Theorem expcn 15672

Description: The power function on complex numbers, for fixed exponent N, is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) Avoid ax-mulf 8302. (Revised by GG, 16-Mar-2025.)

Hypothesis

Ref	Expression
expcn.j	|- J = ( TopOpen ` ℂfld )

Assertion

Ref	Expression
expcn	|- ( N e. NN0 -> ( x e. CC |-> ( x ^ N ) ) e. ( J Cn J ) )

Distinct variable groups:   x, J   x, N

Proof of Theorem expcn

Dummy variables k n u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6093	. . . 4 |- ( n = 0 -> ( x ^ n ) = ( x ^ 0 ) )
2	1	mpteq2dv 4222	. . 3 |- ( n = 0 -> ( x e. CC |-> ( x ^ n ) ) = ( x e. CC |-> ( x ^ 0 ) ) )
3	2	eleq1d 2307	. 2 |- ( n = 0 -> ( ( x e. CC |-> ( x ^ n ) ) e. ( J Cn J ) <-> ( x e. CC |-> ( x ^ 0 ) ) e. ( J Cn J ) ) )
4		oveq2 6093	. . . 4 |- ( n = k -> ( x ^ n ) = ( x ^ k ) )
5	4	mpteq2dv 4222	. . 3 |- ( n = k -> ( x e. CC |-> ( x ^ n ) ) = ( x e. CC |-> ( x ^ k ) ) )
6	5	eleq1d 2307	. 2 |- ( n = k -> ( ( x e. CC |-> ( x ^ n ) ) e. ( J Cn J ) <-> ( x e. CC |-> ( x ^ k ) ) e. ( J Cn J ) ) )
7		oveq2 6093	. . . 4 |- ( n = ( k + 1 ) -> ( x ^ n ) = ( x ^ ( k + 1 ) ) )
8	7	mpteq2dv 4222	. . 3 |- ( n = ( k + 1 ) -> ( x e. CC |-> ( x ^ n ) ) = ( x e. CC |-> ( x ^ ( k + 1 ) ) ) )
9	8	eleq1d 2307	. 2 |- ( n = ( k + 1 ) -> ( ( x e. CC |-> ( x ^ n ) ) e. ( J Cn J ) <-> ( x e. CC |-> ( x ^ ( k + 1 ) ) ) e. ( J Cn J ) ) )
10		oveq2 6093	. . . 4 |- ( n = N -> ( x ^ n ) = ( x ^ N ) )
11	10	mpteq2dv 4222	. . 3 |- ( n = N -> ( x e. CC |-> ( x ^ n ) ) = ( x e. CC |-> ( x ^ N ) ) )
12	11	eleq1d 2307	. 2 |- ( n = N -> ( ( x e. CC |-> ( x ^ n ) ) e. ( J Cn J ) <-> ( x e. CC |-> ( x ^ N ) ) e. ( J Cn J ) ) )
13		exp0 10982	. . . 4 |- ( x e. CC -> ( x ^ 0 ) = 1 )
14	13	mpteq2ia 4217	. . 3 |- ( x e. CC |-> ( x ^ 0 ) ) = ( x e. CC |-> 1 )
15		expcn.j	. . . . . . 7 |- J = ( TopOpen ` ℂfld )
16	15	cnfldtopon 15643	. . . . . 6 |- J e. (TopOn ` CC )
17	16	a1i 9	. . . . 5 |- ( T. -> J e. (TopOn ` CC ) )
18		1cnd 8342	. . . . 5 |- ( T. -> 1 e. CC )
19	17, 17, 18	cnmptc 15385	. . . 4 |- ( T. -> ( x e. CC |-> 1 ) e. ( J Cn J ) )
20	19	mptru 1411	. . 3 |- ( x e. CC |-> 1 ) e. ( J Cn J )
21	14, 20	eqeltri 2311	. 2 |- ( x e. CC |-> ( x ^ 0 ) ) e. ( J Cn J )
22		oveq1 6092	. . . . . 6 |- ( x = n -> ( x ^ ( k + 1 ) ) = ( n ^ ( k + 1 ) ) )
23	22	cbvmptv 4227	. . . . 5 |- ( x e. CC |-> ( x ^ ( k + 1 ) ) ) = ( n e. CC |-> ( n ^ ( k + 1 ) ) )
24		id 19	. . . . . . 7 |- ( n e. CC -> n e. CC )
25		simpl 109	. . . . . . 7 |- ( ( k e. NN0 /\ ( x e. CC |-> ( x ^ k ) ) e. ( J Cn J ) ) -> k e. NN0 )
26		expp1 10985	. . . . . . . 8 |- ( ( n e. CC /\ k e. NN0 ) -> ( n ^ ( k + 1 ) ) = ( ( n ^ k ) x. n ) )
27		expcl 10996	. . . . . . . . 9 |- ( ( n e. CC /\ k e. NN0 ) -> ( n ^ k ) e. CC )
28		simpl 109	. . . . . . . . 9 |- ( ( n e. CC /\ k e. NN0 ) -> n e. CC )
29	27, 28	mulcld 8346	. . . . . . . . 9 |- ( ( n e. CC /\ k e. NN0 ) -> ( ( n ^ k ) x. n ) e. CC )
30		oveq1 6092	. . . . . . . . . 10 |- ( u = ( n ^ k ) -> ( u x. v ) = ( ( n ^ k ) x. v ) )
31		oveq2 6093	. . . . . . . . . 10 |- ( v = n -> ( ( n ^ k ) x. v ) = ( ( n ^ k ) x. n ) )
32		eqid 2238	. . . . . . . . . 10 |- ( u e. CC , v e. CC |-> ( u x. v ) ) = ( u e. CC , v e. CC |-> ( u x. v ) )
33	30, 31, 32	ovmpog 6223	. . . . . . . . 9 |- ( ( ( n ^ k ) e. CC /\ n e. CC /\ ( ( n ^ k ) x. n ) e. CC ) -> ( ( n ^ k ) ( u e. CC , v e. CC |-> ( u x. v ) ) n ) = ( ( n ^ k ) x. n ) )
34	27, 28, 29, 33	syl3anc 1278	. . . . . . . 8 |- ( ( n e. CC /\ k e. NN0 ) -> ( ( n ^ k ) ( u e. CC , v e. CC |-> ( u x. v ) ) n ) = ( ( n ^ k ) x. n ) )
35	26, 34	eqtr4d 2274	. . . . . . 7 |- ( ( n e. CC /\ k e. NN0 ) -> ( n ^ ( k + 1 ) ) = ( ( n ^ k ) ( u e. CC , v e. CC |-> ( u x. v ) ) n ) )
36	24, 25, 35	syl2anr 290	. . . . . 6 |- ( ( ( k e. NN0 /\ ( x e. CC |-> ( x ^ k ) ) e. ( J Cn J ) ) /\ n e. CC ) -> ( n ^ ( k + 1 ) ) = ( ( n ^ k ) ( u e. CC , v e. CC |-> ( u x. v ) ) n ) )
37	36	mpteq2dva 4221	. . . . 5 |- ( ( k e. NN0 /\ ( x e. CC |-> ( x ^ k ) ) e. ( J Cn J ) ) -> ( n e. CC |-> ( n ^ ( k + 1 ) ) ) = ( n e. CC |-> ( ( n ^ k ) ( u e. CC , v e. CC |-> ( u x. v ) ) n ) ) )
38	23, 37	eqtrid 2283	. . . 4 |- ( ( k e. NN0 /\ ( x e. CC |-> ( x ^ k ) ) e. ( J Cn J ) ) -> ( x e. CC |-> ( x ^ ( k + 1 ) ) ) = ( n e. CC |-> ( ( n ^ k ) ( u e. CC , v e. CC |-> ( u x. v ) ) n ) ) )
39	16	a1i 9	. . . . 5 |- ( ( k e. NN0 /\ ( x e. CC |-> ( x ^ k ) ) e. ( J Cn J ) ) -> J e. (TopOn ` CC ) )
40		oveq1 6092	. . . . . . 7 |- ( x = n -> ( x ^ k ) = ( n ^ k ) )
41	40	cbvmptv 4227	. . . . . 6 |- ( x e. CC |-> ( x ^ k ) ) = ( n e. CC |-> ( n ^ k ) )
42		simpr 110	. . . . . 6 |- ( ( k e. NN0 /\ ( x e. CC |-> ( x ^ k ) ) e. ( J Cn J ) ) -> ( x e. CC |-> ( x ^ k ) ) e. ( J Cn J ) )
43	41, 42	eqeltrrid 2326	. . . . 5 |- ( ( k e. NN0 /\ ( x e. CC |-> ( x ^ k ) ) e. ( J Cn J ) ) -> ( n e. CC |-> ( n ^ k ) ) e. ( J Cn J ) )
44	39	cnmptid 15384	. . . . 5 |- ( ( k e. NN0 /\ ( x e. CC |-> ( x ^ k ) ) e. ( J Cn J ) ) -> ( n e. CC |-> n ) e. ( J Cn J ) )
45	15	mpomulcn 15669	. . . . . 6 |- ( u e. CC , v e. CC |-> ( u x. v ) ) e. ( ( J tX J ) Cn J )
46	45	a1i 9	. . . . 5 |- ( ( k e. NN0 /\ ( x e. CC |-> ( x ^ k ) ) e. ( J Cn J ) ) -> ( u e. CC , v e. CC |-> ( u x. v ) ) e. ( ( J tX J ) Cn J ) )
47	39, 43, 44, 46	cnmpt12f 15389	. . . 4 |- ( ( k e. NN0 /\ ( x e. CC |-> ( x ^ k ) ) e. ( J Cn J ) ) -> ( n e. CC |-> ( ( n ^ k ) ( u e. CC , v e. CC |-> ( u x. v ) ) n ) ) e. ( J Cn J ) )
48	38, 47	eqeltrd 2315	. . 3 |- ( ( k e. NN0 /\ ( x e. CC |-> ( x ^ k ) ) e. ( J Cn J ) ) -> ( x e. CC |-> ( x ^ ( k + 1 ) ) ) e. ( J Cn J ) )
49	48	ex 115	. 2 |- ( k e. NN0 -> ( ( x e. CC |-> ( x ^ k ) ) e. ( J Cn J ) -> ( x e. CC |-> ( x ^ ( k + 1 ) ) ) e. ( J Cn J ) ) )
50	3, 6, 9, 12, 21, 49	nn0ind 9762	1 |- ( N e. NN0 -> ( x e. CC |-> ( x ^ N ) ) e. ( J Cn J ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   T. wtru 1403   e. wcel 2209   |-> cmpt 4192   ` cfv 5377  (class class class)co 6085   e. cmpo 6087   CCcc 8177   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   NN0cn0 9565   ^cexp 10977   TopOpenctopn 13596  ℂ_fldccnfld 14895  TopOnctopon 15113   Cn ccn 15288   tX ctx 15355

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-tp 3717  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-map 6924  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8904  df-ap 8911  df-div 9004  df-inn 9306  df-2 9364  df-3 9365  df-4 9366  df-5 9367  df-6 9368  df-7 9369  df-8 9370  df-9 9371  df-n0 9566  df-z 9647  df-dec 9780  df-uz 9924  df-q 10022  df-rp 10057  df-xneg 10176  df-xadd 10177  df-fz 10414  df-seqfrec 10887  df-exp 10978  df-cj 11609  df-re 11610  df-im 11611  df-rsqrt 11766  df-abs 11767  df-struct 13356  df-ndx 13357  df-slot 13358  df-base 13360  df-plusg 13446  df-mulr 13447  df-starv 13448  df-tset 13452  df-ple 13453  df-ds 13455  df-unif 13456  df-rest 13597  df-topn 13598  df-topgen 13616  df-psmet 14882  df-xmet 14883  df-met 14884  df-bl 14885  df-mopn 14886  df-fg 14888  df-metu 14889  df-cnfld 14896  df-top 15101  df-topon 15114  df-topsp 15134  df-bases 15146  df-cn 15291  df-cnp 15292  df-tx 15356  df-xms 15442  df-ms 15443

This theorem is used by:  plycn  15865