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 23530. (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 2977 | . . . . 5 ⊢ Ⅎ𝑡((𝐹‘𝑥)↑𝑁) | |
2 | expcnfg.1 | . . . . . . 7 ⊢ Ⅎ𝑥𝐹 | |
3 | nfcv 2977 | . . . . . . 7 ⊢ Ⅎ𝑥𝑡 | |
4 | 2, 3 | nffv 6680 | . . . . . 6 ⊢ Ⅎ𝑥(𝐹‘𝑡) |
5 | nfcv 2977 | . . . . . 6 ⊢ Ⅎ𝑥↑ | |
6 | nfcv 2977 | . . . . . 6 ⊢ Ⅎ𝑥𝑁 | |
7 | 4, 5, 6 | nfov 7186 | . . . . 5 ⊢ Ⅎ𝑥((𝐹‘𝑡)↑𝑁) |
8 | fveq2 6670 | . . . . . 6 ⊢ (𝑥 = 𝑡 → (𝐹‘𝑥) = (𝐹‘𝑡)) | |
9 | 8 | oveq1d 7171 | . . . . 5 ⊢ (𝑥 = 𝑡 → ((𝐹‘𝑥)↑𝑁) = ((𝐹‘𝑡)↑𝑁)) |
10 | 1, 7, 9 | cbvmpt 5167 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)↑𝑁)) = (𝑡 ∈ 𝐴 ↦ ((𝐹‘𝑡)↑𝑁)) |
11 | expcnfg.2 | . . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→ℂ)) | |
12 | cncff 23501 | . . . . . . . . 9 ⊢ (𝐹 ∈ (𝐴–cn→ℂ) → 𝐹:𝐴⟶ℂ) | |
13 | 11, 12 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
14 | 13 | ffvelrnda 6851 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐴) → (𝐹‘𝑡) ∈ ℂ) |
15 | expcnfg.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
16 | 15 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐴) → 𝑁 ∈ ℕ0) |
17 | 14, 16 | expcld 13511 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐴) → ((𝐹‘𝑡)↑𝑁) ∈ ℂ) |
18 | oveq1 7163 | . . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑡) → (𝑥↑𝑁) = ((𝐹‘𝑡)↑𝑁)) | |
19 | eqid 2821 | . . . . . . . 8 ⊢ (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) = (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) | |
20 | 4, 7, 18, 19 | fvmptf 6789 | . . . . . . 7 ⊢ (((𝐹‘𝑡) ∈ ℂ ∧ ((𝐹‘𝑡)↑𝑁) ∈ ℂ) → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑡)) = ((𝐹‘𝑡)↑𝑁)) |
21 | 14, 17, 20 | syl2anc 586 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐴) → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑡)) = ((𝐹‘𝑡)↑𝑁)) |
22 | 21 | eqcomd 2827 | . . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐴) → ((𝐹‘𝑡)↑𝑁) = ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑡))) |
23 | 22 | mpteq2dva 5161 | . . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝐴 ↦ ((𝐹‘𝑡)↑𝑁)) = (𝑡 ∈ 𝐴 ↦ ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑡)))) |
24 | 10, 23 | syl5eq 2868 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)↑𝑁)) = (𝑡 ∈ 𝐴 ↦ ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑡)))) |
25 | simpr 487 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ) | |
26 | 15 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝑁 ∈ ℕ0) |
27 | 25, 26 | expcld 13511 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑥↑𝑁) ∈ ℂ) |
28 | 27 | fmpttd 6879 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (𝑥↑𝑁)):ℂ⟶ℂ) |
29 | fcompt 6895 | . . . 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 586 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) = (𝑡 ∈ 𝐴 ↦ ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁))‘(𝐹‘𝑡)))) |
31 | 24, 30 | eqtr4d 2859 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)↑𝑁)) = ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹)) |
32 | expcncf 23530 | . . . 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 23515 | . 2 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥↑𝑁)) ∘ 𝐹) ∈ (𝐴–cn→ℂ)) |
35 | 31, 34 | eqeltrd 2913 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)↑𝑁)) ∈ (𝐴–cn→ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Ⅎwnfc 2961 ↦ cmpt 5146 ∘ ccom 5559 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℕ0cn0 11898 ↑cexp 13430 –cn→ccncf 23484 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-mulf 10617 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-icc 12746 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cn 21835 df-cnp 21836 df-tx 22170 df-hmeo 22363 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 |
This theorem is referenced by: ibliccsinexp 42256 itgsinexplem1 42259 itgsinexp 42260 |
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