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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.) Avoid ax-mulf 11238. (Revised by GG, 16-Mar-2025.) |
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 26666 | . . . . . 6 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ℂ ∧ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ+))) |
3 | 2 | simplbi 496 | . . . . 5 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ℂ) |
4 | 3 | adantr 479 | . . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → 𝑥 ∈ ℂ) |
5 | 1 | logdmn0 26667 | . . . . 5 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ≠ 0) |
6 | 5 | adantr 479 | . . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → 𝑥 ≠ 0) |
7 | simpr 483 | . . . 4 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ) | |
8 | 4, 6, 7 | cxpefd 26739 | . . 3 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → (𝑥↑𝑐𝑦) = (exp‘(𝑦 · (log‘𝑥)))) |
9 | 8 | mpoeq3ia 7503 | . 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 24790 | . . . . . . 7 ⊢ 𝐽 ∈ (TopOn‘ℂ) |
13 | 12 | a1i 11 | . . . . . 6 ⊢ (⊤ → 𝐽 ∈ (TopOn‘ℂ)) |
14 | 3 | ssriv 3983 | . . . . . 6 ⊢ 𝐷 ⊆ ℂ |
15 | resttopon 23156 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝐷 ⊆ ℂ) → (𝐽 ↾t 𝐷) ∈ (TopOn‘𝐷)) | |
16 | 13, 14, 15 | sylancl 584 | . . . . 5 ⊢ (⊤ → (𝐽 ↾t 𝐷) ∈ (TopOn‘𝐷)) |
17 | 10, 16 | eqeltrid 2830 | . . . 4 ⊢ (⊤ → 𝐾 ∈ (TopOn‘𝐷)) |
18 | 17, 13 | cnmpt2nd 23664 | . . . . 5 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ 𝑦) ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) |
19 | fvres 6920 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → ((log ↾ 𝐷)‘𝑥) = (log‘𝑥)) | |
20 | 19 | adantr 479 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ ℂ) → ((log ↾ 𝐷)‘𝑥) = (log‘𝑥)) |
21 | 20 | mpoeq3ia 7503 | . . . . . 6 ⊢ (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ ((log ↾ 𝐷)‘𝑥)) = (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (log‘𝑥)) |
22 | 17, 13 | cnmpt1st 23663 | . . . . . . 7 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ 𝑥) ∈ ((𝐾 ×t 𝐽) Cn 𝐾)) |
23 | 1 | logcn 26674 | . . . . . . . . 9 ⊢ (log ↾ 𝐷) ∈ (𝐷–cn→ℂ) |
24 | ssid 4002 | . . . . . . . . . 10 ⊢ ℂ ⊆ ℂ | |
25 | 12 | toponrestid 22914 | . . . . . . . . . . 11 ⊢ 𝐽 = (𝐽 ↾t ℂ) |
26 | 11, 10, 25 | cncfcn 24921 | . . . . . . . . . 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 2824 | . . . . . . . 8 ⊢ (log ↾ 𝐷) ∈ (𝐾 Cn 𝐽) |
29 | 28 | a1i 11 | . . . . . . 7 ⊢ (⊤ → (log ↾ 𝐷) ∈ (𝐾 Cn 𝐽)) |
30 | 17, 13, 22, 29 | cnmpt21f 23667 | . . . . . 6 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ ((log ↾ 𝐷)‘𝑥)) ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) |
31 | 21, 30 | eqeltrrid 2831 | . . . . 5 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (log‘𝑥)) ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) |
32 | 11 | mpomulcn 24876 | . . . . . 6 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ ((𝐽 ×t 𝐽) Cn 𝐽) |
33 | 32 | a1i 11 | . . . . 5 ⊢ (⊤ → (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
34 | oveq12 7433 | . . . . 5 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = (log‘𝑥)) → (𝑢 · 𝑣) = (𝑦 · (log‘𝑥))) | |
35 | 17, 13, 18, 31, 13, 13, 33, 34 | cnmpt22 23669 | . . . 4 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (𝑦 · (log‘𝑥))) ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) |
36 | efcn 26473 | . . . . . 6 ⊢ exp ∈ (ℂ–cn→ℂ) | |
37 | 11 | cncfcn1 24922 | . . . . . 6 ⊢ (ℂ–cn→ℂ) = (𝐽 Cn 𝐽) |
38 | 36, 37 | eleqtri 2824 | . . . . 5 ⊢ exp ∈ (𝐽 Cn 𝐽) |
39 | 38 | a1i 11 | . . . 4 ⊢ (⊤ → exp ∈ (𝐽 Cn 𝐽)) |
40 | 17, 13, 35, 39 | cnmpt21f 23667 | . . 3 ⊢ (⊤ → (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (exp‘(𝑦 · (log‘𝑥)))) ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) |
41 | 40 | mptru 1541 | . 2 ⊢ (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (exp‘(𝑦 · (log‘𝑥)))) ∈ ((𝐾 ×t 𝐽) Cn 𝐽) |
42 | 9, 41 | eqeltri 2822 | 1 ⊢ (𝑥 ∈ 𝐷, 𝑦 ∈ ℂ ↦ (𝑥↑𝑐𝑦)) ∈ ((𝐾 ×t 𝐽) Cn 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ⊤wtru 1535 ∈ wcel 2099 ≠ wne 2930 ∖ cdif 3944 ⊆ wss 3947 ↾ cres 5684 ‘cfv 6554 (class class class)co 7424 ∈ cmpo 7426 ℂcc 11156 ℝcr 11157 0cc0 11158 · cmul 11163 -∞cmnf 11296 ℝ+crp 13028 (,]cioc 13379 expce 16063 ↾t crest 17435 TopOpenctopn 17436 ℂfldccnfld 21343 TopOnctopon 22903 Cn ccn 23219 ×t ctx 23555 –cn→ccncf 24887 logclog 26581 ↑𝑐ccxp 26582 |
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 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 ax-addf 11237 |
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 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-om 7877 df-1st 8003 df-2nd 8004 df-supp 8175 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-er 8734 df-map 8857 df-pm 8858 df-ixp 8927 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fsupp 9406 df-fi 9454 df-sup 9485 df-inf 9486 df-oi 9553 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12611 df-dec 12730 df-uz 12875 df-q 12985 df-rp 13029 df-xneg 13146 df-xadd 13147 df-xmul 13148 df-ioo 13382 df-ioc 13383 df-ico 13384 df-icc 13385 df-fz 13539 df-fzo 13682 df-fl 13812 df-mod 13890 df-seq 14022 df-exp 14082 df-fac 14291 df-bc 14320 df-hash 14348 df-shft 15072 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-limsup 15473 df-clim 15490 df-rlim 15491 df-sum 15691 df-ef 16069 df-sin 16071 df-cos 16072 df-tan 16073 df-pi 16074 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-starv 17281 df-sca 17282 df-vsca 17283 df-ip 17284 df-tset 17285 df-ple 17286 df-ds 17288 df-unif 17289 df-hom 17290 df-cco 17291 df-rest 17437 df-topn 17438 df-0g 17456 df-gsum 17457 df-topgen 17458 df-pt 17459 df-prds 17462 df-xrs 17517 df-qtop 17522 df-imas 17523 df-xps 17525 df-mre 17599 df-mrc 17600 df-acs 17602 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-submnd 18774 df-mulg 19062 df-cntz 19311 df-cmn 19780 df-psmet 21335 df-xmet 21336 df-met 21337 df-bl 21338 df-mopn 21339 df-fbas 21340 df-fg 21341 df-cnfld 21344 df-top 22887 df-topon 22904 df-topsp 22926 df-bases 22940 df-cld 23014 df-ntr 23015 df-cls 23016 df-nei 23093 df-lp 23131 df-perf 23132 df-cn 23222 df-cnp 23223 df-haus 23310 df-cmp 23382 df-tx 23557 df-hmeo 23750 df-fil 23841 df-fm 23933 df-flim 23934 df-flf 23935 df-xms 24317 df-ms 24318 df-tms 24319 df-cncf 24889 df-limc 25886 df-dv 25887 df-log 26583 df-cxp 26584 |
This theorem is referenced by: cxpcn2 26774 sqrtcn 26778 cxpcncf1 34441 cxpcncf2 45520 |
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