Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > expcnfg | Structured version Visualization version GIF version |
Description: If 𝐹 is a complex continuous function and N is a fixed number, then F^N is continuous too. A generalization of expcncf 23823. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Ref | Expression |
---|---|
expcnfg.1 | ⊢ Ⅎ𝑥𝐹 |
expcnfg.2 | ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→ℂ)) |
expcnfg.3 | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
Ref | Expression |
---|---|
expcnfg | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)↑𝑁)) ∈ (𝐴–cn→ℂ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑡((𝐹‘𝑥)↑𝑁) | |
2 | expcnfg.1 | . . . . . . 7 ⊢ Ⅎ𝑥𝐹 | |
3 | nfcv 2904 | . . . . . . 7 ⊢ Ⅎ𝑥𝑡 | |
4 | 2, 3 | nffv 6727 | . . . . . 6 ⊢ Ⅎ𝑥(𝐹‘𝑡) |
5 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑥↑ | |
6 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑥𝑁 | |
7 | 4, 5, 6 | nfov 7243 | . . . . 5 ⊢ Ⅎ𝑥((𝐹‘𝑡)↑𝑁) |
8 | fveq2 6717 | . . . . . 6 ⊢ (𝑥 = 𝑡 → (𝐹‘𝑥) = (𝐹‘𝑡)) | |
9 | 8 | oveq1d 7228 | . . . . 5 ⊢ (𝑥 = 𝑡 → ((𝐹‘𝑥)↑𝑁) = ((𝐹‘𝑡)↑𝑁)) |
10 | 1, 7, 9 | cbvmpt 5156 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)↑𝑁)) = (𝑡 ∈ 𝐴 ↦ ((𝐹‘𝑡)↑𝑁)) |
11 | expcnfg.2 | . . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→ℂ)) | |
12 | cncff 23790 | . . . . . . . . 9 ⊢ (𝐹 ∈ (𝐴–cn→ℂ) → 𝐹:𝐴⟶ℂ) | |
13 | 11, 12 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
14 | 13 | ffvelrnda 6904 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐴) → (𝐹‘𝑡) ∈ ℂ) |
15 | expcnfg.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
16 | 15 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐴) → 𝑁 ∈ ℕ0) |
17 | 14, 16 | expcld 13716 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐴) → ((𝐹‘𝑡)↑𝑁) ∈ ℂ) |
18 | oveq1 7220 | . . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑡) → (𝑥↑𝑁) = ((𝐹‘𝑡)↑𝑁)) | |
19 | eqid 2737 | . . . . . . . 8 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) | |
20 | 4, 7, 18, 19 | fvmptf 6839 | . . . . . . 7 ⊢ (((𝐹‘𝑡) ∈ ℂ ∧ ((𝐹‘𝑡)↑𝑁) ∈ ℂ) → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑡)) = ((𝐹‘𝑡)↑𝑁)) |
21 | 14, 17, 20 | syl2anc 587 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐴) → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑡)) = ((𝐹‘𝑡)↑𝑁)) |
22 | 21 | eqcomd 2743 | . . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐴) → ((𝐹‘𝑡)↑𝑁) = ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑡))) |
23 | 22 | mpteq2dva 5150 | . . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝐴 ↦ ((𝐹‘𝑡)↑𝑁)) = (𝑡 ∈ 𝐴 ↦ ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑡)))) |
24 | 10, 23 | syl5eq 2790 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)↑𝑁)) = (𝑡 ∈ 𝐴 ↦ ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑡)))) |
25 | simpr 488 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ) | |
26 | 15 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝑁 ∈ ℕ0) |
27 | 25, 26 | expcld 13716 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑥↑𝑁) ∈ ℂ) |
28 | 27 | fmpttd 6932 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)):ℂ⟶ℂ) |
29 | fcompt 6948 | . . . 4 ⊢ (((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)):ℂ⟶ℂ ∧ 𝐹:𝐴⟶ℂ) → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) = (𝑡 ∈ 𝐴 ↦ ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑡)))) | |
30 | 28, 13, 29 | syl2anc 587 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) = (𝑡 ∈ 𝐴 ↦ ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑡)))) |
31 | 24, 30 | eqtr4d 2780 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)↑𝑁)) = ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹)) |
32 | expcncf 23823 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (ℂ–cn→ℂ)) | |
33 | 15, 32 | syl 17 | . . 3 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∈ (ℂ–cn→ℂ)) |
34 | 11, 33 | cncfco 23804 | . 2 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) ∈ (𝐴–cn→ℂ)) |
35 | 31, 34 | eqeltrd 2838 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)↑𝑁)) ∈ (𝐴–cn→ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 Ⅎwnfc 2884 ↦ cmpt 5135 ∘ ccom 5555 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 ℂcc 10727 ℕ0cn0 12090 ↑cexp 13635 –cn→ccncf 23773 |
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 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 ax-mulf 10809 |
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 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-er 8391 df-map 8510 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-fi 9027 df-sup 9058 df-inf 9059 df-oi 9126 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-q 12545 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-icc 12942 df-fz 13096 df-fzo 13239 df-seq 13575 df-exp 13636 df-hash 13897 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-starv 16817 df-sca 16818 df-vsca 16819 df-ip 16820 df-tset 16821 df-ple 16822 df-ds 16824 df-unif 16825 df-hom 16826 df-cco 16827 df-rest 16927 df-topn 16928 df-0g 16946 df-gsum 16947 df-topgen 16948 df-pt 16949 df-prds 16952 df-xrs 17007 df-qtop 17012 df-imas 17013 df-xps 17015 df-mre 17089 df-mrc 17090 df-acs 17092 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-mulg 18489 df-cntz 18711 df-cmn 19172 df-psmet 20355 df-xmet 20356 df-met 20357 df-bl 20358 df-mopn 20359 df-cnfld 20364 df-top 21791 df-topon 21808 df-topsp 21830 df-bases 21843 df-cn 22124 df-cnp 22125 df-tx 22459 df-hmeo 22652 df-xms 23218 df-ms 23219 df-tms 23220 df-cncf 23775 |
This theorem is referenced by: ibliccsinexp 43167 itgsinexplem1 43170 itgsinexp 43171 |
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