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Theorem expcn 24864

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 11116. (Revised by GG, 16-Mar-2025.)

**Hypothesis**
Ref	Expression
expcn.j	⊢ 𝐽 = (TopOpen‘ℂ_fld)

**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 7371	. . . 4 ⊢ (𝑛 = 0 → (𝑥↑𝑛) = (𝑥↑0))
2	1	mpteq2dv 5173	. . 3 ⊢ (𝑛 = 0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑0)))
3	2	eleq1d 2825	. 2 ⊢ (𝑛 = 0 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) ∈ (𝐽 Cn 𝐽) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑0)) ∈ (𝐽 Cn 𝐽)))
4		oveq2 7371	. . . 4 ⊢ (𝑛 = 𝑘 → (𝑥↑𝑛) = (𝑥↑𝑘))
5	4	mpteq2dv 5173	. . 3 ⊢ (𝑛 = 𝑘 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)))
6	5	eleq1d 2825	. 2 ⊢ (𝑛 = 𝑘 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) ∈ (𝐽 Cn 𝐽) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)))
7		oveq2 7371	. . . 4 ⊢ (𝑛 = (𝑘 + 1) → (𝑥↑𝑛) = (𝑥↑(𝑘 + 1)))
8	7	mpteq2dv 5173	. . 3 ⊢ (𝑛 = (𝑘 + 1) → (𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))))
9	8	eleq1d 2825	. 2 ⊢ (𝑛 = (𝑘 + 1) → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) ∈ (𝐽 Cn 𝐽) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) ∈ (𝐽 Cn 𝐽)))
10		oveq2 7371	. . . 4 ⊢ (𝑛 = 𝑁 → (𝑥↑𝑛) = (𝑥↑𝑁))
11	10	mpteq2dv 5173	. . 3 ⊢ (𝑛 = 𝑁 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)))
12	11	eleq1d 2825	. 2 ⊢ (𝑛 = 𝑁 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) ∈ (𝐽 Cn 𝐽) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (𝐽 Cn 𝐽)))
13		exp0 14025	. . . 4 ⊢ (𝑥 ∈ ℂ → (𝑥↑0) = 1)
14	13	mpteq2ia 5174	. . 3 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑0)) = (𝑥 ∈ ℂ ↦ 1)
15		expcn.j	. . . . . . 7 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
16	15	cnfldtopon 24772	. . . . . 6 ⊢ 𝐽 ∈ (TopOn‘ℂ)
17	16	a1i 11	. . . . 5 ⊢ (⊤ → 𝐽 ∈ (TopOn‘ℂ))
18		1cnd 11137	. . . . 5 ⊢ (⊤ → 1 ∈ ℂ)
19	17, 17, 18	cnmptc 23652	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ 1) ∈ (𝐽 Cn 𝐽))
20	19	mptru 1554	. . 3 ⊢ (𝑥 ∈ ℂ ↦ 1) ∈ (𝐽 Cn 𝐽)
21	14, 20	eqeltri 2836	. 2 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑0)) ∈ (𝐽 Cn 𝐽)
22		oveq1 7370	. . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑥↑(𝑘 + 1)) = (𝑛↑(𝑘 + 1)))
23	22	cbvmptv 5183	. . . . 5 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) = (𝑛 ∈ ℂ ↦ (𝑛↑(𝑘 + 1)))
24		id 22	. . . . . . 7 ⊢ (𝑛 ∈ ℂ → 𝑛 ∈ ℂ)
25		simpl 483	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ₀ ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → 𝑘 ∈ ℕ₀)
26		expp1 14028	. . . . . . . 8 ⊢ ((𝑛 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝑛↑(𝑘 + 1)) = ((𝑛↑𝑘) · 𝑛))
27		expcl 14039	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝑛↑𝑘) ∈ ℂ)
28		simpl 483	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → 𝑛 ∈ ℂ)
29		ovmpot 7524	. . . . . . . . 9 ⊢ (((𝑛↑𝑘) ∈ ℂ ∧ 𝑛 ∈ ℂ) → ((𝑛↑𝑘)(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑛) = ((𝑛↑𝑘) · 𝑛))
30	27, 28, 29	syl2anc 590	. . . . . . . 8 ⊢ ((𝑛 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → ((𝑛↑𝑘)(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑛) = ((𝑛↑𝑘) · 𝑛))
31	26, 30	eqtr4d 2778	. . . . . . 7 ⊢ ((𝑛 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝑛↑(𝑘 + 1)) = ((𝑛↑𝑘)(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑛))
32	24, 25, 31	syl2anr 603	. . . . . 6 ⊢ (((𝑘 ∈ ℕ₀ ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) ∧ 𝑛 ∈ ℂ) → (𝑛↑(𝑘 + 1)) = ((𝑛↑𝑘)(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑛))
33	32	mpteq2dva 5172	. . . . 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)))
34	23, 33	eqtrid 2787	. . . 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)))
35	16	a1i 11	. . . . 5 ⊢ ((𝑘 ∈ ℕ₀ ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → 𝐽 ∈ (TopOn‘ℂ))
36		oveq1 7370	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (𝑥↑𝑘) = (𝑛↑𝑘))
37	36	cbvmptv 5183	. . . . . 6 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) = (𝑛 ∈ ℂ ↦ (𝑛↑𝑘))
38		simpr 485	. . . . . 6 ⊢ ((𝑘 ∈ ℕ₀ ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽))
39	37, 38	eqeltrrid 2845	. . . . 5 ⊢ ((𝑘 ∈ ℕ₀ ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → (𝑛 ∈ ℂ ↦ (𝑛↑𝑘)) ∈ (𝐽 Cn 𝐽))
40	35	cnmptid 23651	. . . . 5 ⊢ ((𝑘 ∈ ℕ₀ ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → (𝑛 ∈ ℂ ↦ 𝑛) ∈ (𝐽 Cn 𝐽))
41	15	mpomulcn 24859	. . . . . 6 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽)
42	41	a1i 11	. . . . 5 ⊢ ((𝑘 ∈ ℕ₀ ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
43	35, 39, 40, 42	cnmpt12f 23656	. . . 4 ⊢ ((𝑘 ∈ ℕ₀ ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → (𝑛 ∈ ℂ ↦ ((𝑛↑𝑘)(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑛)) ∈ (𝐽 Cn 𝐽))
44	34, 43	eqeltrd 2840	. . 3 ⊢ ((𝑘 ∈ ℕ₀ ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) ∈ (𝐽 Cn 𝐽))
45	44	ex 413	. 2 ⊢ (𝑘 ∈ ℕ₀ → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽) → (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) ∈ (𝐽 Cn 𝐽)))
46	3, 6, 9, 12, 21, 45	nn0ind 12622	1 ⊢ (𝑁 ∈ ℕ₀ → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (𝐽 Cn 𝐽))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1547 ⊤wtru 1548 ∈ wcel 2119 ↦ cmpt 5160 ‘cfv 6492 (class class class)co 7363 ∈ cmpo 7365 ℂcc 11034 0cc0 11036 1c1 11037 + caddc 11039 · cmul 11041 ℕ₀cn0 12435 ↑cexp 14021 TopOpenctopn 17382 ℂ_fldccnfld 21354 TopOnctopon 22900 Cn ccn 23214 ×_t ctx 23550

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-pre-sup 11114

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-iin 4931 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-of 7627 df-om 7814 df-1st 7938 df-2nd 7939 df-supp 8108 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-er 8640 df-map 8772 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9272 df-fi 9321 df-sup 9352 df-inf 9353 df-oi 9422 df-card 9861 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-div 11806 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-z 12523 df-dec 12643 df-uz 12787 df-q 12897 df-rp 12941 df-xneg 13061 df-xadd 13062 df-xmul 13063 df-icc 13303 df-fz 13460 df-fzo 13607 df-seq 13962 df-exp 14022 df-hash 14291 df-cj 15059 df-re 15060 df-im 15061 df-sqrt 15195 df-abs 15196 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-ress 17199 df-plusg 17231 df-mulr 17232 df-starv 17233 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-unif 17241 df-hom 17242 df-cco 17243 df-rest 17383 df-topn 17384 df-0g 17402 df-gsum 17403 df-topgen 17404 df-pt 17405 df-prds 17408 df-xrs 17464 df-qtop 17469 df-imas 17470 df-xps 17472 df-mre 17546 df-mrc 17547 df-acs 17549 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-submnd 18750 df-mulg 19042 df-cntz 19290 df-cmn 19755 df-psmet 21346 df-xmet 21347 df-met 21348 df-bl 21349 df-mopn 21350 df-cnfld 21355 df-top 22884 df-topon 22901 df-topsp 22923 df-bases 22936 df-cn 23217 df-cnp 23218 df-tx 23552 df-hmeo 23745 df-xms 24310 df-ms 24311 df-tms 24312

This theorem is referenced by: sqcn 24866 expcncf 24918 plycn 26251 psercn2 26413 atansopn 26921 pntlem3 27597 climexp 46057

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