cxpcn - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > cxpcn		Structured version Visualization version GIF version

Theorem cxpcn 25021

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 24917	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ℂ ∧ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ⁺)))
3	2	simplbi 490	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ℂ)
4	3	adantr 473	. . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → 𝑥 ∈ ℂ)
5	1	logdmn0 24918	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ≠ 0)
6	5	adantr 473	. . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → 𝑥 ≠ 0)
7		simpr 477	. . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
8	4, 6, 7	cxpefd 24990	. . 3 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → (𝑥↑_𝑐𝑦) = (exp‘(𝑦 · (log‘𝑥))))
9	8	mpoeq3ia 7044	. 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 23088	. . . . . . 7 ⊢ 𝐽 ∈ (TopOn‘ℂ)
13	12	a1i 11	. . . . . 6 ⊢ (⊤ → 𝐽 ∈ (TopOn‘ℂ))
14	3	ssriv 3856	. . . . . 6 ⊢ 𝐷 ⊆ ℂ
15		resttopon 21467	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝐷 ⊆ ℂ) → (𝐽 ↾_t 𝐷) ∈ (TopOn‘𝐷))
16	13, 14, 15	sylancl 577	. . . . 5 ⊢ (⊤ → (𝐽 ↾_t 𝐷) ∈ (TopOn‘𝐷))
17	10, 16	syl5eqel 2864	. . . 4 ⊢ (⊤ → 𝐾 ∈ (TopOn‘𝐷))
18	17, 13	cnmpt2nd 21975	. . . . 5 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ 𝑦) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽))
19		fvres 6512	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → ((log ↾ 𝐷)‘𝑥) = (log‘𝑥))
20	19	adantr 473	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → ((log ↾ 𝐷)‘𝑥) = (log‘𝑥))
21	20	mpoeq3ia 7044	. . . . . 6 ⊢ (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ ((log ↾ 𝐷)‘𝑥)) = (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (log‘𝑥))
22	17, 13	cnmpt1st 21974	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ 𝑥) ∈ ((𝐾 ×_t 𝐽) Cn 𝐾))
23	1	logcn 24925	. . . . . . . . 9 ⊢ (log ↾ 𝐷) ∈ (𝐷–cn→ℂ)
24		ssid 3873	. . . . . . . . . 10 ⊢ ℂ ⊆ ℂ
25	12	toponrestid 21227	. . . . . . . . . . 11 ⊢ 𝐽 = (𝐽 ↾_t ℂ)
26	11, 10, 25	cncfcn 23214	. . . . . . . . . 10 ⊢ ((𝐷 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝐷–cn→ℂ) = (𝐾 Cn 𝐽))
27	14, 24, 26	mp2an 679	. . . . . . . . 9 ⊢ (𝐷–cn→ℂ) = (𝐾 Cn 𝐽)
28	23, 27	eleqtri 2858	. . . . . . . 8 ⊢ (log ↾ 𝐷) ∈ (𝐾 Cn 𝐽)
29	28	a1i 11	. . . . . . 7 ⊢ (⊤ → (log ↾ 𝐷) ∈ (𝐾 Cn 𝐽))
30	17, 13, 22, 29	cnmpt21f 21978	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ ((log ↾ 𝐷)‘𝑥)) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽))
31	21, 30	syl5eqelr 2865	. . . . 5 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (log‘𝑥)) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽))
32	11	mulcn 23172	. . . . . 6 ⊢ · ∈ ((𝐽 ×_t 𝐽) Cn 𝐽)
33	32	a1i 11	. . . . 5 ⊢ (⊤ → · ∈ ((𝐽 ×_t 𝐽) Cn 𝐽))
34	17, 13, 18, 31, 33	cnmpt22f 21981	. . . 4 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (𝑦 · (log‘𝑥))) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽))
35		efcn 24728	. . . . . 6 ⊢ exp ∈ (ℂ–cn→ℂ)
36	11	cncfcn1 23215	. . . . . 6 ⊢ (ℂ–cn→ℂ) = (𝐽 Cn 𝐽)
37	35, 36	eleqtri 2858	. . . . 5 ⊢ exp ∈ (𝐽 Cn 𝐽)
38	37	a1i 11	. . . 4 ⊢ (⊤ → exp ∈ (𝐽 Cn 𝐽))
39	17, 13, 34, 38	cnmpt21f 21978	. . 3 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (exp‘(𝑦 · (log‘𝑥)))) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽))
40	39	mptru 1514	. 2 ⊢ (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (exp‘(𝑦 · (log‘𝑥)))) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽)
41	9, 40	eqeltri 2856	1 ⊢ (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (𝑥↑_𝑐𝑦)) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ⊤wtru 1508 ∈ wcel 2050 ≠ wne 2961 ∖ cdif 3820 ⊆ wss 3823 ↾ cres 5403 ‘cfv 6182 (class class class)co 6970 ∈ cmpo 6972 ℂcc 10327 ℝcr 10328 0cc0 10329 · cmul 10334 -∞cmnf 10466 ℝ⁺crp 12198 (,]cioc 12549 expce 15269 ↾_t crest 16544 TopOpenctopn 16545 ℂ_fldccnfld 20241 TopOnctopon 21216 Cn ccn 21530 ×_t ctx 21866 –cn→ccncf 23181 logclog 24833 ↑_𝑐ccxp 24834

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-inf2 8892 ax-cnex 10385 ax-resscn 10386 ax-1cn 10387 ax-icn 10388 ax-addcl 10389 ax-addrcl 10390 ax-mulcl 10391 ax-mulrcl 10392 ax-mulcom 10393 ax-addass 10394 ax-mulass 10395 ax-distr 10396 ax-i2m1 10397 ax-1ne0 10398 ax-1rid 10399 ax-rnegex 10400 ax-rrecex 10401 ax-cnre 10402 ax-pre-lttri 10403 ax-pre-lttrn 10404 ax-pre-ltadd 10405 ax-pre-mulgt0 10406 ax-pre-sup 10407 ax-addf 10408 ax-mulf 10409

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-iin 4789 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5306 df-eprel 5311 df-po 5320 df-so 5321 df-fr 5360 df-se 5361 df-we 5362 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-isom 6191 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-of 7221 df-om 7391 df-1st 7495 df-2nd 7496 df-supp 7628 df-wrecs 7744 df-recs 7806 df-rdg 7844 df-1o 7899 df-2o 7900 df-oadd 7903 df-er 8083 df-map 8202 df-pm 8203 df-ixp 8254 df-en 8301 df-dom 8302 df-sdom 8303 df-fin 8304 df-fsupp 8623 df-fi 8664 df-sup 8695 df-inf 8696 df-oi 8763 df-card 9156 df-cda 9382 df-pnf 10470 df-mnf 10471 df-xr 10472 df-ltxr 10473 df-le 10474 df-sub 10666 df-neg 10667 df-div 11093 df-nn 11434 df-2 11497 df-3 11498 df-4 11499 df-5 11500 df-6 11501 df-7 11502 df-8 11503 df-9 11504 df-n0 11702 df-z 11788 df-dec 11906 df-uz 12053 df-q 12157 df-rp 12199 df-xneg 12318 df-xadd 12319 df-xmul 12320 df-ioo 12552 df-ioc 12553 df-ico 12554 df-icc 12555 df-fz 12703 df-fzo 12844 df-fl 12971 df-mod 13047 df-seq 13179 df-exp 13239 df-fac 13443 df-bc 13472 df-hash 13500 df-shft 14281 df-cj 14313 df-re 14314 df-im 14315 df-sqrt 14449 df-abs 14450 df-limsup 14683 df-clim 14700 df-rlim 14701 df-sum 14898 df-ef 15275 df-sin 15277 df-cos 15278 df-tan 15279 df-pi 15280 df-struct 16335 df-ndx 16336 df-slot 16337 df-base 16339 df-sets 16340 df-ress 16341 df-plusg 16428 df-mulr 16429 df-starv 16430 df-sca 16431 df-vsca 16432 df-ip 16433 df-tset 16434 df-ple 16435 df-ds 16437 df-unif 16438 df-hom 16439 df-cco 16440 df-rest 16546 df-topn 16547 df-0g 16565 df-gsum 16566 df-topgen 16567 df-pt 16568 df-prds 16571 df-xrs 16625 df-qtop 16630 df-imas 16631 df-xps 16633 df-mre 16709 df-mrc 16710 df-acs 16712 df-mgm 17704 df-sgrp 17746 df-mnd 17757 df-submnd 17798 df-mulg 18006 df-cntz 18212 df-cmn 18662 df-psmet 20233 df-xmet 20234 df-met 20235 df-bl 20236 df-mopn 20237 df-fbas 20238 df-fg 20239 df-cnfld 20242 df-top 21200 df-topon 21217 df-topsp 21239 df-bases 21252 df-cld 21325 df-ntr 21326 df-cls 21327 df-nei 21404 df-lp 21442 df-perf 21443 df-cn 21533 df-cnp 21534 df-haus 21621 df-cmp 21693 df-tx 21868 df-hmeo 22061 df-fil 22152 df-fm 22244 df-flim 22245 df-flf 22246 df-xms 22627 df-ms 22628 df-tms 22629 df-cncf 23183 df-limc 24161 df-dv 24162 df-log 24835 df-cxp 24836

This theorem is referenced by: cxpcn2 25022 sqrtcn 25026 cxpcncf1 31514 cxpcncf2 41613

W3C validator