Theorem expcn 14748

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 7997. (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 5927	. . . 4 |- ( n = 0 -> ( x ^ n ) = ( x ^ 0 ) )
2	1	mpteq2dv 4121	. . 3 |- ( n = 0 -> ( x e. CC |-> ( x ^ n ) ) = ( x e. CC |-> ( x ^ 0 ) ) )
3	2	eleq1d 2262	. 2 |- ( n = 0 -> ( ( x e. CC |-> ( x ^ n ) ) e. ( J Cn J ) <-> ( x e. CC |-> ( x ^ 0 ) ) e. ( J Cn J ) ) )
4		oveq2 5927	. . . 4 |- ( n = k -> ( x ^ n ) = ( x ^ k ) )
5	4	mpteq2dv 4121	. . 3 |- ( n = k -> ( x e. CC |-> ( x ^ n ) ) = ( x e. CC |-> ( x ^ k ) ) )
6	5	eleq1d 2262	. 2 |- ( n = k -> ( ( x e. CC |-> ( x ^ n ) ) e. ( J Cn J ) <-> ( x e. CC |-> ( x ^ k ) ) e. ( J Cn J ) ) )
7		oveq2 5927	. . . 4 |- ( n = ( k + 1 ) -> ( x ^ n ) = ( x ^ ( k + 1 ) ) )
8	7	mpteq2dv 4121	. . 3 |- ( n = ( k + 1 ) -> ( x e. CC |-> ( x ^ n ) ) = ( x e. CC |-> ( x ^ ( k + 1 ) ) ) )
9	8	eleq1d 2262	. 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 5927	. . . 4 |- ( n = N -> ( x ^ n ) = ( x ^ N ) )
11	10	mpteq2dv 4121	. . 3 |- ( n = N -> ( x e. CC |-> ( x ^ n ) ) = ( x e. CC |-> ( x ^ N ) ) )
12	11	eleq1d 2262	. 2 |- ( n = N -> ( ( x e. CC |-> ( x ^ n ) ) e. ( J Cn J ) <-> ( x e. CC |-> ( x ^ N ) ) e. ( J Cn J ) ) )
13		exp0 10617	. . . 4 |- ( x e. CC -> ( x ^ 0 ) = 1 )
14	13	mpteq2ia 4116	. . 3 |- ( x e. CC |-> ( x ^ 0 ) ) = ( x e. CC |-> 1 )
15		expcn.j	. . . . . . 7 |- J = ( TopOpen ` ℂfld )
16	15	cnfldtopon 14719	. . . . . 6 |- J e. (TopOn ` CC )
17	16	a1i 9	. . . . 5 |- ( T. -> J e. (TopOn ` CC ) )
18		1cnd 8037	. . . . 5 |- ( T. -> 1 e. CC )
19	17, 17, 18	cnmptc 14461	. . . 4 |- ( T. -> ( x e. CC |-> 1 ) e. ( J Cn J ) )
20	19	mptru 1373	. . 3 |- ( x e. CC |-> 1 ) e. ( J Cn J )
21	14, 20	eqeltri 2266	. 2 |- ( x e. CC |-> ( x ^ 0 ) ) e. ( J Cn J )
22		oveq1 5926	. . . . . 6 |- ( x = n -> ( x ^ ( k + 1 ) ) = ( n ^ ( k + 1 ) ) )
23	22	cbvmptv 4126	. . . . 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 10620	. . . . . . . 8 |- ( ( n e. CC /\ k e. NN0 ) -> ( n ^ ( k + 1 ) ) = ( ( n ^ k ) x. n ) )
27		expcl 10631	. . . . . . . . 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 8042	. . . . . . . . 9 |- ( ( n e. CC /\ k e. NN0 ) -> ( ( n ^ k ) x. n ) e. CC )
30		oveq1 5926	. . . . . . . . . 10 |- ( u = ( n ^ k ) -> ( u x. v ) = ( ( n ^ k ) x. v ) )
31		oveq2 5927	. . . . . . . . . 10 |- ( v = n -> ( ( n ^ k ) x. v ) = ( ( n ^ k ) x. n ) )
32		eqid 2193	. . . . . . . . . 10 |- ( u e. CC , v e. CC |-> ( u x. v ) ) = ( u e. CC , v e. CC |-> ( u x. v ) )
33	30, 31, 32	ovmpog 6054	. . . . . . . . 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 1249	. . . . . . . 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 2229	. . . . . . 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 4120	. . . . 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 2238	. . . 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 5926	. . . . . . 7 |- ( x = n -> ( x ^ k ) = ( n ^ k ) )
41	40	cbvmptv 4126	. . . . . 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 2281	. . . . 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 14460	. . . . 5 |- ( ( k e. NN0 /\ ( x e. CC |-> ( x ^ k ) ) e. ( J Cn J ) ) -> ( n e. CC |-> n ) e. ( J Cn J ) )
45	15	mpomulcn 14745	. . . . . 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 14465	. . . 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 2270	. . 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 9434	1 |- ( N e. NN0 -> ( x e. CC |-> ( x ^ N ) ) e. ( J Cn J ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   T. wtru 1365   e. wcel 2164   |-> cmpt 4091   ` cfv 5255  (class class class)co 5919   e. cmpo 5921   CCcc 7872   0cc0 7874   1c1 7875   + caddc 7877   x. cmul 7879   NN0cn0 9243   ^cexp 10612   TopOpenctopn 12854  ℂ_fldccnfld 14055  TopOnctopon 14189   Cn ccn 14364   tX ctx 14431

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993  ax-caucvg 7994

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-tp 3627  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-frec 6446  df-map 6706  df-sup 7045  df-inf 7046  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-5 9046  df-6 9047  df-7 9048  df-8 9049  df-9 9050  df-n0 9244  df-z 9321  df-dec 9452  df-uz 9596  df-q 9688  df-rp 9723  df-xneg 9841  df-xadd 9842  df-fz 10078  df-seqfrec 10522  df-exp 10613  df-cj 10989  df-re 10990  df-im 10991  df-rsqrt 11145  df-abs 11146  df-struct 12623  df-ndx 12624  df-slot 12625  df-base 12627  df-plusg 12711  df-mulr 12712  df-starv 12713  df-tset 12717  df-ple 12718  df-ds 12720  df-unif 12721  df-rest 12855  df-topn 12856  df-topgen 12874  df-psmet 14042  df-xmet 14043  df-met 14044  df-bl 14045  df-mopn 14046  df-fg 14048  df-metu 14049  df-cnfld 14056  df-top 14177  df-topon 14190  df-topsp 14210  df-bases 14222  df-cn 14367  df-cnp 14368  df-tx 14432  df-xms 14518  df-ms 14519

This theorem is referenced by:  plycn  14932