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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 24787 | . . . . 5 ⊢ 𝐽 ∈ (TopOn‘ℂ) |
4 | rpcn 13032 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℂ) | |
5 | ax-1 6 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ+ → (𝑥 ∈ ℝ → 𝑥 ∈ ℝ+)) | |
6 | eqid 2726 | . . . . . . . 8 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0)) | |
7 | 6 | ellogdm 26663 | . . . . . . 7 ⊢ (𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↔ (𝑥 ∈ ℂ ∧ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ+))) |
8 | 4, 5, 7 | sylanbrc 581 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ (ℂ ∖ (-∞(,]0))) |
9 | 8 | ssriv 3982 | . . . . 5 ⊢ ℝ+ ⊆ (ℂ ∖ (-∞(,]0)) |
10 | cnex 11230 | . . . . . 6 ⊢ ℂ ∈ V | |
11 | 10 | difexi 5327 | . . . . 5 ⊢ (ℂ ∖ (-∞(,]0)) ∈ V |
12 | restabs 23157 | . . . . 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 1458 | . . . 4 ⊢ ((𝐽 ↾t (ℂ ∖ (-∞(,]0))) ↾t ℝ+) = (𝐽 ↾t ℝ+) |
14 | 1, 13 | eqtr4i 2757 | . . 3 ⊢ 𝐾 = ((𝐽 ↾t (ℂ ∖ (-∞(,]0))) ↾t ℝ+) |
15 | 3 | a1i 11 | . . . 4 ⊢ (⊤ → 𝐽 ∈ (TopOn‘ℂ)) |
16 | difss 4128 | . . . 4 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ ℂ | |
17 | resttopon 23153 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ (ℂ ∖ (-∞(,]0)) ⊆ ℂ) → (𝐽 ↾t (ℂ ∖ (-∞(,]0))) ∈ (TopOn‘(ℂ ∖ (-∞(,]0)))) | |
18 | 15, 16, 17 | sylancl 584 | . . 3 ⊢ (⊤ → (𝐽 ↾t (ℂ ∖ (-∞(,]0))) ∈ (TopOn‘(ℂ ∖ (-∞(,]0)))) |
19 | 9 | a1i 11 | . . 3 ⊢ (⊤ → ℝ+ ⊆ (ℂ ∖ (-∞(,]0))) |
20 | 3 | toponrestid 22911 | . . 3 ⊢ 𝐽 = (𝐽 ↾t ℂ) |
21 | ssidd 4002 | . . 3 ⊢ (⊤ → ℂ ⊆ ℂ) | |
22 | eqid 2726 | . . . . 5 ⊢ (𝐽 ↾t (ℂ ∖ (-∞(,]0))) = (𝐽 ↾t (ℂ ∖ (-∞(,]0))) | |
23 | 6, 2, 22 | cxpcn 26769 | . . . 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 23669 | . 2 ⊢ (⊤ → (𝑥 ∈ ℝ+, 𝑦 ∈ ℂ ↦ (𝑥↑𝑐𝑦)) ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) |
26 | 25 | mptru 1541 | 1 ⊢ (𝑥 ∈ ℝ+, 𝑦 ∈ ℂ ↦ (𝑥↑𝑐𝑦)) ∈ ((𝐾 ×t 𝐽) Cn 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ⊤wtru 1535 ∈ wcel 2099 Vcvv 3462 ∖ cdif 3943 ⊆ wss 3946 ‘cfv 6546 (class class class)co 7416 ∈ cmpo 7418 ℂcc 11147 ℝcr 11148 0cc0 11149 -∞cmnf 11287 ℝ+crp 13022 (,]cioc 13373 ↾t crest 17430 TopOpenctopn 17431 ℂfldccnfld 21339 TopOnctopon 22900 Cn ccn 23216 ×t ctx 23552 ↑𝑐ccxp 26579 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-inf2 9677 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 ax-addf 11228 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-iin 4996 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-se 5630 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-isom 6555 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7995 df-2nd 7996 df-supp 8167 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-er 8726 df-map 8849 df-pm 8850 df-ixp 8919 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fsupp 9399 df-fi 9447 df-sup 9478 df-inf 9479 df-oi 9546 df-card 9975 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12519 df-z 12605 df-dec 12724 df-uz 12869 df-q 12979 df-rp 13023 df-xneg 13140 df-xadd 13141 df-xmul 13142 df-ioo 13376 df-ioc 13377 df-ico 13378 df-icc 13379 df-fz 13533 df-fzo 13676 df-fl 13806 df-mod 13884 df-seq 14016 df-exp 14076 df-fac 14286 df-bc 14315 df-hash 14343 df-shft 15067 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-limsup 15468 df-clim 15485 df-rlim 15486 df-sum 15686 df-ef 16064 df-sin 16066 df-cos 16067 df-tan 16068 df-pi 16069 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-starv 17276 df-sca 17277 df-vsca 17278 df-ip 17279 df-tset 17280 df-ple 17281 df-ds 17283 df-unif 17284 df-hom 17285 df-cco 17286 df-rest 17432 df-topn 17433 df-0g 17451 df-gsum 17452 df-topgen 17453 df-pt 17454 df-prds 17457 df-xrs 17512 df-qtop 17517 df-imas 17518 df-xps 17520 df-mre 17594 df-mrc 17595 df-acs 17597 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-submnd 18769 df-mulg 19058 df-cntz 19307 df-cmn 19776 df-psmet 21331 df-xmet 21332 df-met 21333 df-bl 21334 df-mopn 21335 df-fbas 21336 df-fg 21337 df-cnfld 21340 df-top 22884 df-topon 22901 df-topsp 22923 df-bases 22937 df-cld 23011 df-ntr 23012 df-cls 23013 df-nei 23090 df-lp 23128 df-perf 23129 df-cn 23219 df-cnp 23220 df-haus 23307 df-cmp 23379 df-tx 23554 df-hmeo 23747 df-fil 23838 df-fm 23930 df-flim 23931 df-flf 23932 df-xms 24314 df-ms 24315 df-tms 24316 df-cncf 24886 df-limc 25883 df-dv 25884 df-log 26580 df-cxp 26581 |
This theorem is referenced by: cxpcn3 26773 |
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