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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 24289. (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 2907 | . . . . 5 ⊢ Ⅎ𝑡((𝐹‘𝑥)↑𝑁) | |
2 | expcnfg.1 | . . . . . . 7 ⊢ Ⅎ𝑥𝐹 | |
3 | nfcv 2907 | . . . . . . 7 ⊢ Ⅎ𝑥𝑡 | |
4 | 2, 3 | nffv 6852 | . . . . . 6 ⊢ Ⅎ𝑥(𝐹‘𝑡) |
5 | nfcv 2907 | . . . . . 6 ⊢ Ⅎ𝑥↑ | |
6 | nfcv 2907 | . . . . . 6 ⊢ Ⅎ𝑥𝑁 | |
7 | 4, 5, 6 | nfov 7387 | . . . . 5 ⊢ Ⅎ𝑥((𝐹‘𝑡)↑𝑁) |
8 | fveq2 6842 | . . . . . 6 ⊢ (𝑥 = 𝑡 → (𝐹‘𝑥) = (𝐹‘𝑡)) | |
9 | 8 | oveq1d 7372 | . . . . 5 ⊢ (𝑥 = 𝑡 → ((𝐹‘𝑥)↑𝑁) = ((𝐹‘𝑡)↑𝑁)) |
10 | 1, 7, 9 | cbvmpt 5216 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)↑𝑁)) = (𝑡 ∈ 𝐴 ↦ ((𝐹‘𝑡)↑𝑁)) |
11 | expcnfg.2 | . . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→ℂ)) | |
12 | cncff 24256 | . . . . . . . . 9 ⊢ (𝐹 ∈ (𝐴–cn→ℂ) → 𝐹:𝐴⟶ℂ) | |
13 | 11, 12 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
14 | 13 | ffvelcdmda 7035 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐴) → (𝐹‘𝑡) ∈ ℂ) |
15 | expcnfg.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
16 | 15 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐴) → 𝑁 ∈ ℕ0) |
17 | 14, 16 | expcld 14051 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐴) → ((𝐹‘𝑡)↑𝑁) ∈ ℂ) |
18 | oveq1 7364 | . . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑡) → (𝑥↑𝑁) = ((𝐹‘𝑡)↑𝑁)) | |
19 | eqid 2736 | . . . . . . . 8 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) | |
20 | 4, 7, 18, 19 | fvmptf 6969 | . . . . . . 7 ⊢ (((𝐹‘𝑡) ∈ ℂ ∧ ((𝐹‘𝑡)↑𝑁) ∈ ℂ) → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑡)) = ((𝐹‘𝑡)↑𝑁)) |
21 | 14, 17, 20 | syl2anc 584 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐴) → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑡)) = ((𝐹‘𝑡)↑𝑁)) |
22 | 21 | eqcomd 2742 | . . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐴) → ((𝐹‘𝑡)↑𝑁) = ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑡))) |
23 | 22 | mpteq2dva 5205 | . . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝐴 ↦ ((𝐹‘𝑡)↑𝑁)) = (𝑡 ∈ 𝐴 ↦ ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑡)))) |
24 | 10, 23 | eqtrid 2788 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)↑𝑁)) = (𝑡 ∈ 𝐴 ↦ ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑡)))) |
25 | simpr 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ) | |
26 | 15 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝑁 ∈ ℕ0) |
27 | 25, 26 | expcld 14051 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑥↑𝑁) ∈ ℂ) |
28 | 27 | fmpttd 7063 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)):ℂ⟶ℂ) |
29 | fcompt 7079 | . . . 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 584 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) = (𝑡 ∈ 𝐴 ↦ ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑡)))) |
31 | 24, 30 | eqtr4d 2779 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)↑𝑁)) = ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹)) |
32 | expcncf 24289 | . . . 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 24270 | . 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 396 = wceq 1541 ∈ wcel 2106 Ⅎwnfc 2887 ↦ cmpt 5188 ∘ ccom 5637 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 ℕ0cn0 12413 ↑cexp 13967 –cn→ccncf 24239 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-mulf 11131 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-icc 13271 df-fz 13425 df-fzo 13568 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cn 22578 df-cnp 22579 df-tx 22913 df-hmeo 23106 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241 |
This theorem is referenced by: ibliccsinexp 44182 itgsinexplem1 44185 itgsinexp 44186 |
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