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Mirrors > Home > MPE Home > Th. List > cxpcn | Structured version Visualization version GIF version |
Description: Domain of continuity of the complex power function. (Contributed by Mario Carneiro, 1-May-2016.) |
Ref | Expression |
---|---|
cxpcn.d | ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) |
cxpcn.j | ⊢ 𝐽 = (TopOpen‘ℂfld) |
cxpcn.k | ⊢ 𝐾 = (𝐽 ↾t 𝐷) |
Ref | Expression |
---|---|
cxpcn | ⊢ (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (𝑥↑𝑐𝑦)) ∈ ((𝐾 ×t 𝐽) Cn 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cxpcn.d | . . . . . . 7 ⊢ 𝐷 = (ℂ ∖ (-∞(,]0)) | |
2 | 1 | ellogdm 25527 | . . . . . 6 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ℂ ∧ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ+))) |
3 | 2 | simplbi 501 | . . . . 5 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ℂ) |
4 | 3 | adantr 484 | . . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → 𝑥 ∈ ℂ) |
5 | 1 | logdmn0 25528 | . . . . 5 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ≠ 0) |
6 | 5 | adantr 484 | . . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → 𝑥 ≠ 0) |
7 | simpr 488 | . . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ) | |
8 | 4, 6, 7 | cxpefd 25600 | . . 3 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → (𝑥↑𝑐𝑦) = (exp‘(𝑦 · (log‘𝑥)))) |
9 | 8 | mpoeq3ia 7289 | . 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‘ℂfld) | |
12 | 11 | cnfldtopon 23680 | . . . . . . 7 ⊢ 𝐽 ∈ (TopOn‘ℂ) |
13 | 12 | a1i 11 | . . . . . 6 ⊢ (⊤ → 𝐽 ∈ (TopOn‘ℂ)) |
14 | 3 | ssriv 3905 | . . . . . 6 ⊢ 𝐷 ⊆ ℂ |
15 | resttopon 22058 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝐷 ⊆ ℂ) → (𝐽 ↾t 𝐷) ∈ (TopOn‘𝐷)) | |
16 | 13, 14, 15 | sylancl 589 | . . . . 5 ⊢ (⊤ → (𝐽 ↾t 𝐷) ∈ (TopOn‘𝐷)) |
17 | 10, 16 | eqeltrid 2842 | . . . 4 ⊢ (⊤ → 𝐾 ∈ (TopOn‘𝐷)) |
18 | 17, 13 | cnmpt2nd 22566 | . . . . 5 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ 𝑦) ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) |
19 | fvres 6736 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → ((log ↾ 𝐷)‘𝑥) = (log‘𝑥)) | |
20 | 19 | adantr 484 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → ((log ↾ 𝐷)‘𝑥) = (log‘𝑥)) |
21 | 20 | mpoeq3ia 7289 | . . . . . 6 ⊢ (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ ((log ↾ 𝐷)‘𝑥)) = (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (log‘𝑥)) |
22 | 17, 13 | cnmpt1st 22565 | . . . . . . 7 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ 𝑥) ∈ ((𝐾 ×t 𝐽) Cn 𝐾)) |
23 | 1 | logcn 25535 | . . . . . . . . 9 ⊢ (log ↾ 𝐷) ∈ (𝐷–cn→ℂ) |
24 | ssid 3923 | . . . . . . . . . 10 ⊢ ℂ ⊆ ℂ | |
25 | 12 | toponrestid 21818 | . . . . . . . . . . 11 ⊢ 𝐽 = (𝐽 ↾t ℂ) |
26 | 11, 10, 25 | cncfcn 23807 | . . . . . . . . . 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 2836 | . . . . . . . 8 ⊢ (log ↾ 𝐷) ∈ (𝐾 Cn 𝐽) |
29 | 28 | a1i 11 | . . . . . . 7 ⊢ (⊤ → (log ↾ 𝐷) ∈ (𝐾 Cn 𝐽)) |
30 | 17, 13, 22, 29 | cnmpt21f 22569 | . . . . . 6 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ ((log ↾ 𝐷)‘𝑥)) ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) |
31 | 21, 30 | eqeltrrid 2843 | . . . . 5 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (log‘𝑥)) ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) |
32 | 11 | mulcn 23764 | . . . . . 6 ⊢ · ∈ ((𝐽 ×t 𝐽) Cn 𝐽) |
33 | 32 | a1i 11 | . . . . 5 ⊢ (⊤ → · ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
34 | 17, 13, 18, 31, 33 | cnmpt22f 22572 | . . . 4 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (𝑦 · (log‘𝑥))) ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) |
35 | efcn 25335 | . . . . . 6 ⊢ exp ∈ (ℂ–cn→ℂ) | |
36 | 11 | cncfcn1 23808 | . . . . . 6 ⊢ (ℂ–cn→ℂ) = (𝐽 Cn 𝐽) |
37 | 35, 36 | eleqtri 2836 | . . . . 5 ⊢ exp ∈ (𝐽 Cn 𝐽) |
38 | 37 | a1i 11 | . . . 4 ⊢ (⊤ → exp ∈ (𝐽 Cn 𝐽)) |
39 | 17, 13, 34, 38 | cnmpt21f 22569 | . . 3 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (exp‘(𝑦 · (log‘𝑥)))) ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) |
40 | 39 | mptru 1550 | . 2 ⊢ (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (exp‘(𝑦 · (log‘𝑥)))) ∈ ((𝐾 ×t 𝐽) Cn 𝐽) |
41 | 9, 40 | eqeltri 2834 | 1 ⊢ (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (𝑥↑𝑐𝑦)) ∈ ((𝐾 ×t 𝐽) Cn 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ⊤wtru 1544 ∈ wcel 2110 ≠ wne 2940 ∖ cdif 3863 ⊆ wss 3866 ↾ cres 5553 ‘cfv 6380 (class class class)co 7213 ∈ cmpo 7215 ℂcc 10727 ℝcr 10728 0cc0 10729 · cmul 10734 -∞cmnf 10865 ℝ+crp 12586 (,]cioc 12936 expce 15623 ↾t crest 16925 TopOpenctopn 16926 ℂfldccnfld 20363 TopOnctopon 21807 Cn ccn 22121 ×t ctx 22457 –cn→ccncf 23773 logclog 25443 ↑𝑐ccxp 25444 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 ax-addf 10808 ax-mulf 10809 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-er 8391 df-map 8510 df-pm 8511 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-fi 9027 df-sup 9058 df-inf 9059 df-oi 9126 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-q 12545 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-ioo 12939 df-ioc 12940 df-ico 12941 df-icc 12942 df-fz 13096 df-fzo 13239 df-fl 13367 df-mod 13443 df-seq 13575 df-exp 13636 df-fac 13840 df-bc 13869 df-hash 13897 df-shft 14630 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-limsup 15032 df-clim 15049 df-rlim 15050 df-sum 15250 df-ef 15629 df-sin 15631 df-cos 15632 df-tan 15633 df-pi 15634 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-starv 16817 df-sca 16818 df-vsca 16819 df-ip 16820 df-tset 16821 df-ple 16822 df-ds 16824 df-unif 16825 df-hom 16826 df-cco 16827 df-rest 16927 df-topn 16928 df-0g 16946 df-gsum 16947 df-topgen 16948 df-pt 16949 df-prds 16952 df-xrs 17007 df-qtop 17012 df-imas 17013 df-xps 17015 df-mre 17089 df-mrc 17090 df-acs 17092 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-mulg 18489 df-cntz 18711 df-cmn 19172 df-psmet 20355 df-xmet 20356 df-met 20357 df-bl 20358 df-mopn 20359 df-fbas 20360 df-fg 20361 df-cnfld 20364 df-top 21791 df-topon 21808 df-topsp 21830 df-bases 21843 df-cld 21916 df-ntr 21917 df-cls 21918 df-nei 21995 df-lp 22033 df-perf 22034 df-cn 22124 df-cnp 22125 df-haus 22212 df-cmp 22284 df-tx 22459 df-hmeo 22652 df-fil 22743 df-fm 22835 df-flim 22836 df-flf 22837 df-xms 23218 df-ms 23219 df-tms 23220 df-cncf 23775 df-limc 24763 df-dv 24764 df-log 25445 df-cxp 25446 |
This theorem is referenced by: cxpcn2 25632 sqrtcn 25636 cxpcncf1 32287 cxpcncf2 43115 |
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