Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cxpcncf1 | Structured version Visualization version GIF version |
Description: The power function on complex numbers, for fixed exponent A, is continuous. Similar to cxpcn 24486. (Contributed by Thierry Arnoux, 20-Dec-2021.) |
Ref | Expression |
---|---|
cxpcncf1.a | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
cxpcncf1.d | ⊢ (𝜑 → 𝐷 ⊆ (ℂ ∖ (-∞(,]0))) |
Ref | Expression |
---|---|
cxpcncf1 | ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (𝑥↑𝑐𝐴)) ∈ (𝐷–cn→ℂ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cxpcncf1.d | . . 3 ⊢ (𝜑 → 𝐷 ⊆ (ℂ ∖ (-∞(,]0))) | |
2 | resmpt 5449 | . . 3 ⊢ (𝐷 ⊆ (ℂ ∖ (-∞(,]0)) → ((𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ (𝑥↑𝑐𝐴)) ↾ 𝐷) = (𝑥 ∈ 𝐷 ↦ (𝑥↑𝑐𝐴))) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → ((𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ (𝑥↑𝑐𝐴)) ↾ 𝐷) = (𝑥 ∈ 𝐷 ↦ (𝑥↑𝑐𝐴))) |
4 | eqid 2622 | . . . . . . . 8 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
5 | 4 | cnfldtopon 22586 | . . . . . . 7 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
6 | difss 3737 | . . . . . . 7 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ ℂ | |
7 | resttopon 20965 | . . . . . . 7 ⊢ (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ (ℂ ∖ (-∞(,]0)) ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t (ℂ ∖ (-∞(,]0))) ∈ (TopOn‘(ℂ ∖ (-∞(,]0)))) | |
8 | 5, 6, 7 | mp2an 708 | . . . . . 6 ⊢ ((TopOpen‘ℂfld) ↾t (ℂ ∖ (-∞(,]0))) ∈ (TopOn‘(ℂ ∖ (-∞(,]0))) |
9 | 8 | a1i 11 | . . . . 5 ⊢ (𝜑 → ((TopOpen‘ℂfld) ↾t (ℂ ∖ (-∞(,]0))) ∈ (TopOn‘(ℂ ∖ (-∞(,]0)))) |
10 | 9 | cnmptid 21464 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ 𝑥) ∈ (((TopOpen‘ℂfld) ↾t (ℂ ∖ (-∞(,]0))) Cn ((TopOpen‘ℂfld) ↾t (ℂ ∖ (-∞(,]0))))) |
11 | 5 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) |
12 | cxpcncf1.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
13 | 9, 11, 12 | cnmptc 21465 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ 𝐴) ∈ (((TopOpen‘ℂfld) ↾t (ℂ ∖ (-∞(,]0))) Cn (TopOpen‘ℂfld))) |
14 | eqid 2622 | . . . . . . 7 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0)) | |
15 | eqid 2622 | . . . . . . 7 ⊢ ((TopOpen‘ℂfld) ↾t (ℂ ∖ (-∞(,]0))) = ((TopOpen‘ℂfld) ↾t (ℂ ∖ (-∞(,]0))) | |
16 | 14, 4, 15 | cxpcn 24486 | . . . . . 6 ⊢ (𝑦 ∈ (ℂ ∖ (-∞(,]0)), 𝑧 ∈ ℂ ↦ (𝑦↑𝑐𝑧)) ∈ ((((TopOpen‘ℂfld) ↾t (ℂ ∖ (-∞(,]0))) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld)) |
17 | 16 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝑦 ∈ (ℂ ∖ (-∞(,]0)), 𝑧 ∈ ℂ ↦ (𝑦↑𝑐𝑧)) ∈ ((((TopOpen‘ℂfld) ↾t (ℂ ∖ (-∞(,]0))) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld))) |
18 | oveq12 6659 | . . . . 5 ⊢ ((𝑦 = 𝑥 ∧ 𝑧 = 𝐴) → (𝑦↑𝑐𝑧) = (𝑥↑𝑐𝐴)) | |
19 | 9, 10, 13, 9, 11, 17, 18 | cnmpt12 21470 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ (𝑥↑𝑐𝐴)) ∈ (((TopOpen‘ℂfld) ↾t (ℂ ∖ (-∞(,]0))) Cn (TopOpen‘ℂfld))) |
20 | ssid 3624 | . . . . . . 7 ⊢ ℂ ⊆ ℂ | |
21 | 5 | elexi 3213 | . . . . . . . . . 10 ⊢ (TopOpen‘ℂfld) ∈ V |
22 | 5 | toponunii 20721 | . . . . . . . . . . 11 ⊢ ℂ = ∪ (TopOpen‘ℂfld) |
23 | 22 | restid 16094 | . . . . . . . . . 10 ⊢ ((TopOpen‘ℂfld) ∈ V → ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld)) |
24 | 21, 23 | ax-mp 5 | . . . . . . . . 9 ⊢ ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld) |
25 | 24 | eqcomi 2631 | . . . . . . . 8 ⊢ (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ) |
26 | 4, 15, 25 | cncfcn 22712 | . . . . . . 7 ⊢ (((ℂ ∖ (-∞(,]0)) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((ℂ ∖ (-∞(,]0))–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (ℂ ∖ (-∞(,]0))) Cn (TopOpen‘ℂfld))) |
27 | 6, 20, 26 | mp2an 708 | . . . . . 6 ⊢ ((ℂ ∖ (-∞(,]0))–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (ℂ ∖ (-∞(,]0))) Cn (TopOpen‘ℂfld)) |
28 | 27 | eqcomi 2631 | . . . . 5 ⊢ (((TopOpen‘ℂfld) ↾t (ℂ ∖ (-∞(,]0))) Cn (TopOpen‘ℂfld)) = ((ℂ ∖ (-∞(,]0))–cn→ℂ) |
29 | 28 | a1i 11 | . . . 4 ⊢ (𝜑 → (((TopOpen‘ℂfld) ↾t (ℂ ∖ (-∞(,]0))) Cn (TopOpen‘ℂfld)) = ((ℂ ∖ (-∞(,]0))–cn→ℂ)) |
30 | 19, 29 | eleqtrd 2703 | . . 3 ⊢ (𝜑 → (𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ (𝑥↑𝑐𝐴)) ∈ ((ℂ ∖ (-∞(,]0))–cn→ℂ)) |
31 | rescncf 22700 | . . . 4 ⊢ (𝐷 ⊆ (ℂ ∖ (-∞(,]0)) → ((𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ (𝑥↑𝑐𝐴)) ∈ ((ℂ ∖ (-∞(,]0))–cn→ℂ) → ((𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ (𝑥↑𝑐𝐴)) ↾ 𝐷) ∈ (𝐷–cn→ℂ))) | |
32 | 31 | imp 445 | . . 3 ⊢ ((𝐷 ⊆ (ℂ ∖ (-∞(,]0)) ∧ (𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ (𝑥↑𝑐𝐴)) ∈ ((ℂ ∖ (-∞(,]0))–cn→ℂ)) → ((𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ (𝑥↑𝑐𝐴)) ↾ 𝐷) ∈ (𝐷–cn→ℂ)) |
33 | 1, 30, 32 | syl2anc 693 | . 2 ⊢ (𝜑 → ((𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ (𝑥↑𝑐𝐴)) ↾ 𝐷) ∈ (𝐷–cn→ℂ)) |
34 | 3, 33 | eqeltrrd 2702 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (𝑥↑𝑐𝐴)) ∈ (𝐷–cn→ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∖ cdif 3571 ⊆ wss 3574 ↦ cmpt 4729 ↾ cres 5116 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 ℂcc 9934 0cc0 9936 -∞cmnf 10072 (,]cioc 12176 ↾t crest 16081 TopOpenctopn 16082 ℂfldccnfld 19746 TopOnctopon 20715 Cn ccn 21028 ×t ctx 21363 –cn→ccncf 22679 ↑𝑐ccxp 24302 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 df-sin 14800 df-cos 14801 df-tan 14802 df-pi 14803 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631 df-log 24303 df-cxp 24304 |
This theorem is referenced by: logdivsqrle 30728 |
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