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Mirrors > Home > MPE Home > Th. List > dvcl | Structured version Visualization version GIF version |
Description: The derivative function takes values in the complex numbers. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.) |
Ref | Expression |
---|---|
dvcl.s | β’ (π β π β β) |
dvcl.f | β’ (π β πΉ:π΄βΆβ) |
dvcl.a | β’ (π β π΄ β π) |
Ref | Expression |
---|---|
dvcl | β’ ((π β§ π΅(π D πΉ)πΆ) β πΆ β β) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limccl 25255 | . 2 β’ ((π§ β (π΄ β {π΅}) β¦ (((πΉβπ§) β (πΉβπ΅)) / (π§ β π΅))) limβ π΅) β β | |
2 | eqid 2737 | . . . 4 β’ ((TopOpenββfld) βΎt π) = ((TopOpenββfld) βΎt π) | |
3 | eqid 2737 | . . . 4 β’ (TopOpenββfld) = (TopOpenββfld) | |
4 | eqid 2737 | . . . 4 β’ (π§ β (π΄ β {π΅}) β¦ (((πΉβπ§) β (πΉβπ΅)) / (π§ β π΅))) = (π§ β (π΄ β {π΅}) β¦ (((πΉβπ§) β (πΉβπ΅)) / (π§ β π΅))) | |
5 | dvcl.s | . . . 4 β’ (π β π β β) | |
6 | dvcl.f | . . . 4 β’ (π β πΉ:π΄βΆβ) | |
7 | dvcl.a | . . . 4 β’ (π β π΄ β π) | |
8 | 2, 3, 4, 5, 6, 7 | eldv 25278 | . . 3 β’ (π β (π΅(π D πΉ)πΆ β (π΅ β ((intβ((TopOpenββfld) βΎt π))βπ΄) β§ πΆ β ((π§ β (π΄ β {π΅}) β¦ (((πΉβπ§) β (πΉβπ΅)) / (π§ β π΅))) limβ π΅)))) |
9 | 8 | simplbda 501 | . 2 β’ ((π β§ π΅(π D πΉ)πΆ) β πΆ β ((π§ β (π΄ β {π΅}) β¦ (((πΉβπ§) β (πΉβπ΅)) / (π§ β π΅))) limβ π΅)) |
10 | 1, 9 | sselid 3947 | 1 β’ ((π β§ π΅(π D πΉ)πΆ) β πΆ β β) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β wcel 2107 β cdif 3912 β wss 3915 {csn 4591 class class class wbr 5110 β¦ cmpt 5193 βΆwf 6497 βcfv 6501 (class class class)co 7362 βcc 11056 β cmin 11392 / cdiv 11819 βΎt crest 17309 TopOpenctopn 17310 βfldccnfld 20812 intcnt 22384 limβ climc 25242 D cdv 25243 |
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 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-map 8774 df-pm 8775 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fi 9354 df-sup 9385 df-inf 9386 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-q 12881 df-rp 12923 df-xneg 13040 df-xadd 13041 df-xmul 13042 df-fz 13432 df-seq 13914 df-exp 13975 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-struct 17026 df-slot 17061 df-ndx 17073 df-base 17091 df-plusg 17153 df-mulr 17154 df-starv 17155 df-tset 17159 df-ple 17160 df-ds 17162 df-unif 17163 df-rest 17311 df-topn 17312 df-topgen 17332 df-psmet 20804 df-xmet 20805 df-met 20806 df-bl 20807 df-mopn 20808 df-cnfld 20813 df-top 22259 df-topon 22276 df-topsp 22298 df-bases 22312 df-cnp 22595 df-xms 23689 df-ms 23690 df-limc 25246 df-dv 25247 |
This theorem is referenced by: perfdvf 25283 dvres 25291 dvres2 25292 dvcnp2 25300 dvaddbr 25318 dvmulbr 25319 dvcobr 25326 |
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