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Mirrors > Home > MPE Home > Th. List > dvf | Structured version Visualization version GIF version |
Description: The derivative is a function. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.) |
Ref | Expression |
---|---|
dvf | ⊢ (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reelprrecn 11142 | . 2 ⊢ ℝ ∈ {ℝ, ℂ} | |
2 | dvfg 25268 | . 2 ⊢ (ℝ ∈ {ℝ, ℂ} → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 {cpr 4588 dom cdm 5633 ⟶wf 6492 (class class class)co 7356 ℂcc 11048 ℝcr 11049 D cdv 25225 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 ax-pre-sup 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7802 df-1st 7920 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-er 8647 df-map 8766 df-pm 8767 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-fi 9346 df-sup 9377 df-inf 9378 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-div 11812 df-nn 12153 df-2 12215 df-3 12216 df-4 12217 df-5 12218 df-6 12219 df-7 12220 df-8 12221 df-9 12222 df-n0 12413 df-z 12499 df-dec 12618 df-uz 12763 df-q 12873 df-rp 12915 df-xneg 13032 df-xadd 13033 df-xmul 13034 df-icc 13270 df-fz 13424 df-seq 13906 df-exp 13967 df-cj 14983 df-re 14984 df-im 14985 df-sqrt 15119 df-abs 15120 df-struct 17018 df-slot 17053 df-ndx 17065 df-base 17083 df-plusg 17145 df-mulr 17146 df-starv 17147 df-tset 17151 df-ple 17152 df-ds 17154 df-unif 17155 df-rest 17303 df-topn 17304 df-topgen 17324 df-psmet 20786 df-xmet 20787 df-met 20788 df-bl 20789 df-mopn 20790 df-fbas 20791 df-fg 20792 df-cnfld 20795 df-top 22241 df-topon 22258 df-topsp 22280 df-bases 22294 df-cld 22368 df-ntr 22369 df-cls 22370 df-nei 22447 df-lp 22485 df-perf 22486 df-cnp 22577 df-haus 22664 df-fil 23195 df-fm 23287 df-flim 23288 df-flf 23289 df-xms 23671 df-ms 23672 df-limc 25228 df-dv 25229 |
This theorem is referenced by: dvcjbr 25311 dvcj 25312 dvfre 25313 dvferm1 25347 dvferm2 25349 rolle 25352 cmvth 25353 mvth 25354 dvlip 25355 c1liplem1 25358 dvivthlem2 25371 lhop1lem 25375 lhop1 25376 lhop2 25377 ftc1cn 25405 ftc1cnnc 36140 fperdvper 44131 fourierdlem80 44398 fouriersw 44443 |
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