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Theorem cxpcn 26670

Description: Domain of continuity of the complex power function. (Contributed by Mario Carneiro, 1-May-2016.) Avoid ax-mulf 11108. (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 26564	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ℂ ∧ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ⁺)))
3	2	simplbi 497	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ℂ)
4	3	adantr 480	. . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → 𝑥 ∈ ℂ)
5	1	logdmn0 26565	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ≠ 0)
6	5	adantr 480	. . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → 𝑥 ≠ 0)
7		simpr 484	. . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
8	4, 6, 7	cxpefd 26637	. . 3 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → (𝑥↑_𝑐𝑦) = (exp‘(𝑦 · (log‘𝑥))))
9	8	mpoeq3ia 7431	. 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 24686	. . . . . . 7 ⊢ 𝐽 ∈ (TopOn‘ℂ)
13	12	a1i 11	. . . . . 6 ⊢ (⊤ → 𝐽 ∈ (TopOn‘ℂ))
14	3	ssriv 3941	. . . . . 6 ⊢ 𝐷 ⊆ ℂ
15		resttopon 23064	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝐷 ⊆ ℂ) → (𝐽 ↾_t 𝐷) ∈ (TopOn‘𝐷))
16	13, 14, 15	sylancl 586	. . . . 5 ⊢ (⊤ → (𝐽 ↾_t 𝐷) ∈ (TopOn‘𝐷))
17	10, 16	eqeltrid 2832	. . . 4 ⊢ (⊤ → 𝐾 ∈ (TopOn‘𝐷))
18	17, 13	cnmpt2nd 23572	. . . . 5 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ 𝑦) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽))
19		fvres 6845	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → ((log ↾ 𝐷)‘𝑥) = (log‘𝑥))
20	19	adantr 480	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → ((log ↾ 𝐷)‘𝑥) = (log‘𝑥))
21	20	mpoeq3ia 7431	. . . . . 6 ⊢ (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ ((log ↾ 𝐷)‘𝑥)) = (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (log‘𝑥))
22	17, 13	cnmpt1st 23571	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ 𝑥) ∈ ((𝐾 ×_t 𝐽) Cn 𝐾))
23	1	logcn 26572	. . . . . . . . 9 ⊢ (log ↾ 𝐷) ∈ (𝐷–cn→ℂ)
24		ssid 3960	. . . . . . . . . 10 ⊢ ℂ ⊆ ℂ
25	12	toponrestid 22824	. . . . . . . . . . 11 ⊢ 𝐽 = (𝐽 ↾_t ℂ)
26	11, 10, 25	cncfcn 24819	. . . . . . . . . 10 ⊢ ((𝐷 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝐷–cn→ℂ) = (𝐾 Cn 𝐽))
27	14, 24, 26	mp2an 692	. . . . . . . . 9 ⊢ (𝐷–cn→ℂ) = (𝐾 Cn 𝐽)
28	23, 27	eleqtri 2826	. . . . . . . 8 ⊢ (log ↾ 𝐷) ∈ (𝐾 Cn 𝐽)
29	28	a1i 11	. . . . . . 7 ⊢ (⊤ → (log ↾ 𝐷) ∈ (𝐾 Cn 𝐽))
30	17, 13, 22, 29	cnmpt21f 23575	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ ((log ↾ 𝐷)‘𝑥)) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽))
31	21, 30	eqeltrrid 2833	. . . . 5 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (log‘𝑥)) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽))
32	11	mpomulcn 24774	. . . . . 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 7362	. . . . 5 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = (log‘𝑥)) → (𝑢 · 𝑣) = (𝑦 · (log‘𝑥)))
35	17, 13, 18, 31, 13, 13, 33, 34	cnmpt22 23577	. . . 4 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (𝑦 · (log‘𝑥))) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽))
36		efcn 26369	. . . . . 6 ⊢ exp ∈ (ℂ–cn→ℂ)
37	11	cncfcn1 24820	. . . . . 6 ⊢ (ℂ–cn→ℂ) = (𝐽 Cn 𝐽)
38	36, 37	eleqtri 2826	. . . . 5 ⊢ exp ∈ (𝐽 Cn 𝐽)
39	38	a1i 11	. . . 4 ⊢ (⊤ → exp ∈ (𝐽 Cn 𝐽))
40	17, 13, 35, 39	cnmpt21f 23575	. . 3 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (exp‘(𝑦 · (log‘𝑥)))) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽))
41	40	mptru 1547	. 2 ⊢ (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (exp‘(𝑦 · (log‘𝑥)))) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽)
42	9, 41	eqeltri 2824	1 ⊢ (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (𝑥↑_𝑐𝑦)) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ⊤wtru 1541 ∈ wcel 2109 ≠ wne 2925 ∖ cdif 3902 ⊆ wss 3905 ↾ cres 5625 ‘cfv 6486 (class class class)co 7353 ∈ cmpo 7355 ℂcc 11026 ℝcr 11027 0cc0 11028 · cmul 11033 -∞cmnf 11166 ℝ⁺crp 12911 (,]cioc 13267 expce 15986 ↾_t crest 17342 TopOpenctopn 17343 ℂ_fldccnfld 21279 TopOnctopon 22813 Cn ccn 23127 ×_t ctx 23463 –cn→ccncf 24785 logclog 26479 ↑_𝑐ccxp 26480

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-inf2 9556 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7617 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8632 df-map 8762 df-pm 8763 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9271 df-fi 9320 df-sup 9351 df-inf 9352 df-oi 9421 df-card 9854 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12610 df-uz 12754 df-q 12868 df-rp 12912 df-xneg 13032 df-xadd 13033 df-xmul 13034 df-ioo 13270 df-ioc 13271 df-ico 13272 df-icc 13273 df-fz 13429 df-fzo 13576 df-fl 13714 df-mod 13792 df-seq 13927 df-exp 13987 df-fac 14199 df-bc 14228 df-hash 14256 df-shft 14992 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-limsup 15396 df-clim 15413 df-rlim 15414 df-sum 15612 df-ef 15992 df-sin 15994 df-cos 15995 df-tan 15996 df-pi 15997 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-hom 17203 df-cco 17204 df-rest 17344 df-topn 17345 df-0g 17363 df-gsum 17364 df-topgen 17365 df-pt 17366 df-prds 17369 df-xrs 17424 df-qtop 17429 df-imas 17430 df-xps 17432 df-mre 17506 df-mrc 17507 df-acs 17509 df-mgm 18532 df-sgrp 18611 df-mnd 18627 df-submnd 18676 df-mulg 18965 df-cntz 19214 df-cmn 19679 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-fbas 21276 df-fg 21277 df-cnfld 21280 df-top 22797 df-topon 22814 df-topsp 22836 df-bases 22849 df-cld 22922 df-ntr 22923 df-cls 22924 df-nei 23001 df-lp 23039 df-perf 23040 df-cn 23130 df-cnp 23131 df-haus 23218 df-cmp 23290 df-tx 23465 df-hmeo 23658 df-fil 23749 df-fm 23841 df-flim 23842 df-flf 23843 df-xms 24224 df-ms 24225 df-tms 24226 df-cncf 24787 df-limc 25783 df-dv 25784 df-log 26481 df-cxp 26482

This theorem is referenced by: cxpcn2 26672 sqrtcn 26676 cxpcncf1 34562 cxpcncf2 45881

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