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Theorem expcncf 15332
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 6025 . . . 4 |- ( w = 0 -> ( x ^ w ) = ( x ^ 0 ) )
21mpteq2dv 4180 . . 3 |- ( w = 0 -> ( x e. CC |-> ( x ^ w ) ) = ( x e. CC |-> ( x ^ 0 ) ) )
32eleq1d 2300 . 2 |- ( w = 0 -> ( ( x e. CC |-> ( x ^ w ) ) e. ( CC -cn-> CC ) <-> ( x e. CC |-> ( x ^ 0 ) ) e. ( CC -cn-> CC ) ) )
4 oveq2 6025 . . . 4 |- ( w = k -> ( x ^ w ) = ( x ^ k ) )
54mpteq2dv 4180 . . 3 |- ( w = k -> ( x e. CC |-> ( x ^ w ) ) = ( x e. CC |-> ( x ^ k ) ) )
65eleq1d 2300 . 2 |- ( w = k -> ( ( x e. CC |-> ( x ^ w ) ) e. ( CC -cn-> CC ) <-> ( x e. CC |-> ( x ^ k ) ) e. ( CC -cn-> CC ) ) )
7 oveq2 6025 . . . 4 |- ( w = ( k + 1 ) -> ( x ^ w ) = ( x ^ ( k + 1 ) ) )
87mpteq2dv 4180 . . 3 |- ( w = ( k + 1 ) -> ( x e. CC |-> ( x ^ w ) ) = ( x e. CC |-> ( x ^ ( k + 1 ) ) ) )
98eleq1d 2300 . 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 6025 . . . 4 |- ( w = N -> ( x ^ w ) = ( x ^ N ) )
1110mpteq2dv 4180 . . 3 |- ( w = N -> ( x e. CC |-> ( x ^ w ) ) = ( x e. CC |-> ( x ^ N ) ) )
1211eleq1d 2300 . 2 |- ( w = N -> ( ( x e. CC |-> ( x ^ w ) ) e. ( CC -cn-> CC ) <-> ( x e. CC |-> ( x ^ N ) ) e. ( CC -cn-> CC ) ) )
13 exp0 10804 . . . 4 |- ( x e. CC -> ( x ^ 0 ) = 1 )
1413mpteq2ia 4175 . . 3 |- ( x e. CC |-> ( x ^ 0 ) ) = ( x e. CC |-> 1 )
15 ax-1cn 8124 . . . 4 |- 1 e. CC
16 ssid 3247 . . . 4 |- CC C_ CC
17 cncfmptc 15319 . . . 4 |- ( ( 1 e. CC /\ CC C_ CC /\ CC C_ CC ) -> ( x e. CC |-> 1 ) e. ( CC -cn-> CC ) )
1815, 16, 16, 17mp3an 1373 . . 3 |- ( x e. CC |-> 1 ) e. ( CC -cn-> CC )
1914, 18eqeltri 2304 . 2 |- ( x e. CC |-> ( x ^ 0 ) ) e. ( CC -cn-> CC )
20 oveq1 6024 . . . . . . 7 |- ( a = x -> ( a ^ k ) = ( x ^ k ) )
2120cbvmptv 4185 . . . . . 6 |- ( a e. CC |-> ( a ^ k ) ) = ( x e. CC |-> ( x ^ k ) )
2221eleq1i 2297 . . . . 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 15320 . . . . . . . 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 15331 . . . . 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 10807 . . . . . . . 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 4179 . . . . . 6 |- ( k e. NN0 -> ( x e. CC |-> ( x ^ ( k + 1 ) ) ) = ( x e. CC |-> ( ( x ^ k ) x. x ) ) )
3332eleq1d 2300 . . . . 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 9593 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 1397   e. wcel 2202   C_ wss 3200   |-> cmpt 4150  (class class class)co 6017   CCcc 8029   0cc0 8031   1c1 8032   + caddc 8034   x. cmul 8036   NN0cn0 9401   ^cexp 10799   -cn->ccncf 15293
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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151
This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-map 6818  df-sup 7182  df-inf 7183  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-z 9479  df-uz 9755  df-rp 9888  df-seqfrec 10709  df-exp 10800  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-cncf 15294
This theorem is referenced by: (None)
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