Theorem expcn 15208

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 8090. (Revised by GG, 16-Mar-2025.)

Hypothesis

Ref	Expression
expcn.j	⊢ 𝐽 = (TopOpen‘ℂfld)

Assertion

Ref	Expression
expcn	⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (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 5982	. . . 4 ⊢ (𝑛 = 0 → (𝑥↑𝑛) = (𝑥↑0))
2	1	mpteq2dv 4154	. . 3 ⊢ (𝑛 = 0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑0)))
3	2	eleq1d 2278	. 2 ⊢ (𝑛 = 0 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) ∈ (𝐽 Cn 𝐽) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑0)) ∈ (𝐽 Cn 𝐽)))
4		oveq2 5982	. . . 4 ⊢ (𝑛 = 𝑘 → (𝑥↑𝑛) = (𝑥↑𝑘))
5	4	mpteq2dv 4154	. . 3 ⊢ (𝑛 = 𝑘 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)))
6	5	eleq1d 2278	. 2 ⊢ (𝑛 = 𝑘 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) ∈ (𝐽 Cn 𝐽) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)))
7		oveq2 5982	. . . 4 ⊢ (𝑛 = (𝑘 + 1) → (𝑥↑𝑛) = (𝑥↑(𝑘 + 1)))
8	7	mpteq2dv 4154	. . 3 ⊢ (𝑛 = (𝑘 + 1) → (𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))))
9	8	eleq1d 2278	. 2 ⊢ (𝑛 = (𝑘 + 1) → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) ∈ (𝐽 Cn 𝐽) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) ∈ (𝐽 Cn 𝐽)))
10		oveq2 5982	. . . 4 ⊢ (𝑛 = 𝑁 → (𝑥↑𝑛) = (𝑥↑𝑁))
11	10	mpteq2dv 4154	. . 3 ⊢ (𝑛 = 𝑁 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)))
12	11	eleq1d 2278	. 2 ⊢ (𝑛 = 𝑁 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) ∈ (𝐽 Cn 𝐽) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (𝐽 Cn 𝐽)))
13		exp0 10732	. . . 4 ⊢ (𝑥 ∈ ℂ → (𝑥↑0) = 1)
14	13	mpteq2ia 4149	. . 3 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑0)) = (𝑥 ∈ ℂ ↦ 1)
15		expcn.j	. . . . . . 7 ⊢ 𝐽 = (TopOpen‘ℂfld)
16	15	cnfldtopon 15179	. . . . . 6 ⊢ 𝐽 ∈ (TopOn‘ℂ)
17	16	a1i 9	. . . . 5 ⊢ (⊤ → 𝐽 ∈ (TopOn‘ℂ))
18		1cnd 8130	. . . . 5 ⊢ (⊤ → 1 ∈ ℂ)
19	17, 17, 18	cnmptc 14921	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ 1) ∈ (𝐽 Cn 𝐽))
20	19	mptru 1384	. . 3 ⊢ (𝑥 ∈ ℂ ↦ 1) ∈ (𝐽 Cn 𝐽)
21	14, 20	eqeltri 2282	. 2 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑0)) ∈ (𝐽 Cn 𝐽)
22		oveq1 5981	. . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑥↑(𝑘 + 1)) = (𝑛↑(𝑘 + 1)))
23	22	cbvmptv 4159	. . . . 5 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) = (𝑛 ∈ ℂ ↦ (𝑛↑(𝑘 + 1)))
24		id 19	. . . . . . 7 ⊢ (𝑛 ∈ ℂ → 𝑛 ∈ ℂ)
25		simpl 109	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → 𝑘 ∈ ℕ0)
26		expp1 10735	. . . . . . . 8 ⊢ ((𝑛 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑛↑(𝑘 + 1)) = ((𝑛↑𝑘) · 𝑛))
27		expcl 10746	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑛↑𝑘) ∈ ℂ)
28		simpl 109	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → 𝑛 ∈ ℂ)
29	27, 28	mulcld 8135	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((𝑛↑𝑘) · 𝑛) ∈ ℂ)
30		oveq1 5981	. . . . . . . . . 10 ⊢ (𝑢 = (𝑛↑𝑘) → (𝑢 · 𝑣) = ((𝑛↑𝑘) · 𝑣))
31		oveq2 5982	. . . . . . . . . 10 ⊢ (𝑣 = 𝑛 → ((𝑛↑𝑘) · 𝑣) = ((𝑛↑𝑘) · 𝑛))
32		eqid 2209	. . . . . . . . . 10 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) = (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))
33	30, 31, 32	ovmpog 6110	. . . . . . . . 9 ⊢ (((𝑛↑𝑘) ∈ ℂ ∧ 𝑛 ∈ ℂ ∧ ((𝑛↑𝑘) · 𝑛) ∈ ℂ) → ((𝑛↑𝑘)(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑛) = ((𝑛↑𝑘) · 𝑛))
34	27, 28, 29, 33	syl3anc 1252	. . . . . . . 8 ⊢ ((𝑛 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((𝑛↑𝑘)(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑛) = ((𝑛↑𝑘) · 𝑛))
35	26, 34	eqtr4d 2245	. . . . . . 7 ⊢ ((𝑛 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑛↑(𝑘 + 1)) = ((𝑛↑𝑘)(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑛))
36	24, 25, 35	syl2anr 290	. . . . . 6 ⊢ (((𝑘 ∈ ℕ0 ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) ∧ 𝑛 ∈ ℂ) → (𝑛↑(𝑘 + 1)) = ((𝑛↑𝑘)(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑛))
37	36	mpteq2dva 4153	. . . . 5 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → (𝑛 ∈ ℂ ↦ (𝑛↑(𝑘 + 1))) = (𝑛 ∈ ℂ ↦ ((𝑛↑𝑘)(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑛)))
38	23, 37	eqtrid 2254	. . . 4 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) = (𝑛 ∈ ℂ ↦ ((𝑛↑𝑘)(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑛)))
39	16	a1i 9	. . . . 5 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → 𝐽 ∈ (TopOn‘ℂ))
40		oveq1 5981	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (𝑥↑𝑘) = (𝑛↑𝑘))
41	40	cbvmptv 4159	. . . . . 6 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) = (𝑛 ∈ ℂ ↦ (𝑛↑𝑘))
42		simpr 110	. . . . . 6 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽))
43	41, 42	eqeltrrid 2297	. . . . 5 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → (𝑛 ∈ ℂ ↦ (𝑛↑𝑘)) ∈ (𝐽 Cn 𝐽))
44	39	cnmptid 14920	. . . . 5 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → (𝑛 ∈ ℂ ↦ 𝑛) ∈ (𝐽 Cn 𝐽))
45	15	mpomulcn 15205	. . . . . 6 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)
46	45	a1i 9	. . . . 5 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
47	39, 43, 44, 46	cnmpt12f 14925	. . . 4 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → (𝑛 ∈ ℂ ↦ ((𝑛↑𝑘)(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑛)) ∈ (𝐽 Cn 𝐽))
48	38, 47	eqeltrd 2286	. . 3 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) ∈ (𝐽 Cn 𝐽))
49	48	ex 115	. 2 ⊢ (𝑘 ∈ ℕ0 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽) → (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) ∈ (𝐽 Cn 𝐽)))
50	3, 6, 9, 12, 21, 49	nn0ind 9529	1 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (𝐽 Cn 𝐽))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1375  ⊤wtru 1376   ∈ wcel 2180   ↦ cmpt 4124  ‘cfv 5294  (class class class)co 5974   ∈ cmpo 5976  ℂcc 7965  0cc0 7967  1c1 7968   + caddc 7970   · cmul 7972  ℕ₀cn0 9337  ↑cexp 10727  TopOpenctopn 13239  ℂ_fldccnfld 14485  TopOnctopon 14649   Cn ccn 14824   ×_t ctx 14891

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-iinf 4657  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-mulrcl 8066  ax-addcom 8067  ax-mulcom 8068  ax-addass 8069  ax-mulass 8070  ax-distr 8071  ax-i2m1 8072  ax-0lt1 8073  ax-1rid 8074  ax-0id 8075  ax-rnegex 8076  ax-precex 8077  ax-cnre 8078  ax-pre-ltirr 8079  ax-pre-ltwlin 8080  ax-pre-lttrn 8081  ax-pre-apti 8082  ax-pre-ltadd 8083  ax-pre-mulgt0 8084  ax-pre-mulext 8085  ax-arch 8086  ax-caucvg 8087

This theorem depends on definitions:  df-bi 117  df-stab 835  df-dc 839  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-nel 2476  df-ral 2493  df-rex 2494  df-reu 2495  df-rmo 2496  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-if 3583  df-pw 3631  df-sn 3652  df-pr 3653  df-tp 3654  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-tr 4162  df-id 4361  df-po 4364  df-iso 4365  df-iord 4434  df-on 4436  df-ilim 4437  df-suc 4439  df-iom 4660  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-isom 5303  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-recs 6421  df-frec 6507  df-map 6767  df-sup 7119  df-inf 7120  df-pnf 8151  df-mnf 8152  df-xr 8153  df-ltxr 8154  df-le 8155  df-sub 8287  df-neg 8288  df-reap 8690  df-ap 8697  df-div 8788  df-inn 9079  df-2 9137  df-3 9138  df-4 9139  df-5 9140  df-6 9141  df-7 9142  df-8 9143  df-9 9144  df-n0 9338  df-z 9415  df-dec 9547  df-uz 9691  df-q 9783  df-rp 9818  df-xneg 9936  df-xadd 9937  df-fz 10173  df-seqfrec 10637  df-exp 10728  df-cj 11319  df-re 11320  df-im 11321  df-rsqrt 11475  df-abs 11476  df-struct 13000  df-ndx 13001  df-slot 13002  df-base 13004  df-plusg 13089  df-mulr 13090  df-starv 13091  df-tset 13095  df-ple 13096  df-ds 13098  df-unif 13099  df-rest 13240  df-topn 13241  df-topgen 13259  df-psmet 14472  df-xmet 14473  df-met 14474  df-bl 14475  df-mopn 14476  df-fg 14478  df-metu 14479  df-cnfld 14486  df-top 14637  df-topon 14650  df-topsp 14670  df-bases 14682  df-cn 14827  df-cnp 14828  df-tx 14892  df-xms 14978  df-ms 14979

This theorem is referenced by:  plycn  15401