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Mirrors > Home > MPE Home > Th. List > dvlt0 | Structured version Unicode version |
Description: A function on a closed interval with negative derivative is decreasing. (Contributed by Mario Carneiro, 19-Feb-2015.) |
Ref | Expression |
---|---|
dvgt0.a |
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dvgt0.b |
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dvgt0.f |
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dvlt0.d |
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Ref | Expression |
---|---|
dvlt0 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvgt0.a |
. 2
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2 | dvgt0.b |
. 2
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3 | dvgt0.f |
. 2
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4 | dvlt0.d |
. 2
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5 | gtso 9570 |
. 2
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6 | 1, 2, 3, 4 | dvgt0lem1 21610 |
. . . . . . . . 9
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7 | eliooord 11469 |
. . . . . . . . 9
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8 | 6, 7 | syl 16 |
. . . . . . . 8
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9 | 8 | simprd 463 |
. . . . . . 7
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10 | cncff 20604 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | 3, 10 | syl 16 |
. . . . . . . . . . 11
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12 | 11 | ad2antrr 725 |
. . . . . . . . . 10
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13 | simplrr 760 |
. . . . . . . . . 10
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14 | 12, 13 | ffvelrnd 5956 |
. . . . . . . . 9
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15 | simplrl 759 |
. . . . . . . . . 10
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16 | 12, 15 | ffvelrnd 5956 |
. . . . . . . . 9
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17 | 14, 16 | resubcld 9890 |
. . . . . . . 8
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18 | 0red 9501 |
. . . . . . . 8
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19 | iccssre 11491 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | 1, 2, 19 | syl2anc 661 |
. . . . . . . . . . 11
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21 | 20 | ad2antrr 725 |
. . . . . . . . . 10
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22 | 21, 13 | sseldd 3468 |
. . . . . . . . 9
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23 | 21, 15 | sseldd 3468 |
. . . . . . . . 9
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24 | 22, 23 | resubcld 9890 |
. . . . . . . 8
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25 | simpr 461 |
. . . . . . . . 9
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26 | 23, 22 | posdifd 10040 |
. . . . . . . . 9
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27 | 25, 26 | mpbid 210 |
. . . . . . . 8
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28 | ltdivmul 10318 |
. . . . . . . 8
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29 | 17, 18, 24, 27, 28 | syl112anc 1223 |
. . . . . . 7
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30 | 9, 29 | mpbid 210 |
. . . . . 6
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31 | 24 | recnd 9526 |
. . . . . . 7
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32 | 31 | mul01d 9682 |
. . . . . 6
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33 | 30, 32 | breqtrd 4427 |
. . . . 5
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34 | 14, 16, 18 | ltsubaddd 10049 |
. . . . 5
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35 | 33, 34 | mpbid 210 |
. . . 4
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36 | 16 | recnd 9526 |
. . . . 5
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37 | 36 | addid2d 9684 |
. . . 4
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38 | 35, 37 | breqtrd 4427 |
. . 3
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39 | fvex 5812 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
40 | fvex 5812 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
41 | 39, 40 | brcnv 5133 |
. . 3
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42 | 38, 41 | sylibr 212 |
. 2
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43 | 1, 2, 3, 4, 5, 42 | dvgt0lem2 21611 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-rep 4514 ax-sep 4524 ax-nul 4532 ax-pow 4581 ax-pr 4642 ax-un 6485 ax-inf2 7961 ax-cnex 9452 ax-resscn 9453 ax-1cn 9454 ax-icn 9455 ax-addcl 9456 ax-addrcl 9457 ax-mulcl 9458 ax-mulrcl 9459 ax-mulcom 9460 ax-addass 9461 ax-mulass 9462 ax-distr 9463 ax-i2m1 9464 ax-1ne0 9465 ax-1rid 9466 ax-rnegex 9467 ax-rrecex 9468 ax-cnre 9469 ax-pre-lttri 9470 ax-pre-lttrn 9471 ax-pre-ltadd 9472 ax-pre-mulgt0 9473 ax-pre-sup 9474 ax-addf 9475 ax-mulf 9476 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-nel 2651 df-ral 2804 df-rex 2805 df-reu 2806 df-rmo 2807 df-rab 2808 df-v 3080 df-sbc 3295 df-csb 3399 df-dif 3442 df-un 3444 df-in 3446 df-ss 3453 df-pss 3455 df-nul 3749 df-if 3903 df-pw 3973 df-sn 3989 df-pr 3991 df-tp 3993 df-op 3995 df-uni 4203 df-int 4240 df-iun 4284 df-iin 4285 df-br 4404 df-opab 4462 df-mpt 4463 df-tr 4497 df-eprel 4743 df-id 4747 df-po 4752 df-so 4753 df-fr 4790 df-se 4791 df-we 4792 df-ord 4833 df-on 4834 df-lim 4835 df-suc 4836 df-xp 4957 df-rel 4958 df-cnv 4959 df-co 4960 df-dm 4961 df-rn 4962 df-res 4963 df-ima 4964 df-iota 5492 df-fun 5531 df-fn 5532 df-f 5533 df-f1 5534 df-fo 5535 df-f1o 5536 df-fv 5537 df-isom 5538 df-riota 6164 df-ov 6206 df-oprab 6207 df-mpt2 6208 df-of 6433 df-om 6590 df-1st 6690 df-2nd 6691 df-supp 6804 df-recs 6945 df-rdg 6979 df-1o 7033 df-2o 7034 df-oadd 7037 df-er 7214 df-map 7329 df-pm 7330 df-ixp 7377 df-en 7424 df-dom 7425 df-sdom 7426 df-fin 7427 df-fsupp 7735 df-fi 7775 df-sup 7805 df-oi 7838 df-card 8223 df-cda 8451 df-pnf 9534 df-mnf 9535 df-xr 9536 df-ltxr 9537 df-le 9538 df-sub 9711 df-neg 9712 df-div 10108 df-nn 10437 df-2 10494 df-3 10495 df-4 10496 df-5 10497 df-6 10498 df-7 10499 df-8 10500 df-9 10501 df-10 10502 df-n0 10694 df-z 10761 df-dec 10870 df-uz 10976 df-q 11068 df-rp 11106 df-xneg 11203 df-xadd 11204 df-xmul 11205 df-ioo 11418 df-ico 11420 df-icc 11421 df-fz 11558 df-fzo 11669 df-seq 11927 df-exp 11986 df-hash 12224 df-cj 12709 df-re 12710 df-im 12711 df-sqr 12845 df-abs 12846 df-struct 14297 df-ndx 14298 df-slot 14299 df-base 14300 df-sets 14301 df-ress 14302 df-plusg 14373 df-mulr 14374 df-starv 14375 df-sca 14376 df-vsca 14377 df-ip 14378 df-tset 14379 df-ple 14380 df-ds 14382 df-unif 14383 df-hom 14384 df-cco 14385 df-rest 14483 df-topn 14484 df-0g 14502 df-gsum 14503 df-topgen 14504 df-pt 14505 df-prds 14508 df-xrs 14562 df-qtop 14567 df-imas 14568 df-xps 14570 df-mre 14646 df-mrc 14647 df-acs 14649 df-mnd 15537 df-submnd 15587 df-mulg 15670 df-cntz 15957 df-cmn 16403 df-psmet 17937 df-xmet 17938 df-met 17939 df-bl 17940 df-mopn 17941 df-fbas 17942 df-fg 17943 df-cnfld 17947 df-top 18638 df-bases 18640 df-topon 18641 df-topsp 18642 df-cld 18758 df-ntr 18759 df-cls 18760 df-nei 18837 df-lp 18875 df-perf 18876 df-cn 18966 df-cnp 18967 df-haus 19054 df-cmp 19125 df-tx 19270 df-hmeo 19463 df-fil 19554 df-fm 19646 df-flim 19647 df-flf 19648 df-xms 20030 df-ms 20031 df-tms 20032 df-cncf 20589 df-limc 21477 df-dv 21478 |
This theorem is referenced by: dvne0 21619 |
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