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Mirrors > Home > MPE Home > Th. List > Mathboxes > fdvposlt | Structured version Visualization version GIF version |
Description: Functions with a positive derivative, i.e. monotonously growing functions, preserve strict ordering. (Contributed by Thierry Arnoux, 20-Dec-2021.) |
Ref | Expression |
---|---|
fdvposlt.d | ⊢ 𝐸 = (𝐶(,)𝐷) |
fdvposlt.a | ⊢ (𝜑 → 𝐴 ∈ 𝐸) |
fdvposlt.b | ⊢ (𝜑 → 𝐵 ∈ 𝐸) |
fdvposlt.f | ⊢ (𝜑 → 𝐹:𝐸⟶ℝ) |
fdvposlt.c | ⊢ (𝜑 → (ℝ D 𝐹) ∈ (𝐸–cn→ℝ)) |
fdvposlt.lt | ⊢ (𝜑 → 𝐴 < 𝐵) |
fdvposlt.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < ((ℝ D 𝐹)‘𝑥)) |
Ref | Expression |
---|---|
fdvposlt | ⊢ (𝜑 → (𝐹‘𝐴) < (𝐹‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fdvposlt.lt | . . . . . 6 ⊢ (𝜑 → 𝐴 < 𝐵) | |
2 | fdvposlt.d | . . . . . . . . 9 ⊢ 𝐸 = (𝐶(,)𝐷) | |
3 | ioossre 12547 | . . . . . . . . 9 ⊢ (𝐶(,)𝐷) ⊆ ℝ | |
4 | 2, 3 | eqsstri 3854 | . . . . . . . 8 ⊢ 𝐸 ⊆ ℝ |
5 | fdvposlt.a | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝐸) | |
6 | 4, 5 | sseldi 3819 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
7 | fdvposlt.b | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ 𝐸) | |
8 | 4, 7 | sseldi 3819 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
9 | 6, 8 | posdifd 10962 | . . . . . 6 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴))) |
10 | 1, 9 | mpbid 224 | . . . . 5 ⊢ (𝜑 → 0 < (𝐵 − 𝐴)) |
11 | 6, 8, 1 | ltled 10524 | . . . . . 6 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
12 | volioo 23773 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol‘(𝐴(,)𝐵)) = (𝐵 − 𝐴)) | |
13 | 6, 8, 11, 12 | syl3anc 1439 | . . . . 5 ⊢ (𝜑 → (vol‘(𝐴(,)𝐵)) = (𝐵 − 𝐴)) |
14 | 10, 13 | breqtrrd 4914 | . . . 4 ⊢ (𝜑 → 0 < (vol‘(𝐴(,)𝐵))) |
15 | ioossicc 12571 | . . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) | |
16 | 15 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)) |
17 | ioombl 23769 | . . . . . 6 ⊢ (𝐴(,)𝐵) ∈ dom vol | |
18 | 17 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ dom vol) |
19 | fdvposlt.c | . . . . . . . 8 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (𝐸–cn→ℝ)) | |
20 | cncff 23104 | . . . . . . . 8 ⊢ ((ℝ D 𝐹) ∈ (𝐸–cn→ℝ) → (ℝ D 𝐹):𝐸⟶ℝ) | |
21 | 19, 20 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹):𝐸⟶ℝ) |
22 | 21 | adantr 474 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (ℝ D 𝐹):𝐸⟶ℝ) |
23 | 2, 5, 7 | fct2relem 31277 | . . . . . . 7 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐸) |
24 | 23 | sselda 3821 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ 𝐸) |
25 | 22, 24 | ffvelrnd 6624 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ) |
26 | ax-resscn 10329 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
27 | ssid 3842 | . . . . . . . 8 ⊢ ℂ ⊆ ℂ | |
28 | cncfss 23110 | . . . . . . . 8 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ)) | |
29 | 26, 27, 28 | mp2an 682 | . . . . . . 7 ⊢ ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ) |
30 | 21, 23 | feqresmpt 6510 | . . . . . . . 8 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) = (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥))) |
31 | rescncf 23108 | . . . . . . . . 9 ⊢ ((𝐴[,]𝐵) ⊆ 𝐸 → ((ℝ D 𝐹) ∈ (𝐸–cn→ℝ) → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))) | |
32 | 23, 19, 31 | sylc 65 | . . . . . . . 8 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ)) |
33 | 30, 32 | eqeltrrd 2860 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℝ)) |
34 | 29, 33 | sseldi 3819 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℂ)) |
35 | cniccibl 24044 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℂ)) → (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ 𝐿1) | |
36 | 6, 8, 34, 35 | syl3anc 1439 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ 𝐿1) |
37 | 16, 18, 25, 36 | iblss 24008 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ 𝐿1) |
38 | 21 | adantr 474 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (ℝ D 𝐹):𝐸⟶ℝ) |
39 | 16 | sselda 3821 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝐴[,]𝐵)) |
40 | 39, 24 | syldan 585 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ 𝐸) |
41 | 38, 40 | ffvelrnd 6624 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ) |
42 | fdvposlt.1 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < ((ℝ D 𝐹)‘𝑥)) | |
43 | elrp 12139 | . . . . 5 ⊢ (((ℝ D 𝐹)‘𝑥) ∈ ℝ+ ↔ (((ℝ D 𝐹)‘𝑥) ∈ ℝ ∧ 0 < ((ℝ D 𝐹)‘𝑥))) | |
44 | 41, 42, 43 | sylanbrc 578 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ+) |
45 | 14, 37, 44 | itggt0 24045 | . . 3 ⊢ (𝜑 → 0 < ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) d𝑥) |
46 | fdvposlt.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝐸⟶ℝ) | |
47 | fss 6304 | . . . . 5 ⊢ ((𝐹:𝐸⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:𝐸⟶ℂ) | |
48 | 46, 26, 47 | sylancl 580 | . . . 4 ⊢ (𝜑 → 𝐹:𝐸⟶ℂ) |
49 | cncfss 23110 | . . . . . 6 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝐸–cn→ℝ) ⊆ (𝐸–cn→ℂ)) | |
50 | 26, 27, 49 | mp2an 682 | . . . . 5 ⊢ (𝐸–cn→ℝ) ⊆ (𝐸–cn→ℂ) |
51 | 50, 19 | sseldi 3819 | . . . 4 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (𝐸–cn→ℂ)) |
52 | 2, 5, 7, 11, 48, 51 | ftc2re 31278 | . . 3 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) d𝑥 = ((𝐹‘𝐵) − (𝐹‘𝐴))) |
53 | 45, 52 | breqtrd 4912 | . 2 ⊢ (𝜑 → 0 < ((𝐹‘𝐵) − (𝐹‘𝐴))) |
54 | 46, 5 | ffvelrnd 6624 | . . 3 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℝ) |
55 | 46, 7 | ffvelrnd 6624 | . . 3 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ℝ) |
56 | 54, 55 | posdifd 10962 | . 2 ⊢ (𝜑 → ((𝐹‘𝐴) < (𝐹‘𝐵) ↔ 0 < ((𝐹‘𝐵) − (𝐹‘𝐴)))) |
57 | 53, 56 | mpbird 249 | 1 ⊢ (𝜑 → (𝐹‘𝐴) < (𝐹‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ⊆ wss 3792 class class class wbr 4886 ↦ cmpt 4965 dom cdm 5355 ↾ cres 5357 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 ℂcc 10270 ℝcr 10271 0cc0 10272 < clt 10411 ≤ cle 10412 − cmin 10606 ℝ+crp 12137 (,)cioo 12487 [,]cicc 12490 –cn→ccncf 23087 volcvol 23667 𝐿1cibl 23821 ∫citg 23822 D cdv 24064 |
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 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cc 9592 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-addf 10351 ax-mulf 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 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-symdif 4067 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-disj 4855 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-ofr 7175 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-omul 7848 df-er 8026 df-map 8142 df-pm 8143 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-fi 8605 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 df-acn 9101 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-ioo 12491 df-ioc 12492 df-ico 12493 df-icc 12494 df-fz 12644 df-fzo 12785 df-fl 12912 df-mod 12988 df-seq 13120 df-exp 13179 df-hash 13436 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-limsup 14610 df-clim 14627 df-rlim 14628 df-sum 14825 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-hom 16362 df-cco 16363 df-rest 16469 df-topn 16470 df-0g 16488 df-gsum 16489 df-topgen 16490 df-pt 16491 df-prds 16494 df-xrs 16548 df-qtop 16553 df-imas 16554 df-xps 16556 df-mre 16632 df-mrc 16633 df-acs 16635 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-mulg 17928 df-cntz 18133 df-cmn 18581 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-fbas 20139 df-fg 20140 df-cnfld 20143 df-top 21106 df-topon 21123 df-topsp 21145 df-bases 21158 df-cld 21231 df-ntr 21232 df-cls 21233 df-nei 21310 df-lp 21348 df-perf 21349 df-cn 21439 df-cnp 21440 df-haus 21527 df-cmp 21599 df-tx 21774 df-hmeo 21967 df-fil 22058 df-fm 22150 df-flim 22151 df-flf 22152 df-xms 22533 df-ms 22534 df-tms 22535 df-cncf 23089 df-ovol 23668 df-vol 23669 df-mbf 23823 df-itg1 23824 df-itg2 23825 df-ibl 23826 df-itg 23827 df-0p 23874 df-limc 24067 df-dv 24068 |
This theorem is referenced by: fdvneggt 31280 |
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