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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 12786 | . . . . . . . . 9 ⊢ (𝐶(,)𝐷) ⊆ ℝ | |
4 | 2, 3 | eqsstri 3949 | . . . . . . . 8 ⊢ 𝐸 ⊆ ℝ |
5 | fdvposlt.a | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝐸) | |
6 | 4, 5 | sseldi 3913 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
7 | fdvposlt.b | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ 𝐸) | |
8 | 4, 7 | sseldi 3913 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
9 | 6, 8 | posdifd 11216 | . . . . . 6 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴))) |
10 | 1, 9 | mpbid 235 | . . . . 5 ⊢ (𝜑 → 0 < (𝐵 − 𝐴)) |
11 | 6, 8, 1 | ltled 10777 | . . . . . 6 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
12 | volioo 24173 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol‘(𝐴(,)𝐵)) = (𝐵 − 𝐴)) | |
13 | 6, 8, 11, 12 | syl3anc 1368 | . . . . 5 ⊢ (𝜑 → (vol‘(𝐴(,)𝐵)) = (𝐵 − 𝐴)) |
14 | 10, 13 | breqtrrd 5058 | . . . 4 ⊢ (𝜑 → 0 < (vol‘(𝐴(,)𝐵))) |
15 | ioossicc 12811 | . . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) | |
16 | 15 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)) |
17 | ioombl 24169 | . . . . . 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 23498 | . . . . . . . 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 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (ℝ D 𝐹):𝐸⟶ℝ) |
23 | 2, 5, 7 | fct2relem 31978 | . . . . . . 7 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐸) |
24 | 23 | sselda 3915 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ 𝐸) |
25 | 22, 24 | ffvelrnd 6829 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ) |
26 | ax-resscn 10583 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
27 | ssid 3937 | . . . . . . . 8 ⊢ ℂ ⊆ ℂ | |
28 | cncfss 23504 | . . . . . . . 8 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ)) | |
29 | 26, 27, 28 | mp2an 691 | . . . . . . 7 ⊢ ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ) |
30 | 21, 23 | feqresmpt 6709 | . . . . . . . 8 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) = (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥))) |
31 | rescncf 23502 | . . . . . . . . 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 2891 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℝ)) |
34 | 29, 33 | sseldi 3913 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℂ)) |
35 | cniccibl 24444 | . . . . . 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 1368 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ 𝐿1) |
37 | 16, 18, 25, 36 | iblss 24408 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ 𝐿1) |
38 | 21 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (ℝ D 𝐹):𝐸⟶ℝ) |
39 | 16 | sselda 3915 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝐴[,]𝐵)) |
40 | 39, 24 | syldan 594 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ 𝐸) |
41 | 38, 40 | ffvelrnd 6829 | . . . . 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 12379 | . . . . 5 ⊢ (((ℝ D 𝐹)‘𝑥) ∈ ℝ+ ↔ (((ℝ D 𝐹)‘𝑥) ∈ ℝ ∧ 0 < ((ℝ D 𝐹)‘𝑥))) | |
44 | 41, 42, 43 | sylanbrc 586 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ+) |
45 | 14, 37, 44 | itggt0 24447 | . . 3 ⊢ (𝜑 → 0 < ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) d𝑥) |
46 | fdvposlt.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝐸⟶ℝ) | |
47 | fss 6501 | . . . . 5 ⊢ ((𝐹:𝐸⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:𝐸⟶ℂ) | |
48 | 46, 26, 47 | sylancl 589 | . . . 4 ⊢ (𝜑 → 𝐹:𝐸⟶ℂ) |
49 | cncfss 23504 | . . . . . 6 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝐸–cn→ℝ) ⊆ (𝐸–cn→ℂ)) | |
50 | 26, 27, 49 | mp2an 691 | . . . . 5 ⊢ (𝐸–cn→ℝ) ⊆ (𝐸–cn→ℂ) |
51 | 50, 19 | sseldi 3913 | . . . 4 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (𝐸–cn→ℂ)) |
52 | 2, 5, 7, 11, 48, 51 | ftc2re 31979 | . . 3 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) d𝑥 = ((𝐹‘𝐵) − (𝐹‘𝐴))) |
53 | 45, 52 | breqtrd 5056 | . 2 ⊢ (𝜑 → 0 < ((𝐹‘𝐵) − (𝐹‘𝐴))) |
54 | 46, 5 | ffvelrnd 6829 | . . 3 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℝ) |
55 | 46, 7 | ffvelrnd 6829 | . . 3 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ℝ) |
56 | 54, 55 | posdifd 11216 | . 2 ⊢ (𝜑 → ((𝐹‘𝐴) < (𝐹‘𝐵) ↔ 0 < ((𝐹‘𝐵) − (𝐹‘𝐴)))) |
57 | 53, 56 | mpbird 260 | 1 ⊢ (𝜑 → (𝐹‘𝐴) < (𝐹‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 class class class wbr 5030 ↦ cmpt 5110 dom cdm 5519 ↾ cres 5521 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 < clt 10664 ≤ cle 10665 − cmin 10859 ℝ+crp 12377 (,)cioo 12726 [,]cicc 12729 –cn→ccncf 23481 volcvol 24067 𝐿1cibl 24221 ∫citg 24222 D cdv 24466 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cc 9846 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-symdif 4169 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-disj 4996 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-ofr 7390 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-omul 8090 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-dju 9314 df-card 9352 df-acn 9355 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-perf 21742 df-cn 21832 df-cnp 21833 df-haus 21920 df-cmp 21992 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-xms 22927 df-ms 22928 df-tms 22929 df-cncf 23483 df-ovol 24068 df-vol 24069 df-mbf 24223 df-itg1 24224 df-itg2 24225 df-ibl 24226 df-itg 24227 df-0p 24274 df-limc 24469 df-dv 24470 |
This theorem is referenced by: fdvneggt 31981 |
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