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Theorem expcncf 13725
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 5876 . . . 4 |- ( w = 0 -> ( x ^ w ) = ( x ^ 0 ) )
21mpteq2dv 4091 . . 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 5876 . . . 4 |- ( w = k -> ( x ^ w ) = ( x ^ k ) )
54mpteq2dv 4091 . . 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 5876 . . . 4 |- ( w = ( k + 1 ) -> ( x ^ w ) = ( x ^ ( k + 1 ) ) )
87mpteq2dv 4091 . . 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 5876 . . . 4 |- ( w = N -> ( x ^ w ) = ( x ^ N ) )
1110mpteq2dv 4091 . . 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 10497 . . . 4 |- ( x e. CC -> ( x ^ 0 ) = 1 )
1413mpteq2ia 4086 . . 3 |- ( x e. CC |-> ( x ^ 0 ) ) = ( x e. CC |-> 1 )
15 ax-1cn 7882 . . . 4 |- 1 e. CC
16 ssid 3175 . . . 4 |- CC C_ CC
17 cncfmptc 13715 . . . 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 5875 . . . . . . 7 |- ( a = x -> ( a ^ k ) = ( x ^ k ) )
2120cbvmptv 4096 . . . . . 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 13716 . . . . . . . 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 13724 . . . . 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 10500 . . . . . . . 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 4090 . . . . . 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 9343 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 3129   |-> cmpt 4061  (class class class)co 5868   CCcc 7787   0cc0 7789   1c1 7790   + caddc 7792   x. cmul 7794   NN0cn0 9152   ^cexp 10492   -cn->ccncf 13690
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 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-mulrcl 7888  ax-addcom 7889  ax-mulcom 7890  ax-addass 7891  ax-mulass 7892  ax-distr 7893  ax-i2m1 7894  ax-0lt1 7895  ax-1rid 7896  ax-0id 7897  ax-rnegex 7898  ax-precex 7899  ax-cnre 7900  ax-pre-ltirr 7901  ax-pre-ltwlin 7902  ax-pre-lttrn 7903  ax-pre-apti 7904  ax-pre-ltadd 7905  ax-pre-mulgt0 7906  ax-pre-mulext 7907  ax-arch 7908  ax-caucvg 7909
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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4289  df-po 4292  df-iso 4293  df-iord 4362  df-on 4364  df-ilim 4365  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-isom 5220  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-recs 6299  df-frec 6385  df-map 6643  df-sup 6976  df-inf 6977  df-pnf 7971  df-mnf 7972  df-xr 7973  df-ltxr 7974  df-le 7975  df-sub 8107  df-neg 8108  df-reap 8509  df-ap 8516  df-div 8606  df-inn 8896  df-2 8954  df-3 8955  df-4 8956  df-n0 9153  df-z 9230  df-uz 9505  df-rp 9628  df-seqfrec 10419  df-exp 10493  df-cj 10822  df-re 10823  df-im 10824  df-rsqrt 10978  df-abs 10979  df-cncf 13691
This theorem is referenced by: (None)
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