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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 22956 | . . . . 5 ⊢ 𝐽 ∈ (TopOn‘ℂ) |
4 | rpcn 12124 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℂ) | |
5 | ax-1 6 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ+ → (𝑥 ∈ ℝ → 𝑥 ∈ ℝ+)) | |
6 | eqid 2825 | . . . . . . . 8 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0)) | |
7 | 6 | ellogdm 24784 | . . . . . . 7 ⊢ (𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↔ (𝑥 ∈ ℂ ∧ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ+))) |
8 | 4, 5, 7 | sylanbrc 580 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ (ℂ ∖ (-∞(,]0))) |
9 | 8 | ssriv 3831 | . . . . 5 ⊢ ℝ+ ⊆ (ℂ ∖ (-∞(,]0)) |
10 | cnex 10333 | . . . . . 6 ⊢ ℂ ∈ V | |
11 | difss 3964 | . . . . . 6 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ ℂ | |
12 | 10, 11 | ssexi 5028 | . . . . 5 ⊢ (ℂ ∖ (-∞(,]0)) ∈ V |
13 | restabs 21340 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ ℝ+ ⊆ (ℂ ∖ (-∞(,]0)) ∧ (ℂ ∖ (-∞(,]0)) ∈ V) → ((𝐽 ↾t (ℂ ∖ (-∞(,]0))) ↾t ℝ+) = (𝐽 ↾t ℝ+)) | |
14 | 3, 9, 12, 13 | mp3an 1591 | . . . 4 ⊢ ((𝐽 ↾t (ℂ ∖ (-∞(,]0))) ↾t ℝ+) = (𝐽 ↾t ℝ+) |
15 | 1, 14 | eqtr4i 2852 | . . 3 ⊢ 𝐾 = ((𝐽 ↾t (ℂ ∖ (-∞(,]0))) ↾t ℝ+) |
16 | 3 | a1i 11 | . . . 4 ⊢ (⊤ → 𝐽 ∈ (TopOn‘ℂ)) |
17 | resttopon 21336 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ (ℂ ∖ (-∞(,]0)) ⊆ ℂ) → (𝐽 ↾t (ℂ ∖ (-∞(,]0))) ∈ (TopOn‘(ℂ ∖ (-∞(,]0)))) | |
18 | 16, 11, 17 | sylancl 582 | . . 3 ⊢ (⊤ → (𝐽 ↾t (ℂ ∖ (-∞(,]0))) ∈ (TopOn‘(ℂ ∖ (-∞(,]0)))) |
19 | 9 | a1i 11 | . . 3 ⊢ (⊤ → ℝ+ ⊆ (ℂ ∖ (-∞(,]0))) |
20 | 3 | toponrestid 21096 | . . 3 ⊢ 𝐽 = (𝐽 ↾t ℂ) |
21 | ssidd 3849 | . . 3 ⊢ (⊤ → ℂ ⊆ ℂ) | |
22 | eqid 2825 | . . . . 5 ⊢ (𝐽 ↾t (ℂ ∖ (-∞(,]0))) = (𝐽 ↾t (ℂ ∖ (-∞(,]0))) | |
23 | 6, 2, 22 | cxpcn 24888 | . . . 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 | 15, 18, 19, 20, 16, 21, 24 | cnmpt2res 21851 | . 2 ⊢ (⊤ → (𝑥 ∈ ℝ+, 𝑦 ∈ ℂ ↦ (𝑥↑𝑐𝑦)) ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) |
26 | 25 | mptru 1666 | 1 ⊢ (𝑥 ∈ ℝ+, 𝑦 ∈ ℂ ↦ (𝑥↑𝑐𝑦)) ∈ ((𝐾 ×t 𝐽) Cn 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ⊤wtru 1659 ∈ wcel 2166 Vcvv 3414 ∖ cdif 3795 ⊆ wss 3798 ‘cfv 6123 (class class class)co 6905 ↦ cmpt2 6907 ℂcc 10250 ℝcr 10251 0cc0 10252 -∞cmnf 10389 ℝ+crp 12112 (,]cioc 12464 ↾t crest 16434 TopOpenctopn 16435 ℂfldccnfld 20106 TopOnctopon 21085 Cn ccn 21399 ×t ctx 21734 ↑𝑐ccxp 24701 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-inf2 8815 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330 ax-addf 10331 ax-mulf 10332 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-iin 4743 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-se 5302 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-of 7157 df-om 7327 df-1st 7428 df-2nd 7429 df-supp 7560 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-2o 7827 df-oadd 7830 df-er 8009 df-map 8124 df-pm 8125 df-ixp 8176 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-fsupp 8545 df-fi 8586 df-sup 8617 df-inf 8618 df-oi 8684 df-card 9078 df-cda 9305 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-n0 11619 df-z 11705 df-dec 11822 df-uz 11969 df-q 12072 df-rp 12113 df-xneg 12232 df-xadd 12233 df-xmul 12234 df-ioo 12467 df-ioc 12468 df-ico 12469 df-icc 12470 df-fz 12620 df-fzo 12761 df-fl 12888 df-mod 12964 df-seq 13096 df-exp 13155 df-fac 13354 df-bc 13383 df-hash 13411 df-shft 14184 df-cj 14216 df-re 14217 df-im 14218 df-sqrt 14352 df-abs 14353 df-limsup 14579 df-clim 14596 df-rlim 14597 df-sum 14794 df-ef 15170 df-sin 15172 df-cos 15173 df-tan 15174 df-pi 15175 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-mulr 16319 df-starv 16320 df-sca 16321 df-vsca 16322 df-ip 16323 df-tset 16324 df-ple 16325 df-ds 16327 df-unif 16328 df-hom 16329 df-cco 16330 df-rest 16436 df-topn 16437 df-0g 16455 df-gsum 16456 df-topgen 16457 df-pt 16458 df-prds 16461 df-xrs 16515 df-qtop 16520 df-imas 16521 df-xps 16523 df-mre 16599 df-mrc 16600 df-acs 16602 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-submnd 17689 df-mulg 17895 df-cntz 18100 df-cmn 18548 df-psmet 20098 df-xmet 20099 df-met 20100 df-bl 20101 df-mopn 20102 df-fbas 20103 df-fg 20104 df-cnfld 20107 df-top 21069 df-topon 21086 df-topsp 21108 df-bases 21121 df-cld 21194 df-ntr 21195 df-cls 21196 df-nei 21273 df-lp 21311 df-perf 21312 df-cn 21402 df-cnp 21403 df-haus 21490 df-cmp 21561 df-tx 21736 df-hmeo 21929 df-fil 22020 df-fm 22112 df-flim 22113 df-flf 22114 df-xms 22495 df-ms 22496 df-tms 22497 df-cncf 23051 df-limc 24029 df-dv 24030 df-log 24702 df-cxp 24703 |
This theorem is referenced by: cxpcn3 24891 |
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