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Theorem expcn 24808

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 11224. (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 7432	. . . 4 ⊢ (𝑛 = 0 → (𝑥↑𝑛) = (𝑥↑0))
2	1	mpteq2dv 5252	. . 3 ⊢ (𝑛 = 0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑0)))
3	2	eleq1d 2813	. 2 ⊢ (𝑛 = 0 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) ∈ (𝐽 Cn 𝐽) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑0)) ∈ (𝐽 Cn 𝐽)))
4		oveq2 7432	. . . 4 ⊢ (𝑛 = 𝑘 → (𝑥↑𝑛) = (𝑥↑𝑘))
5	4	mpteq2dv 5252	. . 3 ⊢ (𝑛 = 𝑘 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)))
6	5	eleq1d 2813	. 2 ⊢ (𝑛 = 𝑘 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) ∈ (𝐽 Cn 𝐽) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)))
7		oveq2 7432	. . . 4 ⊢ (𝑛 = (𝑘 + 1) → (𝑥↑𝑛) = (𝑥↑(𝑘 + 1)))
8	7	mpteq2dv 5252	. . 3 ⊢ (𝑛 = (𝑘 + 1) → (𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))))
9	8	eleq1d 2813	. 2 ⊢ (𝑛 = (𝑘 + 1) → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) ∈ (𝐽 Cn 𝐽) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) ∈ (𝐽 Cn 𝐽)))
10		oveq2 7432	. . . 4 ⊢ (𝑛 = 𝑁 → (𝑥↑𝑛) = (𝑥↑𝑁))
11	10	mpteq2dv 5252	. . 3 ⊢ (𝑛 = 𝑁 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)))
12	11	eleq1d 2813	. 2 ⊢ (𝑛 = 𝑁 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) ∈ (𝐽 Cn 𝐽) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (𝐽 Cn 𝐽)))
13		exp0 14068	. . . 4 ⊢ (𝑥 ∈ ℂ → (𝑥↑0) = 1)
14	13	mpteq2ia 5253	. . 3 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑0)) = (𝑥 ∈ ℂ ↦ 1)
15		expcn.j	. . . . . . 7 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
16	15	cnfldtopon 24717	. . . . . 6 ⊢ 𝐽 ∈ (TopOn‘ℂ)
17	16	a1i 11	. . . . 5 ⊢ (⊤ → 𝐽 ∈ (TopOn‘ℂ))
18		1cnd 11245	. . . . 5 ⊢ (⊤ → 1 ∈ ℂ)
19	17, 17, 18	cnmptc 23584	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ 1) ∈ (𝐽 Cn 𝐽))
20	19	mptru 1540	. . 3 ⊢ (𝑥 ∈ ℂ ↦ 1) ∈ (𝐽 Cn 𝐽)
21	14, 20	eqeltri 2824	. 2 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑0)) ∈ (𝐽 Cn 𝐽)
22		oveq1 7431	. . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑥↑(𝑘 + 1)) = (𝑛↑(𝑘 + 1)))
23	22	cbvmptv 5263	. . . . 5 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) = (𝑛 ∈ ℂ ↦ (𝑛↑(𝑘 + 1)))
24		id 22	. . . . . . 7 ⊢ (𝑛 ∈ ℂ → 𝑛 ∈ ℂ)
25		simpl 481	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ₀ ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → 𝑘 ∈ ℕ₀)
26		expp1 14071	. . . . . . . 8 ⊢ ((𝑛 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝑛↑(𝑘 + 1)) = ((𝑛↑𝑘) · 𝑛))
27		expcl 14082	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝑛↑𝑘) ∈ ℂ)
28		simpl 481	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → 𝑛 ∈ ℂ)
29		ovmpot 7586	. . . . . . . . 9 ⊢ (((𝑛↑𝑘) ∈ ℂ ∧ 𝑛 ∈ ℂ) → ((𝑛↑𝑘)(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑛) = ((𝑛↑𝑘) · 𝑛))
30	27, 28, 29	syl2anc 582	. . . . . . . 8 ⊢ ((𝑛 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → ((𝑛↑𝑘)(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑛) = ((𝑛↑𝑘) · 𝑛))
31	26, 30	eqtr4d 2770	. . . . . . 7 ⊢ ((𝑛 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝑛↑(𝑘 + 1)) = ((𝑛↑𝑘)(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))𝑛))
32	24, 25, 31	syl2anr 595	. . . . . 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 5250	. . . . 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 2779	. . . 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 7431	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (𝑥↑𝑘) = (𝑛↑𝑘))
37	36	cbvmptv 5263	. . . . . 6 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) = (𝑛 ∈ ℂ ↦ (𝑛↑𝑘))
38		simpr 483	. . . . . 6 ⊢ ((𝑘 ∈ ℕ₀ ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽))
39	37, 38	eqeltrrid 2833	. . . . 5 ⊢ ((𝑘 ∈ ℕ₀ ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → (𝑛 ∈ ℂ ↦ (𝑛↑𝑘)) ∈ (𝐽 Cn 𝐽))
40	35	cnmptid 23583	. . . . 5 ⊢ ((𝑘 ∈ ℕ₀ ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → (𝑛 ∈ ℂ ↦ 𝑛) ∈ (𝐽 Cn 𝐽))
41	15	mpomulcn 24803	. . . . . 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 23588	. . . 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 2828	. . 3 ⊢ ((𝑘 ∈ ℕ₀ ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) ∈ (𝐽 Cn 𝐽))
45	44	ex 411	. 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 12693	1 ⊢ (𝑁 ∈ ℕ₀ → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (𝐽 Cn 𝐽))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 ↦ cmpt 5233 ‘cfv 6551 (class class class)co 7424 ∈ cmpo 7426 ℂcc 11142 0cc0 11144 1c1 11145 + caddc 11147 · cmul 11149 ℕ₀cn0 12508 ↑cexp 14064 TopOpenctopn 17408 ℂ_fldccnfld 21284 TopOnctopon 22830 Cn ccn 23146 ×_t ctx 23482

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-pre-sup 11222

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-iin 5001 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-se 5636 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-isom 6560 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7689 df-om 7875 df-1st 7997 df-2nd 7998 df-supp 8170 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-2o 8492 df-er 8729 df-map 8851 df-ixp 8921 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-fsupp 9392 df-fi 9440 df-sup 9471 df-inf 9472 df-oi 9539 df-card 9968 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-div 11908 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-n0 12509 df-z 12595 df-dec 12714 df-uz 12859 df-q 12969 df-rp 13013 df-xneg 13130 df-xadd 13131 df-xmul 13132 df-icc 13369 df-fz 13523 df-fzo 13666 df-seq 14005 df-exp 14065 df-hash 14328 df-cj 15084 df-re 15085 df-im 15086 df-sqrt 15220 df-abs 15221 df-struct 17121 df-sets 17138 df-slot 17156 df-ndx 17168 df-base 17186 df-ress 17215 df-plusg 17251 df-mulr 17252 df-starv 17253 df-sca 17254 df-vsca 17255 df-ip 17256 df-tset 17257 df-ple 17258 df-ds 17260 df-unif 17261 df-hom 17262 df-cco 17263 df-rest 17409 df-topn 17410 df-0g 17428 df-gsum 17429 df-topgen 17430 df-pt 17431 df-prds 17434 df-xrs 17489 df-qtop 17494 df-imas 17495 df-xps 17497 df-mre 17571 df-mrc 17572 df-acs 17574 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-submnd 18746 df-mulg 19029 df-cntz 19273 df-cmn 19742 df-psmet 21276 df-xmet 21277 df-met 21278 df-bl 21279 df-mopn 21280 df-cnfld 21285 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22867 df-cn 23149 df-cnp 23150 df-tx 23484 df-hmeo 23677 df-xms 24244 df-ms 24245 df-tms 24246

This theorem is referenced by: sqcn 24812 expcncf 24865 plycn 26213 plycnOLD 26214 psercn2 26377 psercn2OLD 26378 atansopn 26882 pntlem3 27560 climexp 44995

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