![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > expcnfg | Structured version Visualization version GIF version |
Description: If πΉ is a complex continuous function and N is a fixed number, then F^N is continuous too. A generalization of expcncf 24865. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Ref | Expression |
---|---|
expcnfg.1 | β’ β²π₯πΉ |
expcnfg.2 | β’ (π β πΉ β (π΄βcnββ)) |
expcnfg.3 | β’ (π β π β β0) |
Ref | Expression |
---|---|
expcnfg | β’ (π β (π₯ β π΄ β¦ ((πΉβπ₯)βπ)) β (π΄βcnββ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2898 | . . . . 5 β’ β²π‘((πΉβπ₯)βπ) | |
2 | expcnfg.1 | . . . . . . 7 β’ β²π₯πΉ | |
3 | nfcv 2898 | . . . . . . 7 β’ β²π₯π‘ | |
4 | 2, 3 | nffv 6910 | . . . . . 6 β’ β²π₯(πΉβπ‘) |
5 | nfcv 2898 | . . . . . 6 β’ β²π₯β | |
6 | nfcv 2898 | . . . . . 6 β’ β²π₯π | |
7 | 4, 5, 6 | nfov 7454 | . . . . 5 β’ β²π₯((πΉβπ‘)βπ) |
8 | fveq2 6900 | . . . . . 6 β’ (π₯ = π‘ β (πΉβπ₯) = (πΉβπ‘)) | |
9 | 8 | oveq1d 7439 | . . . . 5 β’ (π₯ = π‘ β ((πΉβπ₯)βπ) = ((πΉβπ‘)βπ)) |
10 | 1, 7, 9 | cbvmpt 5261 | . . . 4 β’ (π₯ β π΄ β¦ ((πΉβπ₯)βπ)) = (π‘ β π΄ β¦ ((πΉβπ‘)βπ)) |
11 | expcnfg.2 | . . . . . . . . 9 β’ (π β πΉ β (π΄βcnββ)) | |
12 | cncff 24831 | . . . . . . . . 9 β’ (πΉ β (π΄βcnββ) β πΉ:π΄βΆβ) | |
13 | 11, 12 | syl 17 | . . . . . . . 8 β’ (π β πΉ:π΄βΆβ) |
14 | 13 | ffvelcdmda 7097 | . . . . . . 7 β’ ((π β§ π‘ β π΄) β (πΉβπ‘) β β) |
15 | expcnfg.3 | . . . . . . . . 9 β’ (π β π β β0) | |
16 | 15 | adantr 479 | . . . . . . . 8 β’ ((π β§ π‘ β π΄) β π β β0) |
17 | 14, 16 | expcld 14148 | . . . . . . 7 β’ ((π β§ π‘ β π΄) β ((πΉβπ‘)βπ) β β) |
18 | oveq1 7431 | . . . . . . . 8 β’ (π₯ = (πΉβπ‘) β (π₯βπ) = ((πΉβπ‘)βπ)) | |
19 | eqid 2727 | . . . . . . . 8 β’ (π₯ β β β¦ (π₯βπ)) = (π₯ β β β¦ (π₯βπ)) | |
20 | 4, 7, 18, 19 | fvmptf 7029 | . . . . . . 7 β’ (((πΉβπ‘) β β β§ ((πΉβπ‘)βπ) β β) β ((π₯ β β β¦ (π₯βπ))β(πΉβπ‘)) = ((πΉβπ‘)βπ)) |
21 | 14, 17, 20 | syl2anc 582 | . . . . . 6 β’ ((π β§ π‘ β π΄) β ((π₯ β β β¦ (π₯βπ))β(πΉβπ‘)) = ((πΉβπ‘)βπ)) |
22 | 21 | eqcomd 2733 | . . . . 5 β’ ((π β§ π‘ β π΄) β ((πΉβπ‘)βπ) = ((π₯ β β β¦ (π₯βπ))β(πΉβπ‘))) |
23 | 22 | mpteq2dva 5250 | . . . 4 β’ (π β (π‘ β π΄ β¦ ((πΉβπ‘)βπ)) = (π‘ β π΄ β¦ ((π₯ β β β¦ (π₯βπ))β(πΉβπ‘)))) |
24 | 10, 23 | eqtrid 2779 | . . 3 β’ (π β (π₯ β π΄ β¦ ((πΉβπ₯)βπ)) = (π‘ β π΄ β¦ ((π₯ β β β¦ (π₯βπ))β(πΉβπ‘)))) |
25 | simpr 483 | . . . . . 6 β’ ((π β§ π₯ β β) β π₯ β β) | |
26 | 15 | adantr 479 | . . . . . 6 β’ ((π β§ π₯ β β) β π β β0) |
27 | 25, 26 | expcld 14148 | . . . . 5 β’ ((π β§ π₯ β β) β (π₯βπ) β β) |
28 | 27 | fmpttd 7128 | . . . 4 β’ (π β (π₯ β β β¦ (π₯βπ)):ββΆβ) |
29 | fcompt 7146 | . . . 4 β’ (((π₯ β β β¦ (π₯βπ)):ββΆβ β§ πΉ:π΄βΆβ) β ((π₯ β β β¦ (π₯βπ)) β πΉ) = (π‘ β π΄ β¦ ((π₯ β β β¦ (π₯βπ))β(πΉβπ‘)))) | |
30 | 28, 13, 29 | syl2anc 582 | . . 3 β’ (π β ((π₯ β β β¦ (π₯βπ)) β πΉ) = (π‘ β π΄ β¦ ((π₯ β β β¦ (π₯βπ))β(πΉβπ‘)))) |
31 | 24, 30 | eqtr4d 2770 | . 2 β’ (π β (π₯ β π΄ β¦ ((πΉβπ₯)βπ)) = ((π₯ β β β¦ (π₯βπ)) β πΉ)) |
32 | expcncf 24865 | . . . 4 β’ (π β β0 β (π₯ β β β¦ (π₯βπ)) β (ββcnββ)) | |
33 | 15, 32 | syl 17 | . . 3 β’ (π β (π₯ β β β¦ (π₯βπ)) β (ββcnββ)) |
34 | 11, 33 | cncfco 24845 | . 2 β’ (π β ((π₯ β β β¦ (π₯βπ)) β πΉ) β (π΄βcnββ)) |
35 | 31, 34 | eqeltrd 2828 | 1 β’ (π β (π₯ β π΄ β¦ ((πΉβπ₯)βπ)) β (π΄βcnββ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β²wnfc 2878 β¦ cmpt 5233 β ccom 5684 βΆwf 6547 βcfv 6551 (class class class)co 7424 βcc 11142 β0cn0 12508 βcexp 14064 βcnβccncf 24814 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-pre-sup 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-iin 5001 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-se 5636 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-isom 6560 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7689 df-om 7875 df-1st 7997 df-2nd 7998 df-supp 8170 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-2o 8492 df-er 8729 df-map 8851 df-ixp 8921 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-fsupp 9392 df-fi 9440 df-sup 9471 df-inf 9472 df-oi 9539 df-card 9968 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-div 11908 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-n0 12509 df-z 12595 df-dec 12714 df-uz 12859 df-q 12969 df-rp 13013 df-xneg 13130 df-xadd 13131 df-xmul 13132 df-icc 13369 df-fz 13523 df-fzo 13666 df-seq 14005 df-exp 14065 df-hash 14328 df-cj 15084 df-re 15085 df-im 15086 df-sqrt 15220 df-abs 15221 df-struct 17121 df-sets 17138 df-slot 17156 df-ndx 17168 df-base 17186 df-ress 17215 df-plusg 17251 df-mulr 17252 df-starv 17253 df-sca 17254 df-vsca 17255 df-ip 17256 df-tset 17257 df-ple 17258 df-ds 17260 df-unif 17261 df-hom 17262 df-cco 17263 df-rest 17409 df-topn 17410 df-0g 17428 df-gsum 17429 df-topgen 17430 df-pt 17431 df-prds 17434 df-xrs 17489 df-qtop 17494 df-imas 17495 df-xps 17497 df-mre 17571 df-mrc 17572 df-acs 17574 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-submnd 18746 df-mulg 19029 df-cntz 19273 df-cmn 19742 df-psmet 21276 df-xmet 21277 df-met 21278 df-bl 21279 df-mopn 21280 df-cnfld 21285 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22867 df-cn 23149 df-cnp 23150 df-tx 23484 df-hmeo 23677 df-xms 24244 df-ms 24245 df-tms 24246 df-cncf 24816 |
This theorem is referenced by: ibliccsinexp 45341 itgsinexplem1 45344 itgsinexp 45345 |
Copyright terms: Public domain | W3C validator |