ( B  e.  CC  /\  C  e.  RR )  ->  ( abs `  ( B  ^ c  C ) )  =  ( ( abs `  B )  ^ c  C ) )
3522, 33, 34syl2anc 643 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  ( abs `  ( B  ^ c  C ) )  =  ( ( abs `  B
)  ^ c  C
) )
3628rpcnd 10582 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  C  e.  CC )
3728rpne0d 10585 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  C  =/=  0 )
3836, 37recid2d 9718 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
( 1  /  C
)  x.  C )  =  1 )
3938oveq2d 6036 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
x  ^ c  ( ( 1  /  C
)  x.  C ) )  =  ( x  ^ c  1 ) )
40 simplr 732 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  x  e.  RR+ )
4117adantr 452 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
1  /  C )  e.  RR )
4240, 41, 36cxpmuld 20492 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
x  ^ c  ( ( 1  /  C
)  x.  C ) )  =  ( ( x  ^ c  ( 1  /  C ) )  ^ c  C
) )
4340rpcnd 10582 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  x  e.  CC )
4443cxp1d 20464 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
x  ^ c  1 )  =  x )
4539, 42, 443eqtr3rd 2428 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  x  =  ( ( x  ^ c  ( 1  /  C ) )  ^ c  C ) )
4635, 45breq12d 4166 . . . . . . . . 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 277 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
( abs `  ( B  -  0 ) )  <  ( x  ^ c  ( 1  /  C ) )  <-> 
( abs `  ( B  ^ c  C ) )  <  x ) )
4847biimpd 199 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
( abs `  ( B  -  0 ) )  <  ( x  ^ c  ( 1  /  C ) )  ->  ( abs `  ( B  ^ c  C ) )  <  x ) )
4948imim2d 50 . . . . . 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 2727 . . . . 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 2760 . . . 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 2732 . 2  |-  ( ph  ->  A. x  e.  RR+  E. y  e.  RR  A. n  e.  A  (
y  <_  n  ->  ( abs `  ( B  ^ c  C ) )  <  x ) )
5415rpcnd 10582 . . . . . 6  |-  ( ph  ->  C  e.  CC )
5554adantr 452 . . . . 5  |-  ( (
ph  /\  n  e.  A )  ->  C  e.  CC )
5621, 55cxpcld 20466 . . . 4  |-  ( (
ph  /\  n  e.  A )  ->  ( B  ^ c  C )  e.  CC )
5756ralrimiva 2732 . . 3  |-  ( ph  ->  A. n  e.  A  ( B  ^ c  C )  e.  CC )
58 rlimss 12223 . . . . 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 3326 . . 3  |-  ( ph  ->  A  C_  RR )
6157, 60rlim0 12229 . 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 224 1  |-  ( ph  ->  ( n  e.  A  |->  ( B  ^ c  C ) )  ~~> r  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649   E.wrex 2650    C_ wss 3263   class class class wbr 4153    e. cmpt 4207   dom cdm 4818   -->wf 5390   ` cfv 5394  (class class class)co 6020   CCcc 8921   RRcr 8922   0cc0 8923   1c1 8924    x. cmul 8928    < clt 9053    <_ cle 9054    - cmin 9223    / cdiv 9609   RR+crp 10544   abscabs 11966    ~~> r crli 12206    ^ c ccxp 20320
This theorem is referenced by:  cxp2lim  20682  cxploglim2  20684
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001  ax-addf 9002  ax-mulf 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6841  df-map 6956  df-pm 6957  df-ixp 7000  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-fi 7351  df-sup 7381  df-oi 7412  df-card 7759  df-cda 7981  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-uz 10421  df-q 10507  df-rp 10545  df-xneg 10642  df-xadd 10643  df-xmul 10644  df-ioo 10852  df-ioc 10853  df-ico 10854  df-icc 10855  df-fz 10976  df-fzo 11066  df-fl 11129  df-mod 11178  df-seq 11251  df-exp 11310  df-fac 11494  df-bc 11521  df-hash 11546  df-shft 11809  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-limsup 12192  df-clim 12209  df-rlim 12210  df-sum 12407  df-ef 12597  df-sin 12599  df-cos 12600  df-pi 12602  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-mulr 13470  df-starv 13471  df-sca 13472  df-vsca 13473  df-tset 13475  df-ple 13476  df-ds 13478  df-unif 13479  df-hom 13480  df-cco 13481  df-rest 13577  df-topn 13578  df-topgen 13594  df-pt 13595  df-prds 13598  df-xrs 13653  df-0g 13654  df-gsum 13655  df-qtop 13660  df-imas 13661  df-xps 13663  df-mre 13738  df-mrc 13739  df-acs 13741  df-mnd 14617  df-submnd 14666  df-mulg 14742  df-cntz 15043  df-cmn 15341  df-xmet 16619  df-met 16620  df-bl 16621  df-mopn 16622  df-fbas 16623  df-fg 16624  df-cnfld 16627  df-top 16886  df-bases 16888  df-topon 16889  df-topsp 16890  df-cld 17006  df-ntr 17007  df-cls 17008  df-nei 17085  df-lp 17123  df-perf 17124  df-cn 17213  df-cnp 17214  df-haus 17301  df-tx 17515  df-hmeo 17708  df-fil 17799  df-fm 17891  df-flim 17892  df-flf 17893  df-xms 18259  df-ms 18260  df-tms 18261  df-cncf 18779  df-limc 19620  df-dv 19621  df-log 20321  df-cxp 20322
  Copyright terms: Public domain W3C validator