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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 25216 | . . . . . 6 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ℂ ∧ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ+))) |
3 | 2 | simplbi 500 | . . . . 5 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ℂ) |
4 | 3 | adantr 483 | . . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → 𝑥 ∈ ℂ) |
5 | 1 | logdmn0 25217 | . . . . 5 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ≠ 0) |
6 | 5 | adantr 483 | . . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → 𝑥 ≠ 0) |
7 | simpr 487 | . . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ) | |
8 | 4, 6, 7 | cxpefd 25289 | . . 3 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → (𝑥↑𝑐𝑦) = (exp‘(𝑦 · (log‘𝑥)))) |
9 | 8 | mpoeq3ia 7226 | . 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 23385 | . . . . . . 7 ⊢ 𝐽 ∈ (TopOn‘ℂ) |
13 | 12 | a1i 11 | . . . . . 6 ⊢ (⊤ → 𝐽 ∈ (TopOn‘ℂ)) |
14 | 3 | ssriv 3971 | . . . . . 6 ⊢ 𝐷 ⊆ ℂ |
15 | resttopon 21763 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝐷 ⊆ ℂ) → (𝐽 ↾t 𝐷) ∈ (TopOn‘𝐷)) | |
16 | 13, 14, 15 | sylancl 588 | . . . . 5 ⊢ (⊤ → (𝐽 ↾t 𝐷) ∈ (TopOn‘𝐷)) |
17 | 10, 16 | eqeltrid 2917 | . . . 4 ⊢ (⊤ → 𝐾 ∈ (TopOn‘𝐷)) |
18 | 17, 13 | cnmpt2nd 22271 | . . . . 5 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ 𝑦) ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) |
19 | fvres 6684 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → ((log ↾ 𝐷)‘𝑥) = (log‘𝑥)) | |
20 | 19 | adantr 483 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → ((log ↾ 𝐷)‘𝑥) = (log‘𝑥)) |
21 | 20 | mpoeq3ia 7226 | . . . . . 6 ⊢ (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ ((log ↾ 𝐷)‘𝑥)) = (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (log‘𝑥)) |
22 | 17, 13 | cnmpt1st 22270 | . . . . . . 7 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ 𝑥) ∈ ((𝐾 ×t 𝐽) Cn 𝐾)) |
23 | 1 | logcn 25224 | . . . . . . . . 9 ⊢ (log ↾ 𝐷) ∈ (𝐷–cn→ℂ) |
24 | ssid 3989 | . . . . . . . . . 10 ⊢ ℂ ⊆ ℂ | |
25 | 12 | toponrestid 21523 | . . . . . . . . . . 11 ⊢ 𝐽 = (𝐽 ↾t ℂ) |
26 | 11, 10, 25 | cncfcn 23511 | . . . . . . . . . 10 ⊢ ((𝐷 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝐷–cn→ℂ) = (𝐾 Cn 𝐽)) |
27 | 14, 24, 26 | mp2an 690 | . . . . . . . . 9 ⊢ (𝐷–cn→ℂ) = (𝐾 Cn 𝐽) |
28 | 23, 27 | eleqtri 2911 | . . . . . . . 8 ⊢ (log ↾ 𝐷) ∈ (𝐾 Cn 𝐽) |
29 | 28 | a1i 11 | . . . . . . 7 ⊢ (⊤ → (log ↾ 𝐷) ∈ (𝐾 Cn 𝐽)) |
30 | 17, 13, 22, 29 | cnmpt21f 22274 | . . . . . 6 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ ((log ↾ 𝐷)‘𝑥)) ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) |
31 | 21, 30 | eqeltrrid 2918 | . . . . 5 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (log‘𝑥)) ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) |
32 | 11 | mulcn 23469 | . . . . . 6 ⊢ · ∈ ((𝐽 ×t 𝐽) Cn 𝐽) |
33 | 32 | a1i 11 | . . . . 5 ⊢ (⊤ → · ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
34 | 17, 13, 18, 31, 33 | cnmpt22f 22277 | . . . 4 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (𝑦 · (log‘𝑥))) ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) |
35 | efcn 25025 | . . . . . 6 ⊢ exp ∈ (ℂ–cn→ℂ) | |
36 | 11 | cncfcn1 23512 | . . . . . 6 ⊢ (ℂ–cn→ℂ) = (𝐽 Cn 𝐽) |
37 | 35, 36 | eleqtri 2911 | . . . . 5 ⊢ exp ∈ (𝐽 Cn 𝐽) |
38 | 37 | a1i 11 | . . . 4 ⊢ (⊤ → exp ∈ (𝐽 Cn 𝐽)) |
39 | 17, 13, 34, 38 | cnmpt21f 22274 | . . 3 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (exp‘(𝑦 · (log‘𝑥)))) ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) |
40 | 39 | mptru 1540 | . 2 ⊢ (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (exp‘(𝑦 · (log‘𝑥)))) ∈ ((𝐾 ×t 𝐽) Cn 𝐽) |
41 | 9, 40 | eqeltri 2909 | 1 ⊢ (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (𝑥↑𝑐𝑦)) ∈ ((𝐾 ×t 𝐽) Cn 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ⊤wtru 1534 ∈ wcel 2110 ≠ wne 3016 ∖ cdif 3933 ⊆ wss 3936 ↾ cres 5552 ‘cfv 6350 (class class class)co 7150 ∈ cmpo 7152 ℂcc 10529 ℝcr 10530 0cc0 10531 · cmul 10536 -∞cmnf 10667 ℝ+crp 12383 (,]cioc 12733 expce 15409 ↾t crest 16688 TopOpenctopn 16689 ℂfldccnfld 20539 TopOnctopon 21512 Cn ccn 21826 ×t ctx 22162 –cn→ccncf 23478 logclog 25132 ↑𝑐ccxp 25133 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-ioc 12737 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-mod 13232 df-seq 13364 df-exp 13424 df-fac 13628 df-bc 13657 df-hash 13685 df-shft 14420 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-limsup 14822 df-clim 14839 df-rlim 14840 df-sum 15037 df-ef 15415 df-sin 15417 df-cos 15418 df-tan 15419 df-pi 15420 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-rest 16690 df-topn 16691 df-0g 16709 df-gsum 16710 df-topgen 16711 df-pt 16712 df-prds 16715 df-xrs 16769 df-qtop 16774 df-imas 16775 df-xps 16777 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-mulg 18219 df-cntz 18441 df-cmn 18902 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-fbas 20536 df-fg 20537 df-cnfld 20540 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-cld 21621 df-ntr 21622 df-cls 21623 df-nei 21700 df-lp 21738 df-perf 21739 df-cn 21829 df-cnp 21830 df-haus 21917 df-cmp 21989 df-tx 22164 df-hmeo 22357 df-fil 22448 df-fm 22540 df-flim 22541 df-flf 22542 df-xms 22924 df-ms 22925 df-tms 22926 df-cncf 23480 df-limc 24458 df-dv 24459 df-log 25134 df-cxp 25135 |
This theorem is referenced by: cxpcn2 25321 sqrtcn 25325 cxpcncf1 31861 cxpcncf2 42175 |
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