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Mirrors > Home > MPE Home > Th. List > dvlt0 | Structured version Visualization version GIF version |
Description: A function on a closed interval with negative derivative is decreasing. (Contributed by Mario Carneiro, 19-Feb-2015.) |
Ref | Expression |
---|---|
dvgt0.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
dvgt0.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
dvgt0.f | ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ)) |
dvlt0.d | ⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶(-∞(,)0)) |
Ref | Expression |
---|---|
dvlt0 | ⊢ (𝜑 → 𝐹 Isom < , ◡ < ((𝐴[,]𝐵), ran 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvgt0.a | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | dvgt0.b | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | dvgt0.f | . 2 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ)) | |
4 | dvlt0.d | . 2 ⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶(-∞(,)0)) | |
5 | gtso 10460 | . 2 ⊢ ◡ < Or ℝ | |
6 | 1, 2, 3, 4 | dvgt0lem1 24206 | . . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥)) ∈ (-∞(,)0)) |
7 | eliooord 12549 | . . . . . . . . 9 ⊢ ((((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥)) ∈ (-∞(,)0) → (-∞ < (((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥)) ∧ (((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥)) < 0)) | |
8 | 6, 7 | syl 17 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (-∞ < (((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥)) ∧ (((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥)) < 0)) |
9 | 8 | simprd 491 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥)) < 0) |
10 | cncff 23108 | . . . . . . . . . . . 12 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ) | |
11 | 3, 10 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℝ) |
12 | 11 | ad2antrr 716 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝐹:(𝐴[,]𝐵)⟶ℝ) |
13 | simplrr 768 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑦 ∈ (𝐴[,]𝐵)) | |
14 | 12, 13 | ffvelrnd 6626 | . . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐹‘𝑦) ∈ ℝ) |
15 | simplrl 767 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑥 ∈ (𝐴[,]𝐵)) | |
16 | 12, 15 | ffvelrnd 6626 | . . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐹‘𝑥) ∈ ℝ) |
17 | 14, 16 | resubcld 10805 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹‘𝑦) − (𝐹‘𝑥)) ∈ ℝ) |
18 | 0red 10382 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 0 ∈ ℝ) | |
19 | iccssre 12571 | . . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
20 | 1, 2, 19 | syl2anc 579 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
21 | 20 | ad2antrr 716 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐴[,]𝐵) ⊆ ℝ) |
22 | 21, 13 | sseldd 3822 | . . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℝ) |
23 | 21, 15 | sseldd 3822 | . . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑥 ∈ ℝ) |
24 | 22, 23 | resubcld 10805 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑦 − 𝑥) ∈ ℝ) |
25 | simpr 479 | . . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 𝑥 < 𝑦) | |
26 | 23, 22 | posdifd 10964 | . . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑥 < 𝑦 ↔ 0 < (𝑦 − 𝑥))) |
27 | 25, 26 | mpbid 224 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → 0 < (𝑦 − 𝑥)) |
28 | ltdivmul 11254 | . . . . . . . 8 ⊢ ((((𝐹‘𝑦) − (𝐹‘𝑥)) ∈ ℝ ∧ 0 ∈ ℝ ∧ ((𝑦 − 𝑥) ∈ ℝ ∧ 0 < (𝑦 − 𝑥))) → ((((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥)) < 0 ↔ ((𝐹‘𝑦) − (𝐹‘𝑥)) < ((𝑦 − 𝑥) · 0))) | |
29 | 17, 18, 24, 27, 28 | syl112anc 1442 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((((𝐹‘𝑦) − (𝐹‘𝑥)) / (𝑦 − 𝑥)) < 0 ↔ ((𝐹‘𝑦) − (𝐹‘𝑥)) < ((𝑦 − 𝑥) · 0))) |
30 | 9, 29 | mpbid 224 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹‘𝑦) − (𝐹‘𝑥)) < ((𝑦 − 𝑥) · 0)) |
31 | 24 | recnd 10407 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝑦 − 𝑥) ∈ ℂ) |
32 | 31 | mul01d 10577 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝑦 − 𝑥) · 0) = 0) |
33 | 30, 32 | breqtrd 4914 | . . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → ((𝐹‘𝑦) − (𝐹‘𝑥)) < 0) |
34 | 14, 16, 18 | ltsubaddd 10973 | . . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (((𝐹‘𝑦) − (𝐹‘𝑥)) < 0 ↔ (𝐹‘𝑦) < (0 + (𝐹‘𝑥)))) |
35 | 33, 34 | mpbid 224 | . . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐹‘𝑦) < (0 + (𝐹‘𝑥))) |
36 | 16 | recnd 10407 | . . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐹‘𝑥) ∈ ℂ) |
37 | 36 | addid2d 10579 | . . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (0 + (𝐹‘𝑥)) = (𝐹‘𝑥)) |
38 | 35, 37 | breqtrd 4914 | . . 3 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐹‘𝑦) < (𝐹‘𝑥)) |
39 | fvex 6461 | . . . 4 ⊢ (𝐹‘𝑥) ∈ V | |
40 | fvex 6461 | . . . 4 ⊢ (𝐹‘𝑦) ∈ V | |
41 | 39, 40 | brcnv 5552 | . . 3 ⊢ ((𝐹‘𝑥)◡ < (𝐹‘𝑦) ↔ (𝐹‘𝑦) < (𝐹‘𝑥)) |
42 | 38, 41 | sylibr 226 | . 2 ⊢ (((𝜑 ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑦 ∈ (𝐴[,]𝐵))) ∧ 𝑥 < 𝑦) → (𝐹‘𝑥)◡ < (𝐹‘𝑦)) |
43 | 1, 2, 3, 4, 5, 42 | dvgt0lem2 24207 | 1 ⊢ (𝜑 → 𝐹 Isom < , ◡ < ((𝐴[,]𝐵), ran 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∈ wcel 2107 ⊆ wss 3792 class class class wbr 4888 ◡ccnv 5356 ran crn 5358 ⟶wf 6133 ‘cfv 6137 Isom wiso 6138 (class class class)co 6924 ℝcr 10273 0cc0 10274 + caddc 10277 · cmul 10279 -∞cmnf 10411 < clt 10413 − cmin 10608 / cdiv 11034 (,)cioo 12491 [,]cicc 12494 –cn→ccncf 23091 D cdv 24068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 ax-addf 10353 ax-mulf 10354 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-of 7176 df-om 7346 df-1st 7447 df-2nd 7448 df-supp 7579 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-2o 7846 df-oadd 7849 df-er 8028 df-map 8144 df-pm 8145 df-ixp 8197 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fsupp 8566 df-fi 8607 df-sup 8638 df-inf 8639 df-oi 8706 df-card 9100 df-cda 9327 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11035 df-nn 11379 df-2 11442 df-3 11443 df-4 11444 df-5 11445 df-6 11446 df-7 11447 df-8 11448 df-9 11449 df-n0 11647 df-z 11733 df-dec 11850 df-uz 11997 df-q 12100 df-rp 12142 df-xneg 12261 df-xadd 12262 df-xmul 12263 df-ioo 12495 df-ico 12497 df-icc 12498 df-fz 12648 df-fzo 12789 df-seq 13124 df-exp 13183 df-hash 13440 df-cj 14250 df-re 14251 df-im 14252 df-sqrt 14386 df-abs 14387 df-struct 16261 df-ndx 16262 df-slot 16263 df-base 16265 df-sets 16266 df-ress 16267 df-plusg 16355 df-mulr 16356 df-starv 16357 df-sca 16358 df-vsca 16359 df-ip 16360 df-tset 16361 df-ple 16362 df-ds 16364 df-unif 16365 df-hom 16366 df-cco 16367 df-rest 16473 df-topn 16474 df-0g 16492 df-gsum 16493 df-topgen 16494 df-pt 16495 df-prds 16498 df-xrs 16552 df-qtop 16557 df-imas 16558 df-xps 16560 df-mre 16636 df-mrc 16637 df-acs 16639 df-mgm 17632 df-sgrp 17674 df-mnd 17685 df-submnd 17726 df-mulg 17932 df-cntz 18137 df-cmn 18585 df-psmet 20138 df-xmet 20139 df-met 20140 df-bl 20141 df-mopn 20142 df-fbas 20143 df-fg 20144 df-cnfld 20147 df-top 21110 df-topon 21127 df-topsp 21149 df-bases 21162 df-cld 21235 df-ntr 21236 df-cls 21237 df-nei 21314 df-lp 21352 df-perf 21353 df-cn 21443 df-cnp 21444 df-haus 21531 df-cmp 21603 df-tx 21778 df-hmeo 21971 df-fil 22062 df-fm 22154 df-flim 22155 df-flf 22156 df-xms 22537 df-ms 22538 df-tms 22539 df-cncf 23093 df-limc 24071 df-dv 24072 |
This theorem is referenced by: dvne0 24215 |
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