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Theorem rlimcxp 22252
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 12963 . . . . . . . . 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 2789 . . . . . . . . . 10  |-  ( ph  ->  A. n  e.  A  B  e.  V )
6 dmmptg 5323 . . . . . . . . . 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 5535 . . . . . . . 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 2433 . . . . . . . 8  |-  ( n  e.  A  |->  B )  =  ( n  e.  A  |->  B )
1110fmpt 5852 . . . . . . 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 462 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  A. n  e.  A  B  e.  CC )
14 simpr 458 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  x  e.  RR+ )
15 rlimcxp.3 . . . . . . . 8  |-  ( ph  ->  C  e.  RR+ )
1615adantr 462 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR+ )  ->  C  e.  RR+ )
1716rprecred 11026 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 1  /  C )  e.  RR )
1814, 17rpcxpcld 22060 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( x  ^c  ( 1  /  C ) )  e.  RR+ )
191adantr 462 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( n  e.  A  |->  B )  ~~> r  0 )
2013, 18, 19rlimi 12975 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  E. y  e.  RR  A. n  e.  A  ( y  <_  n  ->  ( abs `  ( B  -  0 ) )  <  ( x  ^c  ( 1  /  C ) ) ) )
214, 1rlimmptrcl 13069 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  A )  ->  B  e.  CC )
2221adantlr 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  B  e.  CC )
2322abscld 12906 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  ( abs `  B )  e.  RR )
2422absge0d 12914 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  0  <_  ( abs `  B
) )
2518adantr 462 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
x  ^c  ( 1  /  C ) )  e.  RR+ )
2625rpred 11015 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
x  ^c  ( 1  /  C ) )  e.  RR )
2725rpge0d 11019 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  0  <_  ( x  ^c 
( 1  /  C
) ) )
2815ad2antrr 718 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  C  e.  RR+ )
2923, 24, 26, 27, 28cxplt2d 22056 . . . . . . . . 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 9696 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  ( B  -  0 )  =  B )
3130fveq2d 5683 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  ( abs `  ( B  - 
0 ) )  =  ( abs `  B
) )
3231breq1d 4290 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
( abs `  ( B  -  0 ) )  <  ( x  ^c  ( 1  /  C ) )  <-> 
( abs `  B
)  <  ( x  ^c  ( 1  /  C ) ) ) )
3328rpred 11015 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  C  e.  RR )
34 abscxp2 22023 . . . . . . . . . . 11  |-  ( ( B  e.  CC  /\  C  e.  RR )  ->  ( abs `  ( B  ^c  C ) )  =  ( ( abs `  B )  ^c  C ) )
3522, 33, 34syl2anc 654 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  ( abs `  ( B  ^c  C ) )  =  ( ( abs `  B
)  ^c  C ) )
3628rpcnd 11017 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  C  e.  CC )
3728rpne0d 11020 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  C  =/=  0 )
3836, 37recid2d 10091 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
( 1  /  C
)  x.  C )  =  1 )
3938oveq2d 6096 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
x  ^c  ( ( 1  /  C
)  x.  C ) )  =  ( x  ^c  1 ) )
40 simplr 747 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  x  e.  RR+ )
4117adantr 462 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
1  /  C )  e.  RR )
4240, 41, 36cxpmuld 22064 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
x  ^c  ( ( 1  /  C
)  x.  C ) )  =  ( ( x  ^c  ( 1  /  C ) )  ^c  C ) )
4340rpcnd 11017 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  x  e.  CC )
4443cxp1d 22036 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  (
x  ^c  1 )  =  x )
4539, 42, 443eqtr3rd 2474 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  A )  ->  x  =  ( ( x  ^c  ( 1  /  C ) )  ^c  C ) )
4635, 45breq12d 4293 . . . . . . . . 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 2784 . . . . 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 2817 . . . 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 2789 . 2  |-  ( ph  ->  A. x  e.  RR+  E. y  e.  RR  A. n  e.  A  (
y  <_  n  ->  ( abs `  ( B  ^c  C ) )  <  x ) )
5415rpcnd 11017 . . . . . 6  |-  ( ph  ->  C  e.  CC )
5554adantr 462 . . . . 5  |-  ( (
ph  /\  n  e.  A )  ->  C  e.  CC )
5621, 55cxpcld 22038 . . . 4  |-  ( (
ph  /\  n  e.  A )  ->  ( B  ^c  C )  e.  CC )
5756ralrimiva 2789 . . 3  |-  ( ph  ->  A. n  e.  A  ( B  ^c  C )  e.  CC )
58 rlimss 12964 . . . . 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 3379 . . 3  |-  ( ph  ->  A  C_  RR )
6157, 60rlim0 12970 . 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 1362    e. wcel 1755   A.wral 2705   E.wrex 2706    C_ wss 3316   class class class wbr 4280    e. cmpt 4338   dom cdm 4827   -->wf 5402   ` cfv 5406  (class class class)co 6080   CCcc 9268   RRcr 9269   0cc0 9270   1c1 9271    x. cmul 9275    < clt 9406    <_ cle 9407    - cmin 9583    / cdiv 9981   RR+crp 10979   abscabs 12707    ~~> r crli 12947    ^c ccxp 21892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348  ax-addf 9349  ax-mulf 9350
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-ixp 7252  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-cda 8325  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-uz 10850  df-q 10942  df-rp 10980  df-xneg 11077  df-xadd 11078  df-xmul 11079  df-ioo 11292  df-ioc 11293  df-ico 11294  df-icc 11295  df-fz 11425  df-fzo 11533  df-fl 11626  df-mod 11693  df-seq 11791  df-exp 11850  df-fac 12036  df-bc 12063  df-hash 12088  df-shft 12540  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-limsup 12933  df-clim 12950  df-rlim 12951  df-sum 13148  df-ef 13336  df-sin 13338  df-cos 13339  df-pi 13341  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-starv 14236  df-sca 14237  df-vsca 14238  df-ip 14239  df-tset 14240  df-ple 14241  df-ds 14243  df-unif 14244  df-hom 14245  df-cco 14246  df-rest 14344  df-topn 14345  df-0g 14363  df-gsum 14364  df-topgen 14365  df-pt 14366  df-prds 14369  df-xrs 14423  df-qtop 14428  df-imas 14429  df-xps 14431  df-mre 14507  df-mrc 14508  df-acs 14510  df-mnd 15398  df-submnd 15448  df-mulg 15528  df-cntz 15815  df-cmn 16259  df-psmet 17653  df-xmet 17654  df-met 17655  df-bl 17656  df-mopn 17657  df-fbas 17658  df-fg 17659  df-cnfld 17663  df-top 18345  df-bases 18347  df-topon 18348  df-topsp 18349  df-cld 18465  df-ntr 18466  df-cls 18467  df-nei 18544  df-lp 18582  df-perf 18583  df-cn 18673  df-cnp 18674  df-haus 18761  df-tx 18977  df-hmeo 19170  df-fil 19261  df-fm 19353  df-flim 19354  df-flf 19355  df-xms 19737  df-ms 19738  df-tms 19739  df-cncf 20296  df-limc 21183  df-dv 21184  df-log 21893  df-cxp 21894
This theorem is referenced by:  cxp2lim  22255  cxploglim2  22257
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