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Theorem expcncf 14131
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 5885 . . . 4 |- ( w = 0 -> ( x ^ w ) = ( x ^ 0 ) )
21mpteq2dv 4096 . . 3 |- ( w = 0 -> ( x e. CC |-> ( x ^ w ) ) = ( x e. CC |-> ( x ^ 0 ) ) )
32eleq1d 2246 . 2 |- ( w = 0 -> ( ( x e. CC |-> ( x ^ w ) ) e. ( CC -cn-> CC ) <-> ( x e. CC |-> ( x ^ 0 ) ) e. ( CC -cn-> CC ) ) )
4 oveq2 5885 . . . 4 |- ( w = k -> ( x ^ w ) = ( x ^ k ) )
54mpteq2dv 4096 . . 3 |- ( w = k -> ( x e. CC |-> ( x ^ w ) ) = ( x e. CC |-> ( x ^ k ) ) )
65eleq1d 2246 . 2 |- ( w = k -> ( ( x e. CC |-> ( x ^ w ) ) e. ( CC -cn-> CC ) <-> ( x e. CC |-> ( x ^ k ) ) e. ( CC -cn-> CC ) ) )
7 oveq2 5885 . . . 4 |- ( w = ( k + 1 ) -> ( x ^ w ) = ( x ^ ( k + 1 ) ) )
87mpteq2dv 4096 . . 3 |- ( w = ( k + 1 ) -> ( x e. CC |-> ( x ^ w ) ) = ( x e. CC |-> ( x ^ ( k + 1 ) ) ) )
98eleq1d 2246 . 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 5885 . . . 4 |- ( w = N -> ( x ^ w ) = ( x ^ N ) )
1110mpteq2dv 4096 . . 3 |- ( w = N -> ( x e. CC |-> ( x ^ w ) ) = ( x e. CC |-> ( x ^ N ) ) )
1211eleq1d 2246 . 2 |- ( w = N -> ( ( x e. CC |-> ( x ^ w ) ) e. ( CC -cn-> CC ) <-> ( x e. CC |-> ( x ^ N ) ) e. ( CC -cn-> CC ) ) )
13 exp0 10526 . . . 4 |- ( x e. CC -> ( x ^ 0 ) = 1 )
1413mpteq2ia 4091 . . 3 |- ( x e. CC |-> ( x ^ 0 ) ) = ( x e. CC |-> 1 )
15 ax-1cn 7906 . . . 4 |- 1 e. CC
16 ssid 3177 . . . 4 |- CC C_ CC
17 cncfmptc 14121 . . . 4 |- ( ( 1 e. CC /\ CC C_ CC /\ CC C_ CC ) -> ( x e. CC |-> 1 ) e. ( CC -cn-> CC ) )
1815, 16, 16, 17mp3an 1337 . . 3 |- ( x e. CC |-> 1 ) e. ( CC -cn-> CC )
1914, 18eqeltri 2250 . 2 |- ( x e. CC |-> ( x ^ 0 ) ) e. ( CC -cn-> CC )
20 oveq1 5884 . . . . . . 7 |- ( a = x -> ( a ^ k ) = ( x ^ k ) )
2120cbvmptv 4101 . . . . . 6 |- ( a e. CC |-> ( a ^ k ) ) = ( x e. CC |-> ( x ^ k ) )
2221eleq1i 2243 . . . . 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 14122 . . . . . . . 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 14130 . . . . 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 10529 . . . . . . . 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 4095 . . . . . 6 |- ( k e. NN0 -> ( x e. CC |-> ( x ^ ( k + 1 ) ) ) = ( x e. CC |-> ( ( x ^ k ) x. x ) ) )
3332eleq1d 2246 . . . . 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 9369 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 1353   e. wcel 2148   C_ wss 3131   |-> cmpt 4066  (class class class)co 5877   CCcc 7811   0cc0 7813   1c1 7814   + caddc 7816   x. cmul 7818   NN0cn0 9178   ^cexp 10521   -cn->ccncf 14096
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-isom 5227  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-map 6652  df-sup 6985  df-inf 6986  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-n0 9179  df-z 9256  df-uz 9531  df-rp 9656  df-seqfrec 10448  df-exp 10522  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010  df-cncf 14097
This theorem is referenced by: (None)
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