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Theorem rlimcxp 23429
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 13336 . . . . . . . . 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 2871 . . . . . . . . . 10  |-  ( ph  ->  A. n  e.  A  B  e.  V )
6 dmmptg 5510 . . . . . . . . . 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 5724 . . . . . . . 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 2457 . . . . . . . 8  |-  ( n  e.  A  |->  B )  =  ( n  e.  A  |->  B )
1110fmpt 6053 . . . . . . 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 11292 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 1  /  C )  e.  RR )
1814, 17rpcxpcld 23237 . . . . 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 13348 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  E. y  e.  RR  A. n  e.  A  ( y  <_  n  ->  ( abs `  ( B  -  0 ) )  <  ( x  ^c  ( 1  /  C ) ) ) )
214, 1rlimmptrcl 13442 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  A )  ->  B  e.  CC )
2221adantlr 714 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  B  e.  CC )
2322abscld 13279 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  ( abs `  B )  e.  RR )
2422absge0d 13287 . . . . . . . . . 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 11281 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
x  ^c  ( 1  /  C ) )  e.  RR )
2725rpge0d 11285 . . . . . . . . . 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 23233 . . . . . . . . 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 9939 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  ( B  -  0 )  =  B )
3130fveq2d 5876 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  ( abs `  ( B  - 
0 ) )  =  ( abs `  B
) )
3231breq1d 4466 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
( abs `  ( B  -  0 ) )  <  ( x  ^c  ( 1  /  C ) )  <-> 
( abs `  B
)  <  ( x  ^c  ( 1  /  C ) ) ) )
3328rpred 11281 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  C  e.  RR )
34 abscxp2 23200 . . . . . . . . . . 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 11283 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  C  e.  CC )
3728rpne0d 11286 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  C  =/=  0 )
3836, 37recid2d 10337 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
( 1  /  C
)  x.  C )  =  1 )
3938oveq2d 6312 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
x  ^c  ( ( 1  /  C
)  x.  C ) )  =  ( x  ^c  1 ) )
40 simplr 755 . . . . . . . . . . . 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 23241 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
x  ^c  ( ( 1  /  C
)  x.  C ) )  =  ( ( x  ^c  ( 1  /  C ) )  ^c  C ) )
4340rpcnd 11283 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  x  e.  CC )
4443cxp1d 23213 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
x  ^c  1 )  =  x )
4539, 42, 443eqtr3rd 2507 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  x  =  ( ( x  ^c  ( 1  /  C ) )  ^c  C ) )
4635, 45breq12d 4469 . . . . . . . . 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 2865 . . . . 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 2931 . . . 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 2871 . 2  |-  ( ph  ->  A. x  e.  RR+  E. y  e.  RR  A. n  e.  A  (
y  <_  n  ->  ( abs `  ( B  ^c  C ) )  <  x ) )
5415rpcnd 11283 . . . . . 6  |-  ( ph  ->  C  e.  CC )
5554adantr 465 . . . . 5  |-  ( (
ph  /\  n  e.  A )  ->  C  e.  CC )
5621, 55cxpcld 23215 . . . 4  |-  ( (
ph  /\  n  e.  A )  ->  ( B  ^c  C )  e.  CC )
5756ralrimiva 2871 . . 3  |-  ( ph  ->  A. n  e.  A  ( B  ^c  C )  e.  CC )
58 rlimss 13337 . . . . 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 3534 . . 3  |-  ( ph  ->  A  C_  RR )
6157, 60rlim0 13343 . 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 1395    e. wcel 1819   A.wral 2807   E.wrex 2808    C_ wss 3471   class class class wbr 4456    |-> cmpt 4515   dom cdm 5008   -->wf 5590   ` cfv 5594  (class class class)co 6296   CCcc 9507   RRcr 9508   0cc0 9509   1c1 9510    x. cmul 9514    < clt 9645    <_ cle 9646    - cmin 9824    / cdiv 10227   RR+crp 11245   abscabs 13079    ~~> r crli 13320    ^c ccxp 23069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ioc 11559  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11822  df-fl 11932  df-mod 12000  df-seq 12111  df-exp 12170  df-fac 12357  df-bc 12384  df-hash 12409  df-shft 12912  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-limsup 13306  df-clim 13323  df-rlim 13324  df-sum 13521  df-ef 13815  df-sin 13817  df-cos 13818  df-pi 13820  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-starv 14727  df-sca 14728  df-vsca 14729  df-ip 14730  df-tset 14731  df-ple 14732  df-ds 14734  df-unif 14735  df-hom 14736  df-cco 14737  df-rest 14840  df-topn 14841  df-0g 14859  df-gsum 14860  df-topgen 14861  df-pt 14862  df-prds 14865  df-xrs 14919  df-qtop 14924  df-imas 14925  df-xps 14927  df-mre 15003  df-mrc 15004  df-acs 15006  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-submnd 16094  df-mulg 16187  df-cntz 16482  df-cmn 16927  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-fbas 18543  df-fg 18544  df-cnfld 18548  df-top 19526  df-bases 19528  df-topon 19529  df-topsp 19530  df-cld 19647  df-ntr 19648  df-cls 19649  df-nei 19726  df-lp 19764  df-perf 19765  df-cn 19855  df-cnp 19856  df-haus 19943  df-tx 20189  df-hmeo 20382  df-fil 20473  df-fm 20565  df-flim 20566  df-flf 20567  df-xms 20949  df-ms 20950  df-tms 20951  df-cncf 21508  df-limc 22396  df-dv 22397  df-log 23070  df-cxp 23071
This theorem is referenced by:  cxp2lim  23432  cxploglim2  23434
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