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 25888. (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 5943 | . . 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 2740 | . . . . . . . 8 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
5 | 4 | cnfldtopon 23936 | . . . . . . 7 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
6 | difss 4071 | . . . . . . 7 ⊢ (ℂ ∖ (-∞(,]0)) ⊆ ℂ | |
7 | resttopon 22302 | . . . . . . 7 ⊢ (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ (ℂ ∖ (-∞(,]0)) ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t (ℂ ∖ (-∞(,]0))) ∈ (TopOn‘(ℂ ∖ (-∞(,]0)))) | |
8 | 5, 6, 7 | mp2an 689 | . . . . . 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 22802 | . . . . 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 22803 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ 𝐴) ∈ (((TopOpen‘ℂfld) ↾t (ℂ ∖ (-∞(,]0))) Cn (TopOpen‘ℂfld))) |
14 | eqid 2740 | . . . . . . 7 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0)) | |
15 | eqid 2740 | . . . . . . 7 ⊢ ((TopOpen‘ℂfld) ↾t (ℂ ∖ (-∞(,]0))) = ((TopOpen‘ℂfld) ↾t (ℂ ∖ (-∞(,]0))) | |
16 | 14, 4, 15 | cxpcn 25888 | . . . . . 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 7278 | . . . . 5 ⊢ ((𝑦 = 𝑥 ∧ 𝑧 = 𝐴) → (𝑦↑𝑐𝑧) = (𝑥↑𝑐𝐴)) | |
19 | 9, 10, 13, 9, 11, 17, 18 | cnmpt12 22808 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ (𝑥↑𝑐𝐴)) ∈ (((TopOpen‘ℂfld) ↾t (ℂ ∖ (-∞(,]0))) Cn (TopOpen‘ℂfld))) |
20 | ssid 3948 | . . . . . . 7 ⊢ ℂ ⊆ ℂ | |
21 | 5 | toponrestid 22060 | . . . . . . . 8 ⊢ (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ) |
22 | 4, 15, 21 | cncfcn 24063 | . . . . . . 7 ⊢ (((ℂ ∖ (-∞(,]0)) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((ℂ ∖ (-∞(,]0))–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (ℂ ∖ (-∞(,]0))) Cn (TopOpen‘ℂfld))) |
23 | 6, 20, 22 | mp2an 689 | . . . . . 6 ⊢ ((ℂ ∖ (-∞(,]0))–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (ℂ ∖ (-∞(,]0))) Cn (TopOpen‘ℂfld)) |
24 | 23 | eqcomi 2749 | . . . . 5 ⊢ (((TopOpen‘ℂfld) ↾t (ℂ ∖ (-∞(,]0))) Cn (TopOpen‘ℂfld)) = ((ℂ ∖ (-∞(,]0))–cn→ℂ) |
25 | 24 | a1i 11 | . . . 4 ⊢ (𝜑 → (((TopOpen‘ℂfld) ↾t (ℂ ∖ (-∞(,]0))) Cn (TopOpen‘ℂfld)) = ((ℂ ∖ (-∞(,]0))–cn→ℂ)) |
26 | 19, 25 | eleqtrd 2843 | . . 3 ⊢ (𝜑 → (𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ (𝑥↑𝑐𝐴)) ∈ ((ℂ ∖ (-∞(,]0))–cn→ℂ)) |
27 | rescncf 24050 | . . . 4 ⊢ (𝐷 ⊆ (ℂ ∖ (-∞(,]0)) → ((𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ (𝑥↑𝑐𝐴)) ∈ ((ℂ ∖ (-∞(,]0))–cn→ℂ) → ((𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ (𝑥↑𝑐𝐴)) ↾ 𝐷) ∈ (𝐷–cn→ℂ))) | |
28 | 27 | imp 407 | . . 3 ⊢ ((𝐷 ⊆ (ℂ ∖ (-∞(,]0)) ∧ (𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ (𝑥↑𝑐𝐴)) ∈ ((ℂ ∖ (-∞(,]0))–cn→ℂ)) → ((𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ (𝑥↑𝑐𝐴)) ↾ 𝐷) ∈ (𝐷–cn→ℂ)) |
29 | 1, 26, 28 | syl2anc 584 | . 2 ⊢ (𝜑 → ((𝑥 ∈ (ℂ ∖ (-∞(,]0)) ↦ (𝑥↑𝑐𝐴)) ↾ 𝐷) ∈ (𝐷–cn→ℂ)) |
30 | 3, 29 | eqeltrrd 2842 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (𝑥↑𝑐𝐴)) ∈ (𝐷–cn→ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 ∖ cdif 3889 ⊆ wss 3892 ↦ cmpt 5162 ↾ cres 5591 ‘cfv 6431 (class class class)co 7269 ∈ cmpo 7271 ℂcc 10862 0cc0 10864 -∞cmnf 11000 (,]cioc 13071 ↾t crest 17121 TopOpenctopn 17122 ℂfldccnfld 20587 TopOnctopon 22049 Cn ccn 22365 ×t ctx 22701 –cn→ccncf 24029 ↑𝑐ccxp 25701 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-inf2 9369 ax-cnex 10920 ax-resscn 10921 ax-1cn 10922 ax-icn 10923 ax-addcl 10924 ax-addrcl 10925 ax-mulcl 10926 ax-mulrcl 10927 ax-mulcom 10928 ax-addass 10929 ax-mulass 10930 ax-distr 10931 ax-i2m1 10932 ax-1ne0 10933 ax-1rid 10934 ax-rnegex 10935 ax-rrecex 10936 ax-cnre 10937 ax-pre-lttri 10938 ax-pre-lttrn 10939 ax-pre-ltadd 10940 ax-pre-mulgt0 10941 ax-pre-sup 10942 ax-addf 10943 ax-mulf 10944 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-isom 6440 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-of 7525 df-om 7702 df-1st 7818 df-2nd 7819 df-supp 7963 df-frecs 8082 df-wrecs 8113 df-recs 8187 df-rdg 8226 df-1o 8282 df-2o 8283 df-er 8473 df-map 8592 df-pm 8593 df-ixp 8661 df-en 8709 df-dom 8710 df-sdom 8711 df-fin 8712 df-fsupp 9099 df-fi 9140 df-sup 9171 df-inf 9172 df-oi 9239 df-card 9690 df-pnf 11004 df-mnf 11005 df-xr 11006 df-ltxr 11007 df-le 11008 df-sub 11199 df-neg 11200 df-div 11625 df-nn 11966 df-2 12028 df-3 12029 df-4 12030 df-5 12031 df-6 12032 df-7 12033 df-8 12034 df-9 12035 df-n0 12226 df-z 12312 df-dec 12429 df-uz 12574 df-q 12680 df-rp 12722 df-xneg 12839 df-xadd 12840 df-xmul 12841 df-ioo 13074 df-ioc 13075 df-ico 13076 df-icc 13077 df-fz 13231 df-fzo 13374 df-fl 13502 df-mod 13580 df-seq 13712 df-exp 13773 df-fac 13978 df-bc 14007 df-hash 14035 df-shft 14768 df-cj 14800 df-re 14801 df-im 14802 df-sqrt 14936 df-abs 14937 df-limsup 15170 df-clim 15187 df-rlim 15188 df-sum 15388 df-ef 15767 df-sin 15769 df-cos 15770 df-tan 15771 df-pi 15772 df-struct 16838 df-sets 16855 df-slot 16873 df-ndx 16885 df-base 16903 df-ress 16932 df-plusg 16965 df-mulr 16966 df-starv 16967 df-sca 16968 df-vsca 16969 df-ip 16970 df-tset 16971 df-ple 16972 df-ds 16974 df-unif 16975 df-hom 16976 df-cco 16977 df-rest 17123 df-topn 17124 df-0g 17142 df-gsum 17143 df-topgen 17144 df-pt 17145 df-prds 17148 df-xrs 17203 df-qtop 17208 df-imas 17209 df-xps 17211 df-mre 17285 df-mrc 17286 df-acs 17288 df-mgm 18316 df-sgrp 18365 df-mnd 18376 df-submnd 18421 df-mulg 18691 df-cntz 18913 df-cmn 19378 df-psmet 20579 df-xmet 20580 df-met 20581 df-bl 20582 df-mopn 20583 df-fbas 20584 df-fg 20585 df-cnfld 20588 df-top 22033 df-topon 22050 df-topsp 22072 df-bases 22086 df-cld 22160 df-ntr 22161 df-cls 22162 df-nei 22239 df-lp 22277 df-perf 22278 df-cn 22368 df-cnp 22369 df-haus 22456 df-cmp 22528 df-tx 22703 df-hmeo 22896 df-fil 22987 df-fm 23079 df-flim 23080 df-flf 23081 df-xms 23463 df-ms 23464 df-tms 23465 df-cncf 24031 df-limc 25020 df-dv 25021 df-log 25702 df-cxp 25703 |
This theorem is referenced by: logdivsqrle 32618 |
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