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Theorem cxpcn 27055

Description: Domain of continuity of the complex power function. (Contributed by Mario Carneiro, 1-May-2016.) Avoid ax-mulf 11261. (Revised by GG, 16-Mar-2025.)

**Hypotheses**
Ref	Expression
cxpcn.d	⊢ 𝐷 = (ℂ ∖ (-∞(,]0))
cxpcn.j	⊢ 𝐽 = (TopOpen‘ℂ_fld)
cxpcn.k	⊢ 𝐾 = (𝐽 ↾_t 𝐷)

**Assertion**
Ref	Expression
cxpcn	⊢ (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (𝑥↑_𝑐𝑦)) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽)

Distinct variable groups: 𝑥,𝑦,𝐷 𝑥,𝐽,𝑦 𝑥,𝐾,𝑦

Proof of Theorem cxpcn Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cxpcn.d	. . . . . . 7 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0))
2	1	ellogdm 26949	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ℂ ∧ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ⁺)))
3	2	simplbi 502	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ℂ)
4	3	adantr 486	. . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → 𝑥 ∈ ℂ)
5	1	logdmn0 26950	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ≠ 0)
6	5	adantr 486	. . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → 𝑥 ≠ 0)
7		simpr 490	. . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
8	4, 6, 7	cxpefd 27022	. . 3 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → (𝑥↑_𝑐𝑦) = (exp‘(𝑦 · (log‘𝑥))))
9	8	mpoeq3ia 7490	. 2 ⊢ (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (𝑥↑_𝑐𝑦)) = (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (exp‘(𝑦 · (log‘𝑥))))
10		cxpcn.k	. . . . 5 ⊢ 𝐾 = (𝐽 ↾_t 𝐷)
11		cxpcn.j	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
12	11	cnfldtopon 25081	. . . . . . 7 ⊢ 𝐽 ∈ (TopOn‘ℂ)
13	12	a1i 11	. . . . . 6 ⊢ (⊤ → 𝐽 ∈ (TopOn‘ℂ))
14	3	ssriv 3935	. . . . . 6 ⊢ 𝐷 ⊆ ℂ
15		resttopon 23459	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝐷 ⊆ ℂ) → (𝐽 ↾_t 𝐷) ∈ (TopOn‘𝐷))
16	13, 14, 15	sylancl 598	. . . . 5 ⊢ (⊤ → (𝐽 ↾_t 𝐷) ∈ (TopOn‘𝐷))
17	10, 16	eqeltrid 2865	. . . 4 ⊢ (⊤ → 𝐾 ∈ (TopOn‘𝐷))
18	17, 13	cnmpt2nd 23968	. . . . 5 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ 𝑦) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽))
19		fvres 6896	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → ((log ↾ 𝐷)‘𝑥) = (log‘𝑥))
20	19	adantr 486	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → ((log ↾ 𝐷)‘𝑥) = (log‘𝑥))
21	20	mpoeq3ia 7490	. . . . . 6 ⊢ (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ ((log ↾ 𝐷)‘𝑥)) = (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (log‘𝑥))
22	17, 13	cnmpt1st 23967	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ 𝑥) ∈ ((𝐾 ×_t 𝐽) Cn 𝐾))
23	1	logcn 26957	. . . . . . . . 9 ⊢ (log ↾ 𝐷) ∈ (𝐷–cn→ℂ)
24		ssid 3953	. . . . . . . . . 10 ⊢ ℂ ⊆ ℂ
25	12	toponrestid 23219	. . . . . . . . . . 11 ⊢ 𝐽 = (𝐽 ↾_t ℂ)
26	11, 10, 25	cncfcn 25211	. . . . . . . . . 10 ⊢ ((𝐷 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝐷–cn→ℂ) = (𝐾 Cn 𝐽))
27	14, 24, 26	mp2an 705	. . . . . . . . 9 ⊢ (𝐷–cn→ℂ) = (𝐾 Cn 𝐽)
28	23, 27	eleqtri 2859	. . . . . . . 8 ⊢ (log ↾ 𝐷) ∈ (𝐾 Cn 𝐽)
29	28	a1i 11	. . . . . . 7 ⊢ (⊤ → (log ↾ 𝐷) ∈ (𝐾 Cn 𝐽))
30	17, 13, 22, 29	cnmpt21f 23971	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ ((log ↾ 𝐷)‘𝑥)) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽))
31	21, 30	eqeltrrid 2866	. . . . 5 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (log‘𝑥)) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽))
32	11	mpomulcn 25168	. . . . . 6 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽)
33	32	a1i 11	. . . . 5 ⊢ (⊤ → (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
34		oveq12 7421	. . . . 5 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = (log‘𝑥)) → (𝑢 · 𝑣) = (𝑦 · (log‘𝑥)))
35	17, 13, 18, 31, 13, 13, 33, 34	cnmpt22 23973	. . . 4 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (𝑦 · (log‘𝑥))) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽))
36		efcn 26752	. . . . . 6 ⊢ exp ∈ (ℂ–cn→ℂ)
37	11	cncfcn1 25212	. . . . . 6 ⊢ (ℂ–cn→ℂ) = (𝐽 Cn 𝐽)
38	36, 37	eleqtri 2859	. . . . 5 ⊢ exp ∈ (𝐽 Cn 𝐽)
39	38	a1i 11	. . . 4 ⊢ (⊤ → exp ∈ (𝐽 Cn 𝐽))
40	17, 13, 35, 39	cnmpt21f 23971	. . 3 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (exp‘(𝑦 · (log‘𝑥)))) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽))
41	40	mptru 1577	. 2 ⊢ (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (exp‘(𝑦 · (log‘𝑥)))) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽)
42	9, 41	eqeltri 2857	1 ⊢ (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (𝑥↑_𝑐𝑦)) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ⊤wtru 1571 ∈ wcel 2145 ≠ wne 2956 ∖ cdif 3896 ⊆ wss 3899 ↾ cres 5653 ‘cfv 6531 (class class class)co 7412 ∈ cmpo 7414 ℂcc 11179 ℝcr 11180 0cc0 11181 · cmul 11186 -∞cmnf 11322 ℝ⁺crp 13101 (,]cioc 13458 expce 16207 ↾_t crest 17571 TopOpenctopn 17572 ℂ_fldccnfld 21658 TopOnctopon 23208 Cn ccn 23522 ×_t ctx 23859 –cn→ccncf 25177 logclog 26864 ↑_𝑐ccxp 26865

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-inf2 9626 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258 ax-pre-sup 11259 ax-addf 11260

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7682 df-om 7867 df-1st 7990 df-2nd 7991 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-2o 8461 df-er 8701 df-map 8833 df-pm 8834 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9338 df-fi 9387 df-sup 9418 df-inf 9419 df-oi 9488 df-card 10001 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-div 11955 df-nn 12317 df-2 12386 df-3 12387 df-4 12388 df-5 12389 df-6 12390 df-7 12391 df-8 12392 df-9 12393 df-n0 12588 df-z 12675 df-dec 12796 df-uz 12947 df-q 13057 df-rp 13102 df-xneg 13222 df-xadd 13223 df-xmul 13224 df-ioo 13461 df-ioc 13462 df-ico 13463 df-icc 13464 df-fz 13621 df-fzo 13769 df-fl 13912 df-mod 13990 df-seq 14125 df-exp 14185 df-fac 14398 df-bc 14427 df-hash 14455 df-shft 15200 df-cj 15246 df-re 15247 df-im 15248 df-sqrt 15382 df-abs 15383 df-limsup 15618 df-clim 15635 df-rlim 15636 df-sum 15834 df-ef 16213 df-sin 16215 df-cos 16216 df-tan 16217 df-pi 16218 df-struct 17305 df-sets 17322 df-slot 17340 df-ndx 17352 df-base 17368 df-ress 17389 df-plusg 17421 df-mulr 17422 df-starv 17423 df-sca 17424 df-vsca 17425 df-ip 17426 df-tset 17427 df-ple 17428 df-ds 17430 df-unif 17431 df-hom 17432 df-cco 17433 df-rest 17573 df-topn 17574 df-0g 17592 df-gsum 17593 df-topgen 17594 df-pt 17595 df-prds 17598 df-xrs 17654 df-qtop 17659 df-imas 17660 df-xps 17662 df-mre 17736 df-mrc 17737 df-acs 17739 df-mgm 18796 df-sgrp 18888 df-mnd 18904 df-submnd 18959 df-mulg 19258 df-cntz 19511 df-cmn 19976 df-psmet 21650 df-xmet 21651 df-met 21652 df-bl 21653 df-mopn 21654 df-fbas 21655 df-fg 21656 df-cnfld 21659 df-top 23192 df-topon 23209 df-topsp 23231 df-bases 23244 df-cld 23317 df-ntr 23318 df-cls 23319 df-nei 23396 df-lp 23434 df-perf 23435 df-cn 23525 df-cnp 23526 df-haus 23613 df-cmp 23685 df-tx 23861 df-hmeo 24054 df-fil 24145 df-fm 24237 df-flim 24238 df-flf 24239 df-xms 24619 df-ms 24620 df-tms 24621 df-cncf 25179 df-limc 26166 df-dv 26167 df-log 26866 df-cxp 26867

This theorem is used by: cxpcn2 27056 sqrtcn 27060 cxpcncf1 35207 cxpcncf2 46853

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