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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 7166 | . . . 4 ⊢ (𝑛 = 0 → (𝑥↑𝑛) = (𝑥↑0)) | |
2 | 1 | mpteq2dv 5164 | . . 3 ⊢ (𝑛 = 0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑0))) |
3 | 2 | eleq1d 2899 | . 2 ⊢ (𝑛 = 0 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) ∈ (𝐽 Cn 𝐽) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑0)) ∈ (𝐽 Cn 𝐽))) |
4 | oveq2 7166 | . . . 4 ⊢ (𝑛 = 𝑘 → (𝑥↑𝑛) = (𝑥↑𝑘)) | |
5 | 4 | mpteq2dv 5164 | . . 3 ⊢ (𝑛 = 𝑘 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑘))) |
6 | 5 | eleq1d 2899 | . 2 ⊢ (𝑛 = 𝑘 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) ∈ (𝐽 Cn 𝐽) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽))) |
7 | oveq2 7166 | . . . 4 ⊢ (𝑛 = (𝑘 + 1) → (𝑥↑𝑛) = (𝑥↑(𝑘 + 1))) | |
8 | 7 | mpteq2dv 5164 | . . 3 ⊢ (𝑛 = (𝑘 + 1) → (𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1)))) |
9 | 8 | eleq1d 2899 | . 2 ⊢ (𝑛 = (𝑘 + 1) → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) ∈ (𝐽 Cn 𝐽) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) ∈ (𝐽 Cn 𝐽))) |
10 | oveq2 7166 | . . . 4 ⊢ (𝑛 = 𝑁 → (𝑥↑𝑛) = (𝑥↑𝑁)) | |
11 | 10 | mpteq2dv 5164 | . . 3 ⊢ (𝑛 = 𝑁 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑁))) |
12 | 11 | eleq1d 2899 | . 2 ⊢ (𝑛 = 𝑁 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑛)) ∈ (𝐽 Cn 𝐽) ↔ (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (𝐽 Cn 𝐽))) |
13 | exp0 13436 | . . . 4 ⊢ (𝑥 ∈ ℂ → (𝑥↑0) = 1) | |
14 | 13 | mpteq2ia 5159 | . . 3 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑0)) = (𝑥 ∈ ℂ ↦ 1) |
15 | expcn.j | . . . . . . 7 ⊢ 𝐽 = (TopOpen‘ℂfld) | |
16 | 15 | cnfldtopon 23393 | . . . . . 6 ⊢ 𝐽 ∈ (TopOn‘ℂ) |
17 | 16 | a1i 11 | . . . . 5 ⊢ (⊤ → 𝐽 ∈ (TopOn‘ℂ)) |
18 | 1cnd 10638 | . . . . 5 ⊢ (⊤ → 1 ∈ ℂ) | |
19 | 17, 17, 18 | cnmptc 22272 | . . . 4 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ 1) ∈ (𝐽 Cn 𝐽)) |
20 | 19 | mptru 1544 | . . 3 ⊢ (𝑥 ∈ ℂ ↦ 1) ∈ (𝐽 Cn 𝐽) |
21 | 14, 20 | eqeltri 2911 | . 2 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑0)) ∈ (𝐽 Cn 𝐽) |
22 | oveq1 7165 | . . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑥↑(𝑘 + 1)) = (𝑛↑(𝑘 + 1))) | |
23 | 22 | cbvmptv 5171 | . . . . 5 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) = (𝑛 ∈ ℂ ↦ (𝑛↑(𝑘 + 1))) |
24 | id 22 | . . . . . . 7 ⊢ (𝑛 ∈ ℂ → 𝑛 ∈ ℂ) | |
25 | simpl 485 | . . . . . . 7 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → 𝑘 ∈ ℕ0) | |
26 | expp1 13439 | . . . . . . 7 ⊢ ((𝑛 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑛↑(𝑘 + 1)) = ((𝑛↑𝑘) · 𝑛)) | |
27 | 24, 25, 26 | syl2anr 598 | . . . . . 6 ⊢ (((𝑘 ∈ ℕ0 ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) ∧ 𝑛 ∈ ℂ) → (𝑛↑(𝑘 + 1)) = ((𝑛↑𝑘) · 𝑛)) |
28 | 27 | mpteq2dva 5163 | . . . . 5 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → (𝑛 ∈ ℂ ↦ (𝑛↑(𝑘 + 1))) = (𝑛 ∈ ℂ ↦ ((𝑛↑𝑘) · 𝑛))) |
29 | 23, 28 | syl5eq 2870 | . . . 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 7165 | . . . . . . 7 ⊢ (𝑥 = 𝑛 → (𝑥↑𝑘) = (𝑛↑𝑘)) | |
32 | 31 | cbvmptv 5171 | . . . . . 6 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) = (𝑛 ∈ ℂ ↦ (𝑛↑𝑘)) |
33 | simpr 487 | . . . . . 6 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) | |
34 | 32, 33 | eqeltrrid 2920 | . . . . 5 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → (𝑛 ∈ ℂ ↦ (𝑛↑𝑘)) ∈ (𝐽 Cn 𝐽)) |
35 | 30 | cnmptid 22271 | . . . . 5 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → (𝑛 ∈ ℂ ↦ 𝑛) ∈ (𝐽 Cn 𝐽)) |
36 | 15 | mulcn 23477 | . . . . . 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 22276 | . . . 4 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → (𝑛 ∈ ℂ ↦ ((𝑛↑𝑘) · 𝑛)) ∈ (𝐽 Cn 𝐽)) |
39 | 29, 38 | eqeltrd 2915 | . . 3 ⊢ ((𝑘 ∈ ℕ0 ∧ (𝑥 ∈ ℂ ↦ (𝑥↑𝑘)) ∈ (𝐽 Cn 𝐽)) → (𝑥 ∈ ℂ ↦ (𝑥↑(𝑘 + 1))) ∈ (𝐽 Cn 𝐽)) |
40 | 39 | ex 415 | . 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 12080 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (𝐽 Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 ↦ cmpt 5148 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 0cc0 10539 1c1 10540 + caddc 10542 · cmul 10544 ℕ0cn0 11900 ↑cexp 13432 TopOpenctopn 16697 ℂfldccnfld 20547 TopOnctopon 21520 Cn ccn 21834 ×t ctx 22170 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-icc 12748 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-xrs 16777 df-qtop 16782 df-imas 16783 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-mulg 18227 df-cntz 18449 df-cmn 18910 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cn 21837 df-cnp 21838 df-tx 22172 df-hmeo 22365 df-xms 22932 df-ms 22933 df-tms 22934 |
This theorem is referenced by: sqcn 23484 expcncf 23532 plycn 24853 psercn2 25013 atansopn 25512 pntlem3 26187 climexp 41893 |
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