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Mirrors > Home > MPE Home > Th. List > dvnf | Structured version Visualization version Unicode version |
Description: The N-times derivative is a function. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
dvnf |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvnff 22889 |
. . . . 5
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2 | 1 | ffvelrnda 6027 |
. . . 4
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3 | 2 | 3impa 1204 |
. . 3
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4 | cnex 9625 |
. . . 4
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5 | simp1 1009 |
. . . . 5
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6 | simp2 1010 |
. . . . . . 7
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7 | elpm2g 7493 |
. . . . . . . 8
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8 | 4, 5, 7 | sylancr 670 |
. . . . . . 7
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9 | 6, 8 | mpbid 214 |
. . . . . 6
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10 | 9 | simprd 465 |
. . . . 5
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11 | 5, 10 | ssexd 4553 |
. . . 4
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12 | elpm2g 7493 |
. . . 4
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13 | 4, 11, 12 | sylancr 670 |
. . 3
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14 | 3, 13 | mpbid 214 |
. 2
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15 | 14 | simpld 461 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-8 1891 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-rep 4518 ax-sep 4528 ax-nul 4537 ax-pow 4584 ax-pr 4642 ax-un 6588 ax-inf2 8151 ax-cnex 9600 ax-resscn 9601 ax-1cn 9602 ax-icn 9603 ax-addcl 9604 ax-addrcl 9605 ax-mulcl 9606 ax-mulrcl 9607 ax-mulcom 9608 ax-addass 9609 ax-mulass 9610 ax-distr 9611 ax-i2m1 9612 ax-1ne0 9613 ax-1rid 9614 ax-rnegex 9615 ax-rrecex 9616 ax-cnre 9617 ax-pre-lttri 9618 ax-pre-lttrn 9619 ax-pre-ltadd 9620 ax-pre-mulgt0 9621 ax-pre-sup 9622 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 987 df-3an 988 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-eu 2305 df-mo 2306 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-nel 2627 df-ral 2744 df-rex 2745 df-reu 2746 df-rmo 2747 df-rab 2748 df-v 3049 df-sbc 3270 df-csb 3366 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-pss 3422 df-nul 3734 df-if 3884 df-pw 3955 df-sn 3971 df-pr 3973 df-tp 3975 df-op 3977 df-uni 4202 df-int 4238 df-iun 4283 df-iin 4284 df-br 4406 df-opab 4465 df-mpt 4466 df-tr 4501 df-eprel 4748 df-id 4752 df-po 4758 df-so 4759 df-fr 4796 df-we 4798 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-pred 5383 df-ord 5429 df-on 5430 df-lim 5431 df-suc 5432 df-iota 5549 df-fun 5587 df-fn 5588 df-f 5589 df-f1 5590 df-fo 5591 df-f1o 5592 df-fv 5593 df-riota 6257 df-ov 6298 df-oprab 6299 df-mpt2 6300 df-om 6698 df-1st 6798 df-2nd 6799 df-wrecs 7033 df-recs 7095 df-rdg 7133 df-1o 7187 df-oadd 7191 df-er 7368 df-map 7479 df-pm 7480 df-en 7575 df-dom 7576 df-sdom 7577 df-fin 7578 df-fi 7930 df-sup 7961 df-inf 7962 df-pnf 9682 df-mnf 9683 df-xr 9684 df-ltxr 9685 df-le 9686 df-sub 9867 df-neg 9868 df-div 10277 df-nn 10617 df-2 10675 df-3 10676 df-4 10677 df-5 10678 df-6 10679 df-7 10680 df-8 10681 df-9 10682 df-10 10683 df-n0 10877 df-z 10945 df-dec 11059 df-uz 11167 df-q 11272 df-rp 11310 df-xneg 11416 df-xadd 11417 df-xmul 11418 df-icc 11649 df-fz 11792 df-seq 12221 df-exp 12280 df-cj 13174 df-re 13175 df-im 13176 df-sqrt 13310 df-abs 13311 df-struct 15135 df-ndx 15136 df-slot 15137 df-base 15138 df-plusg 15215 df-mulr 15216 df-starv 15217 df-tset 15221 df-ple 15222 df-ds 15224 df-unif 15225 df-rest 15333 df-topn 15334 df-topgen 15354 df-psmet 18974 df-xmet 18975 df-met 18976 df-bl 18977 df-mopn 18978 df-fbas 18979 df-fg 18980 df-cnfld 18983 df-top 19933 df-bases 19934 df-topon 19935 df-topsp 19936 df-cld 20046 df-ntr 20047 df-cls 20048 df-nei 20126 df-lp 20164 df-perf 20165 df-cnp 20256 df-haus 20343 df-fil 20873 df-fm 20965 df-flim 20966 df-flf 20967 df-xms 21347 df-ms 21348 df-limc 22833 df-dv 22834 df-dvn 22835 |
This theorem is referenced by: dvn2bss 22896 dvnres 22897 cpnord 22901 taylfvallem1 23324 tayl0 23329 taylply2 23335 taylply 23336 dvtaylp 23337 dvntaylp 23338 dvntaylp0 23339 taylthlem1 23340 taylthlem2 23341 |
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