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Mirrors > Home > MPE Home > Th. List > dvres | Structured version Visualization version Unicode version |
Description: Restriction of a
derivative. Note that our definition of derivative
df-dv 22822 would still make sense if we demanded that
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Ref | Expression |
---|---|
dvres.k |
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dvres.t |
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Ref | Expression |
---|---|
dvres |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reldv 22825 |
. 2
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2 | relres 5132 |
. 2
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3 | simpll 760 |
. . . . . 6
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4 | simplr 762 |
. . . . . . . 8
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5 | inss1 3652 |
. . . . . . . 8
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6 | fssres 5749 |
. . . . . . . 8
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7 | 4, 5, 6 | sylancl 668 |
. . . . . . 7
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8 | resres 5117 |
. . . . . . . . 9
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9 | ffn 5728 |
. . . . . . . . . . 11
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10 | fnresdm 5685 |
. . . . . . . . . . 11
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11 | 4, 9, 10 | 3syl 18 |
. . . . . . . . . 10
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12 | 11 | reseq1d 5104 |
. . . . . . . . 9
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13 | 8, 12 | syl5eqr 2499 |
. . . . . . . 8
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14 | 13 | feq1d 5714 |
. . . . . . 7
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15 | 7, 14 | mpbid 214 |
. . . . . 6
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16 | simprl 764 |
. . . . . . 7
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17 | 5, 16 | syl5ss 3443 |
. . . . . 6
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18 | 3, 15, 17 | dvcl 22854 |
. . . . 5
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19 | 18 | ex 436 |
. . . 4
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20 | 3, 4, 16 | dvcl 22854 |
. . . . . 6
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21 | 20 | ex 436 |
. . . . 5
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22 | 21 | adantrd 470 |
. . . 4
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23 | dvres.k |
. . . . . 6
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24 | dvres.t |
. . . . . 6
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25 | eqid 2451 |
. . . . . 6
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26 | 3 | adantr 467 |
. . . . . 6
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27 | 4 | adantr 467 |
. . . . . 6
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28 | 16 | adantr 467 |
. . . . . 6
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29 | simplrr 771 |
. . . . . 6
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30 | simpr 463 |
. . . . . 6
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31 | 23, 24, 25, 26, 27, 28, 29, 30 | dvreslem 22864 |
. . . . 5
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32 | 31 | ex 436 |
. . . 4
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33 | 19, 22, 32 | pm5.21ndd 356 |
. . 3
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34 | vex 3048 |
. . . 4
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35 | 34 | brres 5111 |
. . 3
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36 | 33, 35 | syl6bbr 267 |
. 2
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37 | 1, 2, 36 | eqbrrdiv 4933 |
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 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-rep 4515 ax-sep 4525 ax-nul 4534 ax-pow 4581 ax-pr 4639 ax-un 6583 ax-cnex 9595 ax-resscn 9596 ax-1cn 9597 ax-icn 9598 ax-addcl 9599 ax-addrcl 9600 ax-mulcl 9601 ax-mulrcl 9602 ax-mulcom 9603 ax-addass 9604 ax-mulass 9605 ax-distr 9606 ax-i2m1 9607 ax-1ne0 9608 ax-1rid 9609 ax-rnegex 9610 ax-rrecex 9611 ax-cnre 9612 ax-pre-lttri 9613 ax-pre-lttrn 9614 ax-pre-ltadd 9615 ax-pre-mulgt0 9616 ax-pre-sup 9617 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 986 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-nel 2625 df-ral 2742 df-rex 2743 df-reu 2744 df-rmo 2745 df-rab 2746 df-v 3047 df-sbc 3268 df-csb 3364 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-pss 3420 df-nul 3732 df-if 3882 df-pw 3953 df-sn 3969 df-pr 3971 df-tp 3973 df-op 3975 df-uni 4199 df-int 4235 df-iun 4280 df-iin 4281 df-br 4403 df-opab 4462 df-mpt 4463 df-tr 4498 df-eprel 4745 df-id 4749 df-po 4755 df-so 4756 df-fr 4793 df-we 4795 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-rn 4845 df-res 4846 df-ima 4847 df-pred 5380 df-ord 5426 df-on 5427 df-lim 5428 df-suc 5429 df-iota 5546 df-fun 5584 df-fn 5585 df-f 5586 df-f1 5587 df-fo 5588 df-f1o 5589 df-fv 5590 df-riota 6252 df-ov 6293 df-oprab 6294 df-mpt2 6295 df-om 6693 df-1st 6793 df-2nd 6794 df-wrecs 7028 df-recs 7090 df-rdg 7128 df-1o 7182 df-oadd 7186 df-er 7363 df-map 7474 df-pm 7475 df-en 7570 df-dom 7571 df-sdom 7572 df-fin 7573 df-fi 7925 df-sup 7956 df-inf 7957 df-pnf 9677 df-mnf 9678 df-xr 9679 df-ltxr 9680 df-le 9681 df-sub 9862 df-neg 9863 df-div 10270 df-nn 10610 df-2 10668 df-3 10669 df-4 10670 df-5 10671 df-6 10672 df-7 10673 df-8 10674 df-9 10675 df-10 10676 df-n0 10870 df-z 10938 df-dec 11052 df-uz 11160 df-q 11265 df-rp 11303 df-xneg 11409 df-xadd 11410 df-xmul 11411 df-fz 11785 df-seq 12214 df-exp 12273 df-cj 13162 df-re 13163 df-im 13164 df-sqrt 13298 df-abs 13299 df-struct 15123 df-ndx 15124 df-slot 15125 df-base 15126 df-plusg 15203 df-mulr 15204 df-starv 15205 df-tset 15209 df-ple 15210 df-ds 15212 df-unif 15213 df-rest 15321 df-topn 15322 df-topgen 15342 df-psmet 18962 df-xmet 18963 df-met 18964 df-bl 18965 df-mopn 18966 df-cnfld 18971 df-top 19921 df-bases 19922 df-topon 19923 df-topsp 19924 df-cld 20034 df-ntr 20035 df-cls 20036 df-cnp 20244 df-xms 21335 df-ms 21336 df-limc 22821 df-dv 22822 |
This theorem is referenced by: dvcmulf 22899 dvmptres2 22916 dvmptntr 22925 dvlip 22945 dvlipcn 22946 dvlip2 22947 c1liplem1 22948 dvgt0lem1 22954 dvne0 22963 lhop1lem 22965 lhop 22968 dvcnvrelem1 22969 dvcvx 22972 ftc2ditglem 22997 pserdv 23384 efcvx 23404 dvlog 23596 dvlog2 23598 dvresntr 37788 dvmptresicc 37791 dvresioo 37793 dvbdfbdioolem1 37800 itgcoscmulx 37846 itgiccshift 37857 itgperiod 37858 dirkercncflem2 37966 fourierdlem57 38027 fourierdlem58 38028 fourierdlem72 38042 fourierdlem73 38043 fourierdlem74 38044 fourierdlem75 38045 fourierdlem80 38050 fourierdlem94 38064 fourierdlem103 38073 fourierdlem104 38074 fourierdlem113 38083 |
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