ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  expcncf Unicode version

Theorem expcncf 14569
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 5905 . . . 4  |-  ( w  =  0  ->  (
x ^ w )  =  ( x ^
0 ) )
21mpteq2dv 4109 . . 3  |-  ( w  =  0  ->  (
x  e.  CC  |->  ( x ^ w ) )  =  ( x  e.  CC  |->  ( x ^ 0 ) ) )
32eleq1d 2258 . 2  |-  ( w  =  0  ->  (
( x  e.  CC  |->  ( x ^ w
) )  e.  ( CC -cn-> CC )  <->  ( x  e.  CC  |->  ( x ^
0 ) )  e.  ( CC -cn-> CC ) ) )
4 oveq2 5905 . . . 4  |-  ( w  =  k  ->  (
x ^ w )  =  ( x ^
k ) )
54mpteq2dv 4109 . . 3  |-  ( w  =  k  ->  (
x  e.  CC  |->  ( x ^ w ) )  =  ( x  e.  CC  |->  ( x ^ k ) ) )
65eleq1d 2258 . 2  |-  ( w  =  k  ->  (
( x  e.  CC  |->  ( x ^ w
) )  e.  ( CC -cn-> CC )  <->  ( x  e.  CC  |->  ( x ^
k ) )  e.  ( CC -cn-> CC ) ) )
7 oveq2 5905 . . . 4  |-  ( w  =  ( k  +  1 )  ->  (
x ^ w )  =  ( x ^
( k  +  1 ) ) )
87mpteq2dv 4109 . . 3  |-  ( w  =  ( k  +  1 )  ->  (
x  e.  CC  |->  ( x ^ w ) )  =  ( x  e.  CC  |->  ( x ^ ( k  +  1 ) ) ) )
98eleq1d 2258 . 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 5905 . . . 4  |-  ( w  =  N  ->  (
x ^ w )  =  ( x ^ N ) )
1110mpteq2dv 4109 . . 3  |-  ( w  =  N  ->  (
x  e.  CC  |->  ( x ^ w ) )  =  ( x  e.  CC  |->  ( x ^ N ) ) )
1211eleq1d 2258 . 2  |-  ( w  =  N  ->  (
( x  e.  CC  |->  ( x ^ w
) )  e.  ( CC -cn-> CC )  <->  ( x  e.  CC  |->  ( x ^ N ) )  e.  ( CC -cn-> CC ) ) )
13 exp0 10558 . . . 4  |-  ( x  e.  CC  ->  (
x ^ 0 )  =  1 )
1413mpteq2ia 4104 . . 3  |-  ( x  e.  CC  |->  ( x ^ 0 ) )  =  ( x  e.  CC  |->  1 )
15 ax-1cn 7935 . . . 4  |-  1  e.  CC
16 ssid 3190 . . . 4  |-  CC  C_  CC
17 cncfmptc 14559 . . . 4  |-  ( ( 1  e.  CC  /\  CC  C_  CC  /\  CC  C_  CC )  ->  (
x  e.  CC  |->  1 )  e.  ( CC
-cn-> CC ) )
1815, 16, 16, 17mp3an 1348 . . 3  |-  ( x  e.  CC  |->  1 )  e.  ( CC -cn-> CC )
1914, 18eqeltri 2262 . 2  |-  ( x  e.  CC  |->  ( x ^ 0 ) )  e.  ( CC -cn-> CC )
20 oveq1 5904 . . . . . . 7  |-  ( a  =  x  ->  (
a ^ k )  =  ( x ^
k ) )
2120cbvmptv 4114 . . . . . 6  |-  ( a  e.  CC  |->  ( a ^ k ) )  =  ( x  e.  CC  |->  ( x ^
k ) )
2221eleq1i 2255 . . . . 5  |-  ( ( a  e.  CC  |->  ( a ^ k ) )  e.  ( CC
-cn-> CC )  <->  ( x  e.  CC  |->  ( x ^
k ) )  e.  ( CC -cn-> CC ) )
2322biimpi 120 . . . . . . 7  |-  ( ( a  e.  CC  |->  ( a ^ k ) )  e.  ( CC
-cn-> CC )  ->  (
x  e.  CC  |->  ( x ^ k ) )  e.  ( CC
-cn-> CC ) )
2423adantl 277 . . . . . 6  |-  ( ( k  e.  NN0  /\  ( a  e.  CC  |->  ( a ^ k
) )  e.  ( CC -cn-> CC ) )  -> 
( x  e.  CC  |->  ( x ^ k
) )  e.  ( CC -cn-> CC ) )
25 cncfmptid 14560 . . . . . . . 8  |-  ( ( CC  C_  CC  /\  CC  C_  CC )  ->  (
x  e.  CC  |->  x )  e.  ( CC
-cn-> CC ) )
2616, 16, 25mp2an 426 . . . . . . 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 14568 . . . . 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 288 . . . 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 10561 . . . . . . . 8  |-  ( ( x  e.  CC  /\  k  e.  NN0 )  -> 
( x ^ (
k  +  1 ) )  =  ( ( x ^ k )  x.  x ) )
3130ancoms 268 . . . . . . 7  |-  ( ( k  e.  NN0  /\  x  e.  CC )  ->  ( x ^ (
k  +  1 ) )  =  ( ( x ^ k )  x.  x ) )
3231mpteq2dva 4108 . . . . . 6  |-  ( k  e.  NN0  ->  ( x  e.  CC  |->  ( x ^ ( k  +  1 ) ) )  =  ( x  e.  CC  |->  ( ( x ^ k )  x.  x ) ) )
3332eleq1d 2258 . . . . 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 276 . . . 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 167 . . 3  |-  ( ( k  e.  NN0  /\  ( x  e.  CC  |->  ( x ^ k
) )  e.  ( CC -cn-> CC ) )  -> 
( x  e.  CC  |->  ( x ^ (
k  +  1 ) ) )  e.  ( CC -cn-> CC ) )
3635ex 115 . 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 9398 1  |-  ( N  e.  NN0  ->  ( x  e.  CC  |->  ( x ^ N ) )  e.  ( CC -cn-> CC ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364    e. wcel 2160    C_ wss 3144    |-> cmpt 4079  (class class class)co 5897   CCcc 7840   0cc0 7842   1c1 7843    + caddc 7845    x. cmul 7847   NN0cn0 9207   ^cexp 10553   -cn->ccncf 14534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-mulrcl 7941  ax-addcom 7942  ax-mulcom 7943  ax-addass 7944  ax-mulass 7945  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-1rid 7949  ax-0id 7950  ax-rnegex 7951  ax-precex 7952  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958  ax-pre-mulgt0 7959  ax-pre-mulext 7960  ax-arch 7961  ax-caucvg 7962
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-isom 5244  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-frec 6417  df-map 6677  df-sup 7014  df-inf 7015  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-reap 8563  df-ap 8570  df-div 8661  df-inn 8951  df-2 9009  df-3 9010  df-4 9011  df-n0 9208  df-z 9285  df-uz 9560  df-rp 9686  df-seqfrec 10479  df-exp 10554  df-cj 10886  df-re 10887  df-im 10888  df-rsqrt 11042  df-abs 11043  df-cncf 14535
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator