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Theorem cxpcn 25886

Description: Domain of continuity of the complex power function. (Contributed by Mario Carneiro, 1-May-2016.)

**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**
Step	Hyp	Ref	Expression
1		cxpcn.d	. . . . . . 7 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0))
2	1	ellogdm 25782	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ℂ ∧ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ⁺)))
3	2	simplbi 498	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ℂ)
4	3	adantr 481	. . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → 𝑥 ∈ ℂ)
5	1	logdmn0 25783	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ≠ 0)
6	5	adantr 481	. . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → 𝑥 ≠ 0)
7		simpr 485	. . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
8	4, 6, 7	cxpefd 25855	. . 3 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → (𝑥↑_𝑐𝑦) = (exp‘(𝑦 · (log‘𝑥))))
9	8	mpoeq3ia 7344	. 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 23934	. . . . . . 7 ⊢ 𝐽 ∈ (TopOn‘ℂ)
13	12	a1i 11	. . . . . 6 ⊢ (⊤ → 𝐽 ∈ (TopOn‘ℂ))
14	3	ssriv 3925	. . . . . 6 ⊢ 𝐷 ⊆ ℂ
15		resttopon 22300	. . . . . 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 2843	. . . 4 ⊢ (⊤ → 𝐾 ∈ (TopOn‘𝐷))
18	17, 13	cnmpt2nd 22808	. . . . 5 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ 𝑦) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽))
19		fvres 6786	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → ((log ↾ 𝐷)‘𝑥) = (log‘𝑥))
20	19	adantr 481	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → ((log ↾ 𝐷)‘𝑥) = (log‘𝑥))
21	20	mpoeq3ia 7344	. . . . . 6 ⊢ (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ ((log ↾ 𝐷)‘𝑥)) = (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (log‘𝑥))
22	17, 13	cnmpt1st 22807	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ 𝑥) ∈ ((𝐾 ×_t 𝐽) Cn 𝐾))
23	1	logcn 25790	. . . . . . . . 9 ⊢ (log ↾ 𝐷) ∈ (𝐷–cn→ℂ)
24		ssid 3943	. . . . . . . . . 10 ⊢ ℂ ⊆ ℂ
25	12	toponrestid 22058	. . . . . . . . . . 11 ⊢ 𝐽 = (𝐽 ↾_t ℂ)
26	11, 10, 25	cncfcn 24061	. . . . . . . . . 10 ⊢ ((𝐷 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝐷–cn→ℂ) = (𝐾 Cn 𝐽))
27	14, 24, 26	mp2an 689	. . . . . . . . 9 ⊢ (𝐷–cn→ℂ) = (𝐾 Cn 𝐽)
28	23, 27	eleqtri 2837	. . . . . . . 8 ⊢ (log ↾ 𝐷) ∈ (𝐾 Cn 𝐽)
29	28	a1i 11	. . . . . . 7 ⊢ (⊤ → (log ↾ 𝐷) ∈ (𝐾 Cn 𝐽))
30	17, 13, 22, 29	cnmpt21f 22811	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ ((log ↾ 𝐷)‘𝑥)) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽))
31	21, 30	eqeltrrid 2844	. . . . 5 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (log‘𝑥)) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽))
32	11	mulcn 24018	. . . . . 6 ⊢ · ∈ ((𝐽 ×_t 𝐽) Cn 𝐽)
33	32	a1i 11	. . . . 5 ⊢ (⊤ → · ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
34	17, 13, 18, 31, 33	cnmpt22f 22814	. . . 4 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (𝑦 · (log‘𝑥))) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽))
35		efcn 25590	. . . . . 6 ⊢ exp ∈ (ℂ–cn→ℂ)
36	11	cncfcn1 24062	. . . . . 6 ⊢ (ℂ–cn→ℂ) = (𝐽 Cn 𝐽)
37	35, 36	eleqtri 2837	. . . . 5 ⊢ exp ∈ (𝐽 Cn 𝐽)
38	37	a1i 11	. . . 4 ⊢ (⊤ → exp ∈ (𝐽 Cn 𝐽))
39	17, 13, 34, 38	cnmpt21f 22811	. . 3 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (exp‘(𝑦 · (log‘𝑥)))) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽))
40	39	mptru 1546	. 2 ⊢ (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (exp‘(𝑦 · (log‘𝑥)))) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽)
41	9, 40	eqeltri 2835	1 ⊢ (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (𝑥↑_𝑐𝑦)) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ⊤wtru 1540 ∈ wcel 2106 ≠ wne 2943 ∖ cdif 3884 ⊆ wss 3887 ↾ cres 5587 ‘cfv 6427 (class class class)co 7268 ∈ cmpo 7270 ℂcc 10857 ℝcr 10858 0cc0 10859 · cmul 10864 -∞cmnf 10995 ℝ⁺crp 12718 (,]cioc 13068 expce 15759 ↾_t crest 17119 TopOpenctopn 17120 ℂ_fldccnfld 20585 TopOnctopon 22047 Cn ccn 22363 ×_t ctx 22699 –cn→ccncf 24027 logclog 25698 ↑_𝑐ccxp 25699

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 ax-inf2 9387 ax-cnex 10915 ax-resscn 10916 ax-1cn 10917 ax-icn 10918 ax-addcl 10919 ax-addrcl 10920 ax-mulcl 10921 ax-mulrcl 10922 ax-mulcom 10923 ax-addass 10924 ax-mulass 10925 ax-distr 10926 ax-i2m1 10927 ax-1ne0 10928 ax-1rid 10929 ax-rnegex 10930 ax-rrecex 10931 ax-cnre 10932 ax-pre-lttri 10933 ax-pre-lttrn 10934 ax-pre-ltadd 10935 ax-pre-mulgt0 10936 ax-pre-sup 10937 ax-addf 10938 ax-mulf 10939

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4841 df-int 4881 df-iun 4927 df-iin 4928 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-se 5541 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6263 df-on 6264 df-lim 6265 df-suc 6266 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-isom 6436 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-of 7524 df-om 7704 df-1st 7821 df-2nd 7822 df-supp 7966 df-frecs 8085 df-wrecs 8116 df-recs 8190 df-rdg 8229 df-1o 8285 df-2o 8286 df-er 8486 df-map 8605 df-pm 8606 df-ixp 8674 df-en 8722 df-dom 8723 df-sdom 8724 df-fin 8725 df-fsupp 9117 df-fi 9158 df-sup 9189 df-inf 9190 df-oi 9257 df-card 9685 df-pnf 10999 df-mnf 11000 df-xr 11001 df-ltxr 11002 df-le 11003 df-sub 11195 df-neg 11196 df-div 11621 df-nn 11962 df-2 12024 df-3 12025 df-4 12026 df-5 12027 df-6 12028 df-7 12029 df-8 12030 df-9 12031 df-n0 12222 df-z 12308 df-dec 12426 df-uz 12571 df-q 12677 df-rp 12719 df-xneg 12836 df-xadd 12837 df-xmul 12838 df-ioo 13071 df-ioc 13072 df-ico 13073 df-icc 13074 df-fz 13228 df-fzo 13371 df-fl 13500 df-mod 13578 df-seq 13710 df-exp 13771 df-fac 13976 df-bc 14005 df-hash 14033 df-shft 14766 df-cj 14798 df-re 14799 df-im 14800 df-sqrt 14934 df-abs 14935 df-limsup 15168 df-clim 15185 df-rlim 15186 df-sum 15386 df-ef 15765 df-sin 15767 df-cos 15768 df-tan 15769 df-pi 15770 df-struct 16836 df-sets 16853 df-slot 16871 df-ndx 16883 df-base 16901 df-ress 16930 df-plusg 16963 df-mulr 16964 df-starv 16965 df-sca 16966 df-vsca 16967 df-ip 16968 df-tset 16969 df-ple 16970 df-ds 16972 df-unif 16973 df-hom 16974 df-cco 16975 df-rest 17121 df-topn 17122 df-0g 17140 df-gsum 17141 df-topgen 17142 df-pt 17143 df-prds 17146 df-xrs 17201 df-qtop 17206 df-imas 17207 df-xps 17209 df-mre 17283 df-mrc 17284 df-acs 17286 df-mgm 18314 df-sgrp 18363 df-mnd 18374 df-submnd 18419 df-mulg 18689 df-cntz 18911 df-cmn 19376 df-psmet 20577 df-xmet 20578 df-met 20579 df-bl 20580 df-mopn 20581 df-fbas 20582 df-fg 20583 df-cnfld 20586 df-top 22031 df-topon 22048 df-topsp 22070 df-bases 22084 df-cld 22158 df-ntr 22159 df-cls 22160 df-nei 22237 df-lp 22275 df-perf 22276 df-cn 22366 df-cnp 22367 df-haus 22454 df-cmp 22526 df-tx 22701 df-hmeo 22894 df-fil 22985 df-fm 23077 df-flim 23078 df-flf 23079 df-xms 23461 df-ms 23462 df-tms 23463 df-cncf 24029 df-limc 25018 df-dv 25019 df-log 25700 df-cxp 25701

This theorem is referenced by: cxpcn2 25887 sqrtcn 25891 cxpcncf1 32561 cxpcncf2 43399

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