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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 25797 | . 2 β’ ((π§ β (π΄ β {π΅}) β¦ (((πΉβπ§) β (πΉβπ΅)) / (π§ β π΅))) limβ π΅) β β | |
2 | eqid 2728 | . . . 4 β’ ((TopOpenββfld) βΎt π) = ((TopOpenββfld) βΎt π) | |
3 | eqid 2728 | . . . 4 β’ (TopOpenββfld) = (TopOpenββfld) | |
4 | eqid 2728 | . . . 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 25820 | . . 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 499 | . 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 3976 | 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 395 β wcel 2099 β cdif 3942 β wss 3945 {csn 4624 class class class wbr 5142 β¦ cmpt 5225 βΆwf 6538 βcfv 6542 (class class class)co 7414 βcc 11130 β cmin 11468 / cdiv 11895 βΎt crest 17395 TopOpenctopn 17396 βfldccnfld 21272 intcnt 22914 limβ climc 25784 D cdv 25785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-pm 8841 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fi 9428 df-sup 9459 df-inf 9460 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-q 12957 df-rp 13001 df-xneg 13118 df-xadd 13119 df-xmul 13120 df-fz 13511 df-seq 13993 df-exp 14053 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-struct 17109 df-slot 17144 df-ndx 17156 df-base 17174 df-plusg 17239 df-mulr 17240 df-starv 17241 df-tset 17245 df-ple 17246 df-ds 17248 df-unif 17249 df-rest 17397 df-topn 17398 df-topgen 17418 df-psmet 21264 df-xmet 21265 df-met 21266 df-bl 21267 df-mopn 21268 df-cnfld 21273 df-top 22789 df-topon 22806 df-topsp 22828 df-bases 22842 df-cnp 23125 df-xms 24219 df-ms 24220 df-limc 25788 df-dv 25789 |
This theorem is referenced by: perfdvf 25825 dvres 25833 dvres2 25834 dvcnp2 25842 dvcnp2OLD 25843 dvaddbr 25861 dvmulbr 25862 dvmulbrOLD 25863 dvcobr 25870 dvcobrOLD 25871 |
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