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Mirrors > Home > MPE Home > Th. List > expcn | Structured version Visualization version GIF version |
Description: The power function on complex numbers, for fixed exponent 𝑁, is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
Ref | Expression |
---|---|
expcn.j | ⊢ 𝐽 = (TopOpen‘ℂfld) |
Ref | Expression |
---|---|
expcn | ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (𝐽 Cn 𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7226 | . . . 4 ⊢ (𝑛 = 0 → (𝑥↑𝑛) = (𝑥↑0)) | |
2 | 1 | mpteq2dv 5156 | . . 3 ⊢ (𝑛 = 0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑0))) |
3 | 2 | eleq1d 2822 | . 2 ⊢ (𝑛 = 0 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) ∈ (𝐽 Cn 𝐽) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑0)) ∈ (𝐽 Cn 𝐽))) |
4 | oveq2 7226 | . . . 4 ⊢ (𝑛 = 𝑘 → (𝑥↑𝑛) = (𝑥↑𝑘)) | |
5 | 4 | mpteq2dv 5156 | . . 3 ⊢ (𝑛 = 𝑘 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) |
6 | 5 | eleq1d 2822 | . 2 ⊢ (𝑛 = 𝑘 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) ∈ (𝐽 Cn 𝐽) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽))) |
7 | oveq2 7226 | . . . 4 ⊢ (𝑛 = (𝑘 + 1) → (𝑥↑𝑛) = (𝑥↑(𝑘 + 1))) | |
8 | 7 | mpteq2dv 5156 | . . 3 ⊢ (𝑛 = (𝑘 + 1) → (𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1)))) |
9 | 8 | eleq1d 2822 | . 2 ⊢ (𝑛 = (𝑘 + 1) → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) ∈ (𝐽 Cn 𝐽) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) ∈ (𝐽 Cn 𝐽))) |
10 | oveq2 7226 | . . . 4 ⊢ (𝑛 = 𝑁 → (𝑥↑𝑛) = (𝑥↑𝑁)) | |
11 | 10 | mpteq2dv 5156 | . . 3 ⊢ (𝑛 = 𝑁 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) |
12 | 11 | eleq1d 2822 | . 2 ⊢ (𝑛 = 𝑁 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) ∈ (𝐽 Cn 𝐽) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (𝐽 Cn 𝐽))) |
13 | exp0 13644 | . . . 4 ⊢ (𝑥 ∈ ℂ → (𝑥↑0) = 1) | |
14 | 13 | mpteq2ia 5151 | . . 3 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑0)) = (𝑥 ∈ ℂ ↦ 1) |
15 | expcn.j | . . . . . . 7 ⊢ 𝐽 = (TopOpen‘ℂfld) | |
16 | 15 | cnfldtopon 23685 | . . . . . 6 ⊢ 𝐽 ∈ (TopOn‘ℂ) |
17 | 16 | a1i 11 | . . . . 5 ⊢ (⊤ → 𝐽 ∈ (TopOn‘ℂ)) |
18 | 1cnd 10833 | . . . . 5 ⊢ (⊤ → 1 ∈ ℂ) | |
19 | 17, 17, 18 | cnmptc 22564 | . . . 4 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ 1) ∈ (𝐽 Cn 𝐽)) |
20 | 19 | mptru 1550 | . . 3 ⊢ (𝑥 ∈ ℂ ↦ 1) ∈ (𝐽 Cn 𝐽) |
21 | 14, 20 | eqeltri 2834 | . 2 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑0)) ∈ (𝐽 Cn 𝐽) |
22 | oveq1 7225 | . . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑥↑(𝑘 + 1)) = (𝑛↑(𝑘 + 1))) | |
23 | 22 | cbvmptv 5163 | . . . . 5 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) = (𝑛 ∈ ℂ ↦ (𝑛↑(𝑘 + 1))) |
24 | id 22 | . . . . . . 7 ⊢ (𝑛 ∈ ℂ → 𝑛 ∈ ℂ) | |
25 | simpl 486 | . . . . . . 7 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → 𝑘 ∈ ℕ0) | |
26 | expp1 13647 | . . . . . . 7 ⊢ ((𝑛 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑛↑(𝑘 + 1)) = ((𝑛↑𝑘) · 𝑛)) | |
27 | 24, 25, 26 | syl2anr 600 | . . . . . 6 ⊢ (((𝑘 ∈ ℕ0 ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) ∧ 𝑛 ∈ ℂ) → (𝑛↑(𝑘 + 1)) = ((𝑛↑𝑘) · 𝑛)) |
28 | 27 | mpteq2dva 5155 | . . . . 5 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → (𝑛 ∈ ℂ ↦ (𝑛↑(𝑘 + 1))) = (𝑛 ∈ ℂ ↦ ((𝑛↑𝑘) · 𝑛))) |
29 | 23, 28 | syl5eq 2790 | . . . 4 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) = (𝑛 ∈ ℂ ↦ ((𝑛↑𝑘) · 𝑛))) |
30 | 16 | a1i 11 | . . . . 5 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → 𝐽 ∈ (TopOn‘ℂ)) |
31 | oveq1 7225 | . . . . . . 7 ⊢ (𝑥 = 𝑛 → (𝑥↑𝑘) = (𝑛↑𝑘)) | |
32 | 31 | cbvmptv 5163 | . . . . . 6 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) = (𝑛 ∈ ℂ ↦ (𝑛↑𝑘)) |
33 | simpr 488 | . . . . . 6 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) | |
34 | 32, 33 | eqeltrrid 2843 | . . . . 5 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → (𝑛 ∈ ℂ ↦ (𝑛↑𝑘)) ∈ (𝐽 Cn 𝐽)) |
35 | 30 | cnmptid 22563 | . . . . 5 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → (𝑛 ∈ ℂ ↦ 𝑛) ∈ (𝐽 Cn 𝐽)) |
36 | 15 | mulcn 23769 | . . . . . 6 ⊢ · ∈ ((𝐽 ×t 𝐽) Cn 𝐽) |
37 | 36 | a1i 11 | . . . . 5 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → · ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) |
38 | 30, 34, 35, 37 | cnmpt12f 22568 | . . . 4 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → (𝑛 ∈ ℂ ↦ ((𝑛↑𝑘) · 𝑛)) ∈ (𝐽 Cn 𝐽)) |
39 | 29, 38 | eqeltrd 2838 | . . 3 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) ∈ (𝐽 Cn 𝐽)) |
40 | 39 | ex 416 | . 2 ⊢ (𝑘 ∈ ℕ0 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽) → (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) ∈ (𝐽 Cn 𝐽))) |
41 | 3, 6, 9, 12, 21, 40 | nn0ind 12277 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (𝐽 Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ⊤wtru 1544 ∈ wcel 2110 ↦ cmpt 5140 ‘cfv 6385 (class class class)co 7218 ℂcc 10732 0cc0 10734 1c1 10735 + caddc 10737 · cmul 10739 ℕ0cn0 12095 ↑cexp 13640 TopOpenctopn 16931 ℂfldccnfld 20368 TopOnctopon 21812 Cn ccn 22126 ×t ctx 22462 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5184 ax-sep 5197 ax-nul 5204 ax-pow 5263 ax-pr 5327 ax-un 7528 ax-cnex 10790 ax-resscn 10791 ax-1cn 10792 ax-icn 10793 ax-addcl 10794 ax-addrcl 10795 ax-mulcl 10796 ax-mulrcl 10797 ax-mulcom 10798 ax-addass 10799 ax-mulass 10800 ax-distr 10801 ax-i2m1 10802 ax-1ne0 10803 ax-1rid 10804 ax-rnegex 10805 ax-rrecex 10806 ax-cnre 10807 ax-pre-lttri 10808 ax-pre-lttrn 10809 ax-pre-ltadd 10810 ax-pre-mulgt0 10811 ax-pre-sup 10812 ax-mulf 10814 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3415 df-sbc 3700 df-csb 3817 df-dif 3874 df-un 3876 df-in 3878 df-ss 3888 df-pss 3890 df-nul 4243 df-if 4445 df-pw 4520 df-sn 4547 df-pr 4549 df-tp 4551 df-op 4553 df-uni 4825 df-int 4865 df-iun 4911 df-iin 4912 df-br 5059 df-opab 5121 df-mpt 5141 df-tr 5167 df-id 5460 df-eprel 5465 df-po 5473 df-so 5474 df-fr 5514 df-se 5515 df-we 5516 df-xp 5562 df-rel 5563 df-cnv 5564 df-co 5565 df-dm 5566 df-rn 5567 df-res 5568 df-ima 5569 df-pred 6165 df-ord 6221 df-on 6222 df-lim 6223 df-suc 6224 df-iota 6343 df-fun 6387 df-fn 6388 df-f 6389 df-f1 6390 df-fo 6391 df-f1o 6392 df-fv 6393 df-isom 6394 df-riota 7175 df-ov 7221 df-oprab 7222 df-mpo 7223 df-of 7474 df-om 7650 df-1st 7766 df-2nd 7767 df-supp 7909 df-wrecs 8052 df-recs 8113 df-rdg 8151 df-1o 8207 df-2o 8208 df-er 8396 df-map 8515 df-ixp 8584 df-en 8632 df-dom 8633 df-sdom 8634 df-fin 8635 df-fsupp 8991 df-fi 9032 df-sup 9063 df-inf 9064 df-oi 9131 df-card 9560 df-pnf 10874 df-mnf 10875 df-xr 10876 df-ltxr 10877 df-le 10878 df-sub 11069 df-neg 11070 df-div 11495 df-nn 11836 df-2 11898 df-3 11899 df-4 11900 df-5 11901 df-6 11902 df-7 11903 df-8 11904 df-9 11905 df-n0 12096 df-z 12182 df-dec 12299 df-uz 12444 df-q 12550 df-rp 12592 df-xneg 12709 df-xadd 12710 df-xmul 12711 df-icc 12947 df-fz 13101 df-fzo 13244 df-seq 13580 df-exp 13641 df-hash 13902 df-cj 14667 df-re 14668 df-im 14669 df-sqrt 14803 df-abs 14804 df-struct 16705 df-sets 16722 df-slot 16740 df-ndx 16750 df-base 16766 df-ress 16790 df-plusg 16820 df-mulr 16821 df-starv 16822 df-sca 16823 df-vsca 16824 df-ip 16825 df-tset 16826 df-ple 16827 df-ds 16829 df-unif 16830 df-hom 16831 df-cco 16832 df-rest 16932 df-topn 16933 df-0g 16951 df-gsum 16952 df-topgen 16953 df-pt 16954 df-prds 16957 df-xrs 17012 df-qtop 17017 df-imas 17018 df-xps 17020 df-mre 17094 df-mrc 17095 df-acs 17097 df-mgm 18119 df-sgrp 18168 df-mnd 18179 df-submnd 18224 df-mulg 18494 df-cntz 18716 df-cmn 19177 df-psmet 20360 df-xmet 20361 df-met 20362 df-bl 20363 df-mopn 20364 df-cnfld 20369 df-top 21796 df-topon 21813 df-topsp 21835 df-bases 21848 df-cn 22129 df-cnp 22130 df-tx 22464 df-hmeo 22657 df-xms 23223 df-ms 23224 df-tms 23225 |
This theorem is referenced by: sqcn 23776 expcncf 23828 plycn 25160 psercn2 25320 atansopn 25820 pntlem3 26495 climexp 42829 |
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