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Theorem expcncf 15298
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 6015 . . . 4 |- ( w = 0 -> ( x ^ w ) = ( x ^ 0 ) )
21mpteq2dv 4175 . . 3 |- ( w = 0 -> ( x e. CC |-> ( x ^ w ) ) = ( x e. CC |-> ( x ^ 0 ) ) )
32eleq1d 2298 . 2 |- ( w = 0 -> ( ( x e. CC |-> ( x ^ w ) ) e. ( CC -cn-> CC ) <-> ( x e. CC |-> ( x ^ 0 ) ) e. ( CC -cn-> CC ) ) )
4 oveq2 6015 . . . 4 |- ( w = k -> ( x ^ w ) = ( x ^ k ) )
54mpteq2dv 4175 . . 3 |- ( w = k -> ( x e. CC |-> ( x ^ w ) ) = ( x e. CC |-> ( x ^ k ) ) )
65eleq1d 2298 . 2 |- ( w = k -> ( ( x e. CC |-> ( x ^ w ) ) e. ( CC -cn-> CC ) <-> ( x e. CC |-> ( x ^ k ) ) e. ( CC -cn-> CC ) ) )
7 oveq2 6015 . . . 4 |- ( w = ( k + 1 ) -> ( x ^ w ) = ( x ^ ( k + 1 ) ) )
87mpteq2dv 4175 . . 3 |- ( w = ( k + 1 ) -> ( x e. CC |-> ( x ^ w ) ) = ( x e. CC |-> ( x ^ ( k + 1 ) ) ) )
98eleq1d 2298 . 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 6015 . . . 4 |- ( w = N -> ( x ^ w ) = ( x ^ N ) )
1110mpteq2dv 4175 . . 3 |- ( w = N -> ( x e. CC |-> ( x ^ w ) ) = ( x e. CC |-> ( x ^ N ) ) )
1211eleq1d 2298 . 2 |- ( w = N -> ( ( x e. CC |-> ( x ^ w ) ) e. ( CC -cn-> CC ) <-> ( x e. CC |-> ( x ^ N ) ) e. ( CC -cn-> CC ) ) )
13 exp0 10777 . . . 4 |- ( x e. CC -> ( x ^ 0 ) = 1 )
1413mpteq2ia 4170 . . 3 |- ( x e. CC |-> ( x ^ 0 ) ) = ( x e. CC |-> 1 )
15 ax-1cn 8103 . . . 4 |- 1 e. CC
16 ssid 3244 . . . 4 |- CC C_ CC
17 cncfmptc 15285 . . . 4 |- ( ( 1 e. CC /\ CC C_ CC /\ CC C_ CC ) -> ( x e. CC |-> 1 ) e. ( CC -cn-> CC ) )
1815, 16, 16, 17mp3an 1371 . . 3 |- ( x e. CC |-> 1 ) e. ( CC -cn-> CC )
1914, 18eqeltri 2302 . 2 |- ( x e. CC |-> ( x ^ 0 ) ) e. ( CC -cn-> CC )
20 oveq1 6014 . . . . . . 7 |- ( a = x -> ( a ^ k ) = ( x ^ k ) )
2120cbvmptv 4180 . . . . . 6 |- ( a e. CC |-> ( a ^ k ) ) = ( x e. CC |-> ( x ^ k ) )
2221eleq1i 2295 . . . . 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 15286 . . . . . . . 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 15297 . . . . 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 10780 . . . . . . . 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 4174 . . . . . 6 |- ( k e. NN0 -> ( x e. CC |-> ( x ^ ( k + 1 ) ) ) = ( x e. CC |-> ( ( x ^ k ) x. x ) ) )
3332eleq1d 2298 . . . . 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 9572 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 1395   e. wcel 2200   C_ wss 3197   |-> cmpt 4145  (class class class)co 6007   CCcc 8008   0cc0 8010   1c1 8011   + caddc 8013   x. cmul 8015   NN0cn0 9380   ^cexp 10772   -cn->ccncf 15259
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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128  ax-arch 8129  ax-caucvg 8130
This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-map 6805  df-sup 7162  df-inf 7163  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-n0 9381  df-z 9458  df-uz 9734  df-rp 9862  df-seqfrec 10682  df-exp 10773  df-cj 11368  df-re 11369  df-im 11370  df-rsqrt 11524  df-abs 11525  df-cncf 15260
This theorem is referenced by: (None)
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