Theorem expcn 15360

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 8198. (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 6036	. . . 4 |- ( n = 0 -> ( x ^ n ) = ( x ^ 0 ) )
2	1	mpteq2dv 4185	. . 3 |- ( n = 0 -> ( x e. CC |-> ( x ^ n ) ) = ( x e. CC |-> ( x ^ 0 ) ) )
3	2	eleq1d 2300	. 2 |- ( n = 0 -> ( ( x e. CC |-> ( x ^ n ) ) e. ( J Cn J ) <-> ( x e. CC |-> ( x ^ 0 ) ) e. ( J Cn J ) ) )
4		oveq2 6036	. . . 4 |- ( n = k -> ( x ^ n ) = ( x ^ k ) )
5	4	mpteq2dv 4185	. . 3 |- ( n = k -> ( x e. CC |-> ( x ^ n ) ) = ( x e. CC |-> ( x ^ k ) ) )
6	5	eleq1d 2300	. 2 |- ( n = k -> ( ( x e. CC |-> ( x ^ n ) ) e. ( J Cn J ) <-> ( x e. CC |-> ( x ^ k ) ) e. ( J Cn J ) ) )
7		oveq2 6036	. . . 4 |- ( n = ( k + 1 ) -> ( x ^ n ) = ( x ^ ( k + 1 ) ) )
8	7	mpteq2dv 4185	. . 3 |- ( n = ( k + 1 ) -> ( x e. CC |-> ( x ^ n ) ) = ( x e. CC |-> ( x ^ ( k + 1 ) ) ) )
9	8	eleq1d 2300	. 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 6036	. . . 4 |- ( n = N -> ( x ^ n ) = ( x ^ N ) )
11	10	mpteq2dv 4185	. . 3 |- ( n = N -> ( x e. CC |-> ( x ^ n ) ) = ( x e. CC |-> ( x ^ N ) ) )
12	11	eleq1d 2300	. 2 |- ( n = N -> ( ( x e. CC |-> ( x ^ n ) ) e. ( J Cn J ) <-> ( x e. CC |-> ( x ^ N ) ) e. ( J Cn J ) ) )
13		exp0 10849	. . . 4 |- ( x e. CC -> ( x ^ 0 ) = 1 )
14	13	mpteq2ia 4180	. . 3 |- ( x e. CC |-> ( x ^ 0 ) ) = ( x e. CC |-> 1 )
15		expcn.j	. . . . . . 7 |- J = ( TopOpen ` ℂfld )
16	15	cnfldtopon 15331	. . . . . 6 |- J e. (TopOn ` CC )
17	16	a1i 9	. . . . 5 |- ( T. -> J e. (TopOn ` CC ) )
18		1cnd 8238	. . . . 5 |- ( T. -> 1 e. CC )
19	17, 17, 18	cnmptc 15073	. . . 4 |- ( T. -> ( x e. CC |-> 1 ) e. ( J Cn J ) )
20	19	mptru 1407	. . 3 |- ( x e. CC |-> 1 ) e. ( J Cn J )
21	14, 20	eqeltri 2304	. 2 |- ( x e. CC |-> ( x ^ 0 ) ) e. ( J Cn J )
22		oveq1 6035	. . . . . 6 |- ( x = n -> ( x ^ ( k + 1 ) ) = ( n ^ ( k + 1 ) ) )
23	22	cbvmptv 4190	. . . . 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 10852	. . . . . . . 8 |- ( ( n e. CC /\ k e. NN0 ) -> ( n ^ ( k + 1 ) ) = ( ( n ^ k ) x. n ) )
27		expcl 10863	. . . . . . . . 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 8243	. . . . . . . . 9 |- ( ( n e. CC /\ k e. NN0 ) -> ( ( n ^ k ) x. n ) e. CC )
30		oveq1 6035	. . . . . . . . . 10 |- ( u = ( n ^ k ) -> ( u x. v ) = ( ( n ^ k ) x. v ) )
31		oveq2 6036	. . . . . . . . . 10 |- ( v = n -> ( ( n ^ k ) x. v ) = ( ( n ^ k ) x. n ) )
32		eqid 2231	. . . . . . . . . 10 |- ( u e. CC , v e. CC |-> ( u x. v ) ) = ( u e. CC , v e. CC |-> ( u x. v ) )
33	30, 31, 32	ovmpog 6166	. . . . . . . . 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 1274	. . . . . . . 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 2267	. . . . . . 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 4184	. . . . 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 2276	. . . 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 6035	. . . . . . 7 |- ( x = n -> ( x ^ k ) = ( n ^ k ) )
41	40	cbvmptv 4190	. . . . . 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 2319	. . . . 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 15072	. . . . 5 |- ( ( k e. NN0 /\ ( x e. CC |-> ( x ^ k ) ) e. ( J Cn J ) ) -> ( n e. CC |-> n ) e. ( J Cn J ) )
45	15	mpomulcn 15357	. . . . . 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 15077	. . . 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 2308	. . 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 9637	1 |- ( N e. NN0 -> ( x e. CC |-> ( x ^ N ) ) e. ( J Cn J ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   T. wtru 1399   e. wcel 2202   |-> cmpt 4155   ` cfv 5333  (class class class)co 6028   e. cmpo 6030   CCcc 8073   0cc0 8075   1c1 8076   + caddc 8078   x. cmul 8080   NN0cn0 9445   ^cexp 10844   TopOpenctopn 13384  ℂ_fldccnfld 14632  TopOnctopon 14801   Cn ccn 14976   tX ctx 15043

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194  ax-caucvg 8195

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-tp 3681  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-map 6862  df-sup 7226  df-inf 7227  df-pnf 8259  df-mnf 8260  df-xr 8261  df-ltxr 8262  df-le 8263  df-sub 8395  df-neg 8396  df-reap 8798  df-ap 8805  df-div 8896  df-inn 9187  df-2 9245  df-3 9246  df-4 9247  df-5 9248  df-6 9249  df-7 9250  df-8 9251  df-9 9252  df-n0 9446  df-z 9523  df-dec 9655  df-uz 9799  df-q 9897  df-rp 9932  df-xneg 10050  df-xadd 10051  df-fz 10287  df-seqfrec 10754  df-exp 10845  df-cj 11463  df-re 11464  df-im 11465  df-rsqrt 11619  df-abs 11620  df-struct 13145  df-ndx 13146  df-slot 13147  df-base 13149  df-plusg 13234  df-mulr 13235  df-starv 13236  df-tset 13240  df-ple 13241  df-ds 13243  df-unif 13244  df-rest 13385  df-topn 13386  df-topgen 13404  df-psmet 14619  df-xmet 14620  df-met 14621  df-bl 14622  df-mopn 14623  df-fg 14625  df-metu 14626  df-cnfld 14633  df-top 14789  df-topon 14802  df-topsp 14822  df-bases 14834  df-cn 14979  df-cnp 14980  df-tx 15044  df-xms 15130  df-ms 15131

This theorem is referenced by:  plycn  15553