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Mirrors > Home > MPE Home > Th. List > Mathboxes > fdvposle | Structured version Visualization version GIF version |
Description: Functions with a nonnegative derivative, i.e. monotonously growing functions, preserve 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)) |
fdvposle.le | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
fdvposle.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ≤ ((ℝ D 𝐹)‘𝑥)) |
Ref | Expression |
---|---|
fdvposle | ⊢ (𝜑 → (𝐹‘𝐴) ≤ (𝐹‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ioossicc 12259 | . . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)) |
3 | ioombl 23333 | . . . . . 6 ⊢ (𝐴(,)𝐵) ∈ dom vol | |
4 | 3 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ dom vol) |
5 | fdvposlt.c | . . . . . . . 8 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (𝐸–cn→ℝ)) | |
6 | cncff 22696 | . . . . . . . 8 ⊢ ((ℝ D 𝐹) ∈ (𝐸–cn→ℝ) → (ℝ D 𝐹):𝐸⟶ℝ) | |
7 | 5, 6 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹):𝐸⟶ℝ) |
8 | 7 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (ℝ D 𝐹):𝐸⟶ℝ) |
9 | fdvposlt.d | . . . . . . . 8 ⊢ 𝐸 = (𝐶(,)𝐷) | |
10 | fdvposlt.a | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝐸) | |
11 | fdvposlt.b | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ 𝐸) | |
12 | 9, 10, 11 | fct2relem 30675 | . . . . . . 7 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐸) |
13 | 12 | sselda 3603 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ 𝐸) |
14 | 8, 13 | ffvelrnd 6360 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ) |
15 | ioossre 12235 | . . . . . . . 8 ⊢ (𝐶(,)𝐷) ⊆ ℝ | |
16 | 9, 15 | eqsstri 3635 | . . . . . . 7 ⊢ 𝐸 ⊆ ℝ |
17 | 16, 10 | sseldi 3601 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
18 | 16, 11 | sseldi 3601 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
19 | ax-resscn 9993 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
20 | ssid 3624 | . . . . . . . 8 ⊢ ℂ ⊆ ℂ | |
21 | cncfss 22702 | . . . . . . . 8 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ)) | |
22 | 19, 20, 21 | mp2an 708 | . . . . . . 7 ⊢ ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ) |
23 | 7, 12 | feqresmpt 6250 | . . . . . . . 8 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) = (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥))) |
24 | rescncf 22700 | . . . . . . . . 9 ⊢ ((𝐴[,]𝐵) ⊆ 𝐸 → ((ℝ D 𝐹) ∈ (𝐸–cn→ℝ) → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))) | |
25 | 12, 5, 24 | sylc 65 | . . . . . . . 8 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ)) |
26 | 23, 25 | eqeltrrd 2702 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℝ)) |
27 | 22, 26 | sseldi 3601 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℂ)) |
28 | cniccibl 23607 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℂ)) → (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ 𝐿1) | |
29 | 17, 18, 27, 28 | syl3anc 1326 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ 𝐿1) |
30 | 2, 4, 14, 29 | iblss 23571 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ 𝐿1) |
31 | 7 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (ℝ D 𝐹):𝐸⟶ℝ) |
32 | 2 | sselda 3603 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝐴[,]𝐵)) |
33 | 32, 13 | syldan 487 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ 𝐸) |
34 | 31, 33 | ffvelrnd 6360 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ) |
35 | fdvposle.1 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ≤ ((ℝ D 𝐹)‘𝑥)) | |
36 | 30, 34, 35 | itgge0 23577 | . . 3 ⊢ (𝜑 → 0 ≤ ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) d𝑥) |
37 | fdvposle.le | . . . 4 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
38 | fdvposlt.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝐸⟶ℝ) | |
39 | fss 6056 | . . . . 5 ⊢ ((𝐹:𝐸⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:𝐸⟶ℂ) | |
40 | 38, 19, 39 | sylancl 694 | . . . 4 ⊢ (𝜑 → 𝐹:𝐸⟶ℂ) |
41 | cncfss 22702 | . . . . . 6 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝐸–cn→ℝ) ⊆ (𝐸–cn→ℂ)) | |
42 | 19, 20, 41 | mp2an 708 | . . . . 5 ⊢ (𝐸–cn→ℝ) ⊆ (𝐸–cn→ℂ) |
43 | 42, 5 | sseldi 3601 | . . . 4 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (𝐸–cn→ℂ)) |
44 | 9, 10, 11, 37, 40, 43 | ftc2re 30676 | . . 3 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) d𝑥 = ((𝐹‘𝐵) − (𝐹‘𝐴))) |
45 | 36, 44 | breqtrd 4679 | . 2 ⊢ (𝜑 → 0 ≤ ((𝐹‘𝐵) − (𝐹‘𝐴))) |
46 | 38, 11 | ffvelrnd 6360 | . . 3 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ℝ) |
47 | 38, 10 | ffvelrnd 6360 | . . 3 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℝ) |
48 | 46, 47 | subge0d 10617 | . 2 ⊢ (𝜑 → (0 ≤ ((𝐹‘𝐵) − (𝐹‘𝐴)) ↔ (𝐹‘𝐴) ≤ (𝐹‘𝐵))) |
49 | 45, 48 | mpbid 222 | 1 ⊢ (𝜑 → (𝐹‘𝐴) ≤ (𝐹‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 class class class wbr 4653 ↦ cmpt 4729 dom cdm 5114 ↾ cres 5116 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 ≤ cle 10075 − cmin 10266 (,)cioo 12175 [,]cicc 12178 –cn→ccncf 22679 volcvol 23232 𝐿1cibl 23386 ∫citg 23387 D cdv 23627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cc 9257 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-ofr 6898 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-ovol 23233 df-vol 23234 df-mbf 23388 df-itg1 23389 df-itg2 23390 df-ibl 23391 df-itg 23392 df-0p 23437 df-limc 23630 df-dv 23631 |
This theorem is referenced by: fdvnegge 30680 |
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