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Theorem expcncf 13386
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 5861 . . . 4 |- ( w = 0 -> ( x ^ w ) = ( x ^ 0 ) )
21mpteq2dv 4080 . . 3 |- ( w = 0 -> ( x e. CC |-> ( x ^ w ) ) = ( x e. CC |-> ( x ^ 0 ) ) )
32eleq1d 2239 . 2 |- ( w = 0 -> ( ( x e. CC |-> ( x ^ w ) ) e. ( CC -cn-> CC ) <-> ( x e. CC |-> ( x ^ 0 ) ) e. ( CC -cn-> CC ) ) )
4 oveq2 5861 . . . 4 |- ( w = k -> ( x ^ w ) = ( x ^ k ) )
54mpteq2dv 4080 . . 3 |- ( w = k -> ( x e. CC |-> ( x ^ w ) ) = ( x e. CC |-> ( x ^ k ) ) )
65eleq1d 2239 . 2 |- ( w = k -> ( ( x e. CC |-> ( x ^ w ) ) e. ( CC -cn-> CC ) <-> ( x e. CC |-> ( x ^ k ) ) e. ( CC -cn-> CC ) ) )
7 oveq2 5861 . . . 4 |- ( w = ( k + 1 ) -> ( x ^ w ) = ( x ^ ( k + 1 ) ) )
87mpteq2dv 4080 . . 3 |- ( w = ( k + 1 ) -> ( x e. CC |-> ( x ^ w ) ) = ( x e. CC |-> ( x ^ ( k + 1 ) ) ) )
98eleq1d 2239 . 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 5861 . . . 4 |- ( w = N -> ( x ^ w ) = ( x ^ N ) )
1110mpteq2dv 4080 . . 3 |- ( w = N -> ( x e. CC |-> ( x ^ w ) ) = ( x e. CC |-> ( x ^ N ) ) )
1211eleq1d 2239 . 2 |- ( w = N -> ( ( x e. CC |-> ( x ^ w ) ) e. ( CC -cn-> CC ) <-> ( x e. CC |-> ( x ^ N ) ) e. ( CC -cn-> CC ) ) )
13 exp0 10480 . . . 4 |- ( x e. CC -> ( x ^ 0 ) = 1 )
1413mpteq2ia 4075 . . 3 |- ( x e. CC |-> ( x ^ 0 ) ) = ( x e. CC |-> 1 )
15 ax-1cn 7867 . . . 4 |- 1 e. CC
16 ssid 3167 . . . 4 |- CC C_ CC
17 cncfmptc 13376 . . . 4 |- ( ( 1 e. CC /\ CC C_ CC /\ CC C_ CC ) -> ( x e. CC |-> 1 ) e. ( CC -cn-> CC ) )
1815, 16, 16, 17mp3an 1332 . . 3 |- ( x e. CC |-> 1 ) e. ( CC -cn-> CC )
1914, 18eqeltri 2243 . 2 |- ( x e. CC |-> ( x ^ 0 ) ) e. ( CC -cn-> CC )
20 oveq1 5860 . . . . . . 7 |- ( a = x -> ( a ^ k ) = ( x ^ k ) )
2120cbvmptv 4085 . . . . . 6 |- ( a e. CC |-> ( a ^ k ) ) = ( x e. CC |-> ( x ^ k ) )
2221eleq1i 2236 . . . . 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 275 . . . . . 6 |- ( ( k e. NN0 /\ ( a e. CC |-> ( a ^ k ) ) e. ( CC -cn-> CC ) ) -> ( x e. CC |-> ( x ^ k ) ) e. ( CC -cn-> CC ) )
25 cncfmptid 13377 . . . . . . . 8 |- ( ( CC C_ CC /\ CC C_ CC ) -> ( x e. CC |-> x ) e. ( CC -cn-> CC ) )
2616, 16, 25mp2an 424 . . . . . . 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 13385 . . . . 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 286 . . . 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 10483 . . . . . . . 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 4079 . . . . . 6 |- ( k e. NN0 -> ( x e. CC |-> ( x ^ ( k + 1 ) ) ) = ( x e. CC |-> ( ( x ^ k ) x. x ) ) )
3332eleq1d 2239 . . . . 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 274 . . . 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 9326 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 1348   e. wcel 2141   C_ wss 3121   |-> cmpt 4050  (class class class)co 5853   CCcc 7772   0cc0 7774   1c1 7775   + caddc 7777   x. cmul 7779   NN0cn0 9135   ^cexp 10475   -cn->ccncf 13351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-map 6628  df-sup 6961  df-inf 6962  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-rp 9611  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-cncf 13352
This theorem is referenced by: (None)
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