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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvmulcncf | Structured version Visualization version GIF version |
Description: A sufficient condition for the derivative of a product to be continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
dvmulcncf.s | β’ (π β π β {β, β}) |
dvmulcncf.f | β’ (π β πΉ:πβΆβ) |
dvmulcncf.g | β’ (π β πΊ:πβΆβ) |
dvmulcncf.fdv | β’ (π β (π D πΉ) β (πβcnββ)) |
dvmulcncf.gdv | β’ (π β (π D πΊ) β (πβcnββ)) |
Ref | Expression |
---|---|
dvmulcncf | β’ (π β (π D (πΉ βf Β· πΊ)) β (πβcnββ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvmulcncf.s | . . 3 β’ (π β π β {β, β}) | |
2 | dvmulcncf.f | . . 3 β’ (π β πΉ:πβΆβ) | |
3 | dvmulcncf.g | . . 3 β’ (π β πΊ:πβΆβ) | |
4 | dvmulcncf.fdv | . . . 4 β’ (π β (π D πΉ) β (πβcnββ)) | |
5 | cncff 24279 | . . . 4 β’ ((π D πΉ) β (πβcnββ) β (π D πΉ):πβΆβ) | |
6 | fdm 6681 | . . . 4 β’ ((π D πΉ):πβΆβ β dom (π D πΉ) = π) | |
7 | 4, 5, 6 | 3syl 18 | . . 3 β’ (π β dom (π D πΉ) = π) |
8 | dvmulcncf.gdv | . . . 4 β’ (π β (π D πΊ) β (πβcnββ)) | |
9 | cncff 24279 | . . . 4 β’ ((π D πΊ) β (πβcnββ) β (π D πΊ):πβΆβ) | |
10 | fdm 6681 | . . . 4 β’ ((π D πΊ):πβΆβ β dom (π D πΊ) = π) | |
11 | 8, 9, 10 | 3syl 18 | . . 3 β’ (π β dom (π D πΊ) = π) |
12 | 1, 2, 3, 7, 11 | dvmulf 25330 | . 2 β’ (π β (π D (πΉ βf Β· πΊ)) = (((π D πΉ) βf Β· πΊ) βf + ((π D πΊ) βf Β· πΉ))) |
13 | ax-resscn 11116 | . . . . . . . 8 β’ β β β | |
14 | sseq1 3973 | . . . . . . . 8 β’ (π = β β (π β β β β β β)) | |
15 | 13, 14 | mpbiri 258 | . . . . . . 7 β’ (π = β β π β β) |
16 | eqimss 4004 | . . . . . . 7 β’ (π = β β π β β) | |
17 | 15, 16 | pm3.2i 472 | . . . . . 6 β’ ((π = β β π β β) β§ (π = β β π β β)) |
18 | elpri 4612 | . . . . . . 7 β’ (π β {β, β} β (π = β β¨ π = β)) | |
19 | 1, 18 | syl 17 | . . . . . 6 β’ (π β (π = β β¨ π = β)) |
20 | pm3.44 959 | . . . . . 6 β’ (((π = β β π β β) β§ (π = β β π β β)) β ((π = β β¨ π = β) β π β β)) | |
21 | 17, 19, 20 | mpsyl 68 | . . . . 5 β’ (π β π β β) |
22 | dvbsss 25289 | . . . . . 6 β’ dom (π D πΉ) β π | |
23 | 7, 22 | eqsstrrdi 4003 | . . . . 5 β’ (π β π β π) |
24 | dvcn 25308 | . . . . 5 β’ (((π β β β§ πΊ:πβΆβ β§ π β π) β§ dom (π D πΊ) = π) β πΊ β (πβcnββ)) | |
25 | 21, 3, 23, 11, 24 | syl31anc 1374 | . . . 4 β’ (π β πΊ β (πβcnββ)) |
26 | 4, 25 | mulcncff 44201 | . . 3 β’ (π β ((π D πΉ) βf Β· πΊ) β (πβcnββ)) |
27 | dvcn 25308 | . . . . 5 β’ (((π β β β§ πΉ:πβΆβ β§ π β π) β§ dom (π D πΉ) = π) β πΉ β (πβcnββ)) | |
28 | 21, 2, 23, 7, 27 | syl31anc 1374 | . . . 4 β’ (π β πΉ β (πβcnββ)) |
29 | 8, 28 | mulcncff 44201 | . . 3 β’ (π β ((π D πΊ) βf Β· πΉ) β (πβcnββ)) |
30 | 26, 29 | addcncff 44215 | . 2 β’ (π β (((π D πΉ) βf Β· πΊ) βf + ((π D πΊ) βf Β· πΉ)) β (πβcnββ)) |
31 | 12, 30 | eqeltrd 2834 | 1 β’ (π β (π D (πΉ βf Β· πΊ)) β (πβcnββ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β¨ wo 846 = wceq 1542 β wcel 2107 β wss 3914 {cpr 4592 dom cdm 5637 βΆwf 6496 (class class class)co 7361 βf cof 7619 βcc 11057 βcr 11058 + caddc 11062 Β· cmul 11064 βcnβccncf 24262 D cdv 25250 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 ax-addf 11138 ax-mulf 11139 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-om 7807 df-1st 7925 df-2nd 7926 df-supp 8097 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-2o 8417 df-er 8654 df-map 8773 df-pm 8774 df-ixp 8842 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fsupp 9312 df-fi 9355 df-sup 9386 df-inf 9387 df-oi 9454 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-dec 12627 df-uz 12772 df-q 12882 df-rp 12924 df-xneg 13041 df-xadd 13042 df-xmul 13043 df-icc 13280 df-fz 13434 df-fzo 13577 df-seq 13916 df-exp 13977 df-hash 14240 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-starv 17156 df-sca 17157 df-vsca 17158 df-ip 17159 df-tset 17160 df-ple 17161 df-ds 17163 df-unif 17164 df-hom 17165 df-cco 17166 df-rest 17312 df-topn 17313 df-0g 17331 df-gsum 17332 df-topgen 17333 df-pt 17334 df-prds 17337 df-xrs 17392 df-qtop 17397 df-imas 17398 df-xps 17400 df-mre 17474 df-mrc 17475 df-acs 17477 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-submnd 18610 df-mulg 18881 df-cntz 19105 df-cmn 19572 df-psmet 20811 df-xmet 20812 df-met 20813 df-bl 20814 df-mopn 20815 df-fbas 20816 df-fg 20817 df-cnfld 20820 df-top 22266 df-topon 22283 df-topsp 22305 df-bases 22319 df-cld 22393 df-ntr 22394 df-cls 22395 df-nei 22472 df-lp 22510 df-perf 22511 df-cn 22601 df-cnp 22602 df-haus 22689 df-tx 22936 df-hmeo 23129 df-fil 23220 df-fm 23312 df-flim 23313 df-flf 23314 df-xms 23696 df-ms 23697 df-tms 23698 df-cncf 24264 df-limc 25253 df-dv 25254 |
This theorem is referenced by: fourierdlem72 44509 |
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