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Theorem expcncf 12578
Description: The power function on complex numbers, for fixed exponent N, is continuous. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
Assertion
Ref Expression
expcncf  |-  ( N  e.  NN0  ->  ( x  e.  CC  |->  ( x ^ N ) )  e.  ( CC -cn-> CC ) )
Distinct variable group:    x, N

Proof of Theorem expcncf
Dummy variables  w  k  a are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5736 . . . 4  |-  ( w  =  0  ->  (
x ^ w )  =  ( x ^
0 ) )
21mpteq2dv 3979 . . 3  |-  ( w  =  0  ->  (
x  e.  CC  |->  ( x ^ w ) )  =  ( x  e.  CC  |->  ( x ^ 0 ) ) )
32eleq1d 2183 . 2  |-  ( w  =  0  ->  (
( x  e.  CC  |->  ( x ^ w
) )  e.  ( CC -cn-> CC )  <->  ( x  e.  CC  |->  ( x ^
0 ) )  e.  ( CC -cn-> CC ) ) )
4 oveq2 5736 . . . 4  |-  ( w  =  k  ->  (
x ^ w )  =  ( x ^
k ) )
54mpteq2dv 3979 . . 3  |-  ( w  =  k  ->  (
x  e.  CC  |->  ( x ^ w ) )  =  ( x  e.  CC  |->  ( x ^ k ) ) )
65eleq1d 2183 . 2  |-  ( w  =  k  ->  (
( x  e.  CC  |->  ( x ^ w
) )  e.  ( CC -cn-> CC )  <->  ( x  e.  CC  |->  ( x ^
k ) )  e.  ( CC -cn-> CC ) ) )
7 oveq2 5736 . . . 4  |-  ( w  =  ( k  +  1 )  ->  (
x ^ w )  =  ( x ^
( k  +  1 ) ) )
87mpteq2dv 3979 . . 3  |-  ( w  =  ( k  +  1 )  ->  (
x  e.  CC  |->  ( x ^ w ) )  =  ( x  e.  CC  |->  ( x ^ ( k  +  1 ) ) ) )
98eleq1d 2183 . 2  |-  ( w  =  ( k  +  1 )  ->  (
( x  e.  CC  |->  ( x ^ w
) )  e.  ( CC -cn-> CC )  <->  ( x  e.  CC  |->  ( x ^
( k  +  1 ) ) )  e.  ( CC -cn-> CC ) ) )
10 oveq2 5736 . . . 4  |-  ( w  =  N  ->  (
x ^ w )  =  ( x ^ N ) )
1110mpteq2dv 3979 . . 3  |-  ( w  =  N  ->  (
x  e.  CC  |->  ( x ^ w ) )  =  ( x  e.  CC  |->  ( x ^ N ) ) )
1211eleq1d 2183 . 2  |-  ( w  =  N  ->  (
( x  e.  CC  |->  ( x ^ w
) )  e.  ( CC -cn-> CC )  <->  ( x  e.  CC  |->  ( x ^ N ) )  e.  ( CC -cn-> CC ) ) )
13 exp0 10190 . . . 4  |-  ( x  e.  CC  ->  (
x ^ 0 )  =  1 )
1413mpteq2ia 3974 . . 3  |-  ( x  e.  CC  |->  ( x ^ 0 ) )  =  ( x  e.  CC  |->  1 )
15 ax-1cn 7638 . . . 4  |-  1  e.  CC
16 ssid 3083 . . . 4  |-  CC  C_  CC
17 cncfmptc 12568 . . . 4  |-  ( ( 1  e.  CC  /\  CC  C_  CC  /\  CC  C_  CC )  ->  (
x  e.  CC  |->  1 )  e.  ( CC
-cn-> CC ) )
1815, 16, 16, 17mp3an 1298 . . 3  |-  ( x  e.  CC  |->  1 )  e.  ( CC -cn-> CC )
1914, 18eqeltri 2187 . 2  |-  ( x  e.  CC  |->  ( x ^ 0 ) )  e.  ( CC -cn-> CC )
20 oveq1 5735 . . . . . . 7  |-  ( a  =  x  ->  (
a ^ k )  =  ( x ^
k ) )
2120cbvmptv 3984 . . . . . 6  |-  ( a  e.  CC  |->  ( a ^ k ) )  =  ( x  e.  CC  |->  ( x ^
k ) )
2221eleq1i 2180 . . . . 5  |-  ( ( a  e.  CC  |->  ( a ^ k ) )  e.  ( CC
-cn-> CC )  <->  ( x  e.  CC  |->  ( x ^
k ) )  e.  ( CC -cn-> CC ) )
2322biimpi 119 . . . . . . 7  |-  ( ( a  e.  CC  |->  ( a ^ k ) )  e.  ( CC
-cn-> CC )  ->  (
x  e.  CC  |->  ( x ^ k ) )  e.  ( CC
-cn-> CC ) )
2423adantl 273 . . . . . 6  |-  ( ( k  e.  NN0  /\  ( a  e.  CC  |->  ( a ^ k
) )  e.  ( CC -cn-> CC ) )  -> 
( x  e.  CC  |->  ( x ^ k
) )  e.  ( CC -cn-> CC ) )
25 cncfmptid 12569 . . . . . . . 8  |-  ( ( CC  C_  CC  /\  CC  C_  CC )  ->  (
x  e.  CC  |->  x )  e.  ( CC
-cn-> CC ) )
2616, 16, 25mp2an 420 . . . . . . 7  |-  ( x  e.  CC  |->  x )  e.  ( CC -cn-> CC )
2726a1i 9 . . . . . 6  |-  ( ( k  e.  NN0  /\  ( a  e.  CC  |->  ( a ^ k
) )  e.  ( CC -cn-> CC ) )  -> 
( x  e.  CC  |->  x )  e.  ( CC -cn-> CC ) )
2824, 27mulcncf 12577 . . . . 5  |-  ( ( k  e.  NN0  /\  ( a  e.  CC  |->  ( a ^ k
) )  e.  ( CC -cn-> CC ) )  -> 
( x  e.  CC  |->  ( ( x ^
k )  x.  x
) )  e.  ( CC -cn-> CC ) )
2922, 28sylan2br 284 . . . 4  |-  ( ( k  e.  NN0  /\  ( x  e.  CC  |->  ( x ^ k
) )  e.  ( CC -cn-> CC ) )  -> 
( x  e.  CC  |->  ( ( x ^
k )  x.  x
) )  e.  ( CC -cn-> CC ) )
30 expp1 10193 . . . . . . . 8  |-  ( ( x  e.  CC  /\  k  e.  NN0 )  -> 
( x ^ (
k  +  1 ) )  =  ( ( x ^ k )  x.  x ) )
3130ancoms 266 . . . . . . 7  |-  ( ( k  e.  NN0  /\  x  e.  CC )  ->  ( x ^ (
k  +  1 ) )  =  ( ( x ^ k )  x.  x ) )
3231mpteq2dva 3978 . . . . . 6  |-  ( k  e.  NN0  ->  ( x  e.  CC  |->  ( x ^ ( k  +  1 ) ) )  =  ( x  e.  CC  |->  ( ( x ^ k )  x.  x ) ) )
3332eleq1d 2183 . . . . 5  |-  ( k  e.  NN0  ->  ( ( x  e.  CC  |->  ( x ^ ( k  +  1 ) ) )  e.  ( CC
-cn-> CC )  <->  ( x  e.  CC  |->  ( ( x ^ k )  x.  x ) )  e.  ( CC -cn-> CC ) ) )
3433adantr 272 . . . 4  |-  ( ( k  e.  NN0  /\  ( x  e.  CC  |->  ( x ^ k
) )  e.  ( CC -cn-> CC ) )  -> 
( ( x  e.  CC  |->  ( x ^
( k  +  1 ) ) )  e.  ( CC -cn-> CC )  <-> 
( x  e.  CC  |->  ( ( x ^
k )  x.  x
) )  e.  ( CC -cn-> CC ) ) )
3529, 34mpbird 166 . . 3  |-  ( ( k  e.  NN0  /\  ( x  e.  CC  |->  ( x ^ k
) )  e.  ( CC -cn-> CC ) )  -> 
( x  e.  CC  |->  ( x ^ (
k  +  1 ) ) )  e.  ( CC -cn-> CC ) )
3635ex 114 . 2  |-  ( k  e.  NN0  ->  ( ( x  e.  CC  |->  ( x ^ k ) )  e.  ( CC
-cn-> CC )  ->  (
x  e.  CC  |->  ( x ^ ( k  +  1 ) ) )  e.  ( CC
-cn-> CC ) ) )
373, 6, 9, 12, 19, 36nn0ind 9069 1  |-  ( N  e.  NN0  ->  ( x  e.  CC  |->  ( x ^ N ) )  e.  ( CC -cn-> CC ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463    C_ wss 3037    |-> cmpt 3949  (class class class)co 5728   CCcc 7545   0cc0 7547   1c1 7548    + caddc 7550    x. cmul 7552   NN0cn0 8881   ^cexp 10185   -cn->ccncf 12543
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-mulrcl 7644  ax-addcom 7645  ax-mulcom 7646  ax-addass 7647  ax-mulass 7648  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-1rid 7652  ax-0id 7653  ax-rnegex 7654  ax-precex 7655  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-apti 7660  ax-pre-ltadd 7661  ax-pre-mulgt0 7662  ax-pre-mulext 7663  ax-arch 7664  ax-caucvg 7665
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rmo 2398  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-if 3441  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-id 4175  df-po 4178  df-iso 4179  df-iord 4248  df-on 4250  df-ilim 4251  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-isom 5090  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-recs 6156  df-frec 6242  df-map 6498  df-sup 6823  df-inf 6824  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-reap 8255  df-ap 8262  df-div 8346  df-inn 8631  df-2 8689  df-3 8690  df-4 8691  df-n0 8882  df-z 8959  df-uz 9229  df-rp 9344  df-seqfrec 10112  df-exp 10186  df-cj 10507  df-re 10508  df-im 10509  df-rsqrt 10662  df-abs 10663  df-cncf 12544
This theorem is referenced by: (None)
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