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Mirrors > Home > MPE Home > Th. List > cxpcn2 | Structured version Visualization version GIF version |
Description: Continuity of the complex power function, when the base is real. (Contributed by Mario Carneiro, 1-May-2016.) |
Ref | Expression |
---|---|
cxpcn2.j | ⊢ 𝐽 = (TopOpen‘ℂfld) |
cxpcn2.k | ⊢ 𝐾 = (𝐽 ↾t ℝ+) |
Ref | Expression |
---|---|
cxpcn2 | ⊢ (𝑥 ∈ ℝ+, 𝑦 ∈ ℂ ↦ (𝑥↑𝑐𝑦)) ∈ ((𝐾 ×t 𝐽) Cn 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cxpcn2.k | . . . 4 ⊢ 𝐾 = (𝐽 ↾t ℝ+) | |
2 | cxpcn2.j | . . . . . 6 ⊢ 𝐽 = (TopOpen‘ℂfld) | |
3 | 2 | cnfldtopon 23318 | . . . . 5 ⊢ 𝐽 ∈ (TopOn‘ℂ) |
4 | rpcn 12387 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℂ) | |
5 | ax-1 6 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ+ → (𝑥 ∈ ℝ → 𝑥 ∈ ℝ+)) | |
6 | eqid 2818 | . . . . . . . 8 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0)) | |
7 | 6 | ellogdm 25149 | . . . . . . 7 ⊢ (𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↔ (𝑥 ∈ ℂ ∧ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ+))) |
8 | 4, 5, 7 | sylanbrc 583 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ (ℂ ∖ (-∞(,]0))) |
9 | 8 | ssriv 3968 | . . . . 5 ⊢ ℝ+ ⊆ (ℂ ∖ (-∞(,]0)) |
10 | cnex 10606 | . . . . . 6 ⊢ ℂ ∈ V | |
11 | 10 | difexi 5223 | . . . . 5 ⊢ (ℂ ∖ (-∞(,]0)) ∈ V |
12 | restabs 21701 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ ℝ+ ⊆ (ℂ ∖ (-∞(,]0)) ∧ (ℂ ∖ (-∞(,]0)) ∈ V) → ((𝐽 ↾t (ℂ ∖ (-∞(,]0))) ↾t ℝ+) = (𝐽 ↾t ℝ+)) | |
13 | 3, 9, 11, 12 | mp3an 1452 | . . . 4 ⊢ ((𝐽 ↾t (ℂ ∖ (-∞(,]0))) ↾t ℝ+) = (𝐽 ↾t ℝ+) |
14 | 1, 13 | eqtr4i 2844 | . . 3 ⊢ 𝐾 = ((𝐽 ↾t (ℂ ∖ (-∞(,]0))) ↾t ℝ+) |
15 | 3 | a1i 11 | . . . 4 ⊢ (⊤ → 𝐽 ∈ (TopOn‘ℂ)) |
16 | difss 4105 | . . . 4 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ ℂ | |
17 | resttopon 21697 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ (ℂ ∖ (-∞(,]0)) ⊆ ℂ) → (𝐽 ↾t (ℂ ∖ (-∞(,]0))) ∈ (TopOn‘(ℂ ∖ (-∞(,]0)))) | |
18 | 15, 16, 17 | sylancl 586 | . . 3 ⊢ (⊤ → (𝐽 ↾t (ℂ ∖ (-∞(,]0))) ∈ (TopOn‘(ℂ ∖ (-∞(,]0)))) |
19 | 9 | a1i 11 | . . 3 ⊢ (⊤ → ℝ+ ⊆ (ℂ ∖ (-∞(,]0))) |
20 | 3 | toponrestid 21457 | . . 3 ⊢ 𝐽 = (𝐽 ↾t ℂ) |
21 | ssidd 3987 | . . 3 ⊢ (⊤ → ℂ ⊆ ℂ) | |
22 | eqid 2818 | . . . . 5 ⊢ (𝐽 ↾t (ℂ ∖ (-∞(,]0))) = (𝐽 ↾t (ℂ ∖ (-∞(,]0))) | |
23 | 6, 2, 22 | cxpcn 25253 | . . . 4 ⊢ (𝑥 ∈ (ℂ ∖ (-∞(,]0)), 𝑦 ∈ ℂ ↦ (𝑥↑𝑐𝑦)) ∈ (((𝐽 ↾t (ℂ ∖ (-∞(,]0))) ×t 𝐽) Cn 𝐽) |
24 | 23 | a1i 11 | . . 3 ⊢ (⊤ → (𝑥 ∈ (ℂ ∖ (-∞(,]0)), 𝑦 ∈ ℂ ↦ (𝑥↑𝑐𝑦)) ∈ (((𝐽 ↾t (ℂ ∖ (-∞(,]0))) ×t 𝐽) Cn 𝐽)) |
25 | 14, 18, 19, 20, 15, 21, 24 | cnmpt2res 22213 | . 2 ⊢ (⊤ → (𝑥 ∈ ℝ+, 𝑦 ∈ ℂ ↦ (𝑥↑𝑐𝑦)) ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) |
26 | 25 | mptru 1535 | 1 ⊢ (𝑥 ∈ ℝ+, 𝑦 ∈ ℂ ↦ (𝑥↑𝑐𝑦)) ∈ ((𝐾 ×t 𝐽) Cn 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ⊤wtru 1529 ∈ wcel 2105 Vcvv 3492 ∖ cdif 3930 ⊆ wss 3933 ‘cfv 6348 (class class class)co 7145 ∈ cmpo 7147 ℂcc 10523 ℝcr 10524 0cc0 10525 -∞cmnf 10661 ℝ+crp 12377 (,]cioc 12727 ↾t crest 16682 TopOpenctopn 16683 ℂfldccnfld 20473 TopOnctopon 21446 Cn ccn 21760 ×t ctx 22096 ↑𝑐ccxp 25066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12881 df-fzo 13022 df-fl 13150 df-mod 13226 df-seq 13358 df-exp 13418 df-fac 13622 df-bc 13651 df-hash 13679 df-shft 14414 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-limsup 14816 df-clim 14833 df-rlim 14834 df-sum 15031 df-ef 15409 df-sin 15411 df-cos 15412 df-tan 15413 df-pi 15414 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-rest 16684 df-topn 16685 df-0g 16703 df-gsum 16704 df-topgen 16705 df-pt 16706 df-prds 16709 df-xrs 16763 df-qtop 16768 df-imas 16769 df-xps 16771 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-mulg 18163 df-cntz 18385 df-cmn 18837 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-fbas 20470 df-fg 20471 df-cnfld 20474 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-cld 21555 df-ntr 21556 df-cls 21557 df-nei 21634 df-lp 21672 df-perf 21673 df-cn 21763 df-cnp 21764 df-haus 21851 df-cmp 21923 df-tx 22098 df-hmeo 22291 df-fil 22382 df-fm 22474 df-flim 22475 df-flf 22476 df-xms 22857 df-ms 22858 df-tms 22859 df-cncf 23413 df-limc 24391 df-dv 24392 df-log 25067 df-cxp 25068 |
This theorem is referenced by: cxpcn3 25256 |
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