Theorem expcn 15085

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 8055. (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 5959	. . . 4 |- ( n = 0 -> ( x ^ n ) = ( x ^ 0 ) )
2	1	mpteq2dv 4139	. . 3 |- ( n = 0 -> ( x e. CC |-> ( x ^ n ) ) = ( x e. CC |-> ( x ^ 0 ) ) )
3	2	eleq1d 2275	. 2 |- ( n = 0 -> ( ( x e. CC |-> ( x ^ n ) ) e. ( J Cn J ) <-> ( x e. CC |-> ( x ^ 0 ) ) e. ( J Cn J ) ) )
4		oveq2 5959	. . . 4 |- ( n = k -> ( x ^ n ) = ( x ^ k ) )
5	4	mpteq2dv 4139	. . 3 |- ( n = k -> ( x e. CC |-> ( x ^ n ) ) = ( x e. CC |-> ( x ^ k ) ) )
6	5	eleq1d 2275	. 2 |- ( n = k -> ( ( x e. CC |-> ( x ^ n ) ) e. ( J Cn J ) <-> ( x e. CC |-> ( x ^ k ) ) e. ( J Cn J ) ) )
7		oveq2 5959	. . . 4 |- ( n = ( k + 1 ) -> ( x ^ n ) = ( x ^ ( k + 1 ) ) )
8	7	mpteq2dv 4139	. . 3 |- ( n = ( k + 1 ) -> ( x e. CC |-> ( x ^ n ) ) = ( x e. CC |-> ( x ^ ( k + 1 ) ) ) )
9	8	eleq1d 2275	. 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 5959	. . . 4 |- ( n = N -> ( x ^ n ) = ( x ^ N ) )
11	10	mpteq2dv 4139	. . 3 |- ( n = N -> ( x e. CC |-> ( x ^ n ) ) = ( x e. CC |-> ( x ^ N ) ) )
12	11	eleq1d 2275	. 2 |- ( n = N -> ( ( x e. CC |-> ( x ^ n ) ) e. ( J Cn J ) <-> ( x e. CC |-> ( x ^ N ) ) e. ( J Cn J ) ) )
13		exp0 10695	. . . 4 |- ( x e. CC -> ( x ^ 0 ) = 1 )
14	13	mpteq2ia 4134	. . 3 |- ( x e. CC |-> ( x ^ 0 ) ) = ( x e. CC |-> 1 )
15		expcn.j	. . . . . . 7 |- J = ( TopOpen ` ℂfld )
16	15	cnfldtopon 15056	. . . . . 6 |- J e. (TopOn ` CC )
17	16	a1i 9	. . . . 5 |- ( T. -> J e. (TopOn ` CC ) )
18		1cnd 8095	. . . . 5 |- ( T. -> 1 e. CC )
19	17, 17, 18	cnmptc 14798	. . . 4 |- ( T. -> ( x e. CC |-> 1 ) e. ( J Cn J ) )
20	19	mptru 1382	. . 3 |- ( x e. CC |-> 1 ) e. ( J Cn J )
21	14, 20	eqeltri 2279	. 2 |- ( x e. CC |-> ( x ^ 0 ) ) e. ( J Cn J )
22		oveq1 5958	. . . . . 6 |- ( x = n -> ( x ^ ( k + 1 ) ) = ( n ^ ( k + 1 ) ) )
23	22	cbvmptv 4144	. . . . 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 10698	. . . . . . . 8 |- ( ( n e. CC /\ k e. NN0 ) -> ( n ^ ( k + 1 ) ) = ( ( n ^ k ) x. n ) )
27		expcl 10709	. . . . . . . . 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 8100	. . . . . . . . 9 |- ( ( n e. CC /\ k e. NN0 ) -> ( ( n ^ k ) x. n ) e. CC )
30		oveq1 5958	. . . . . . . . . 10 |- ( u = ( n ^ k ) -> ( u x. v ) = ( ( n ^ k ) x. v ) )
31		oveq2 5959	. . . . . . . . . 10 |- ( v = n -> ( ( n ^ k ) x. v ) = ( ( n ^ k ) x. n ) )
32		eqid 2206	. . . . . . . . . 10 |- ( u e. CC , v e. CC |-> ( u x. v ) ) = ( u e. CC , v e. CC |-> ( u x. v ) )
33	30, 31, 32	ovmpog 6087	. . . . . . . . 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 1250	. . . . . . . 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 2242	. . . . . . 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 4138	. . . . 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 2251	. . . 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 5958	. . . . . . 7 |- ( x = n -> ( x ^ k ) = ( n ^ k ) )
41	40	cbvmptv 4144	. . . . . 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 2294	. . . . 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 14797	. . . . 5 |- ( ( k e. NN0 /\ ( x e. CC |-> ( x ^ k ) ) e. ( J Cn J ) ) -> ( n e. CC |-> n ) e. ( J Cn J ) )
45	15	mpomulcn 15082	. . . . . 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 14802	. . . 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 2283	. . 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 9494	1 |- ( N e. NN0 -> ( x e. CC |-> ( x ^ N ) ) e. ( J Cn J ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   T. wtru 1374   e. wcel 2177   |-> cmpt 4109   ` cfv 5276  (class class class)co 5951   e. cmpo 5953   CCcc 7930   0cc0 7932   1c1 7933   + caddc 7935   x. cmul 7937   NN0cn0 9302   ^cexp 10690   TopOpenctopn 13116  ℂ_fldccnfld 14362  TopOnctopon 14526   Cn ccn 14701   tX ctx 14768

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-nul 4174  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-iinf 4640  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-mulrcl 8031  ax-addcom 8032  ax-mulcom 8033  ax-addass 8034  ax-mulass 8035  ax-distr 8036  ax-i2m1 8037  ax-0lt1 8038  ax-1rid 8039  ax-0id 8040  ax-rnegex 8041  ax-precex 8042  ax-cnre 8043  ax-pre-ltirr 8044  ax-pre-ltwlin 8045  ax-pre-lttrn 8046  ax-pre-apti 8047  ax-pre-ltadd 8048  ax-pre-mulgt0 8049  ax-pre-mulext 8050  ax-arch 8051  ax-caucvg 8052

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-if 3573  df-pw 3619  df-sn 3640  df-pr 3641  df-tp 3642  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-tr 4147  df-id 4344  df-po 4347  df-iso 4348  df-iord 4417  df-on 4419  df-ilim 4420  df-suc 4422  df-iom 4643  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-isom 5285  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-recs 6398  df-frec 6484  df-map 6744  df-sup 7093  df-inf 7094  df-pnf 8116  df-mnf 8117  df-xr 8118  df-ltxr 8119  df-le 8120  df-sub 8252  df-neg 8253  df-reap 8655  df-ap 8662  df-div 8753  df-inn 9044  df-2 9102  df-3 9103  df-4 9104  df-5 9105  df-6 9106  df-7 9107  df-8 9108  df-9 9109  df-n0 9303  df-z 9380  df-dec 9512  df-uz 9656  df-q 9748  df-rp 9783  df-xneg 9901  df-xadd 9902  df-fz 10138  df-seqfrec 10600  df-exp 10691  df-cj 11197  df-re 11198  df-im 11199  df-rsqrt 11353  df-abs 11354  df-struct 12878  df-ndx 12879  df-slot 12880  df-base 12882  df-plusg 12966  df-mulr 12967  df-starv 12968  df-tset 12972  df-ple 12973  df-ds 12975  df-unif 12976  df-rest 13117  df-topn 13118  df-topgen 13136  df-psmet 14349  df-xmet 14350  df-met 14351  df-bl 14352  df-mopn 14353  df-fg 14355  df-metu 14356  df-cnfld 14363  df-top 14514  df-topon 14527  df-topsp 14547  df-bases 14559  df-cn 14704  df-cnp 14705  df-tx 14769  df-xms 14855  df-ms 14856

This theorem is referenced by:  plycn  15278