fdvposlt - Mathbox for Thierry Arnoux

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Theorem fdvposlt 34563

Description: Functions with a positive derivative, i.e. monotonously growing functions, preserve strict ordering. (Contributed by Thierry Arnoux, 20-Dec-2021.)

**Hypotheses**
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))

**Assertion**
Ref	Expression
fdvposlt	⊢ (𝜑 → (𝐹‘𝐴) < (𝐹‘𝐵))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝑥,𝐸 𝑥,𝐹 𝜑,𝑥 Allowed substitution hints: 𝐶(𝑥) 𝐷(𝑥)

**Proof of Theorem fdvposlt**
Step	Hyp	Ref	Expression
1		fdvposlt.lt	. . . . . 6 ⊢ (𝜑 → 𝐴 < 𝐵)
2		fdvposlt.d	. . . . . . . . 9 ⊢ 𝐸 = (𝐶(,)𝐷)
3		ioossre 13344	. . . . . . . . 9 ⊢ (𝐶(,)𝐷) ⊆ ℝ
4	2, 3	eqsstri 3990	. . . . . . . 8 ⊢ 𝐸 ⊆ ℝ
5		fdvposlt.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝐸)
6	4, 5	sselid 3941	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
7		fdvposlt.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ 𝐸)
8	4, 7	sselid 3941	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
9	6, 8	posdifd 11741	. . . . . 6 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)))
10	1, 9	mpbid 232	. . . . 5 ⊢ (𝜑 → 0 < (𝐵 − 𝐴))
11	6, 8, 1	ltled 11298	. . . . . 6 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
12		volioo 25446	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol‘(𝐴(,)𝐵)) = (𝐵 − 𝐴))
13	6, 8, 11, 12	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (vol‘(𝐴(,)𝐵)) = (𝐵 − 𝐴))
14	10, 13	breqtrrd 5130	. . . 4 ⊢ (𝜑 → 0 < (vol‘(𝐴(,)𝐵)))
15		ioossicc 13370	. . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
16	15	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
17		ioombl 25442	. . . . . 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 24762	. . . . . . . 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 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (ℝ D 𝐹):𝐸⟶ℝ)
23	2, 5, 7	fct2relem 34561	. . . . . . 7 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐸)
24	23	sselda 3943	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ 𝐸)
25	22, 24	ffvelcdmd 7039	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ)
26		ax-resscn 11101	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
27		ssid 3966	. . . . . . . 8 ⊢ ℂ ⊆ ℂ
28		cncfss 24768	. . . . . . . 8 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ))
29	26, 27, 28	mp2an 692	. . . . . . 7 ⊢ ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ)
30	21, 23	feqresmpt 6912	. . . . . . . 8 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) = (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥)))
31		rescncf 24766	. . . . . . . . 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 2829	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
34	29, 33	sselid 3941	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
35		cniccibl 25718	. . . . . 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 1373	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ 𝐿¹)
37	16, 18, 25, 36	iblss 25682	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ 𝐿¹)
38	21	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (ℝ D 𝐹):𝐸⟶ℝ)
39	16	sselda 3943	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝐴[,]𝐵))
40	39, 24	syldan 591	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ 𝐸)
41	38, 40	ffvelcdmd 7039	. . . . 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 12929	. . . . 5 ⊢ (((ℝ D 𝐹)‘𝑥) ∈ ℝ⁺ ↔ (((ℝ D 𝐹)‘𝑥) ∈ ℝ ∧ 0 < ((ℝ D 𝐹)‘𝑥)))
44	41, 42, 43	sylanbrc 583	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ⁺)
45	14, 37, 44	itggt0 25721	. . 3 ⊢ (𝜑 → 0 < ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) d𝑥)
46		fdvposlt.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝐸⟶ℝ)
47		fss 6686	. . . . 5 ⊢ ((𝐹:𝐸⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:𝐸⟶ℂ)
48	46, 26, 47	sylancl 586	. . . 4 ⊢ (𝜑 → 𝐹:𝐸⟶ℂ)
49		cncfss 24768	. . . . . 6 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝐸–cn→ℝ) ⊆ (𝐸–cn→ℂ))
50	26, 27, 49	mp2an 692	. . . . 5 ⊢ (𝐸–cn→ℝ) ⊆ (𝐸–cn→ℂ)
51	50, 19	sselid 3941	. . . 4 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (𝐸–cn→ℂ))
52	2, 5, 7, 11, 48, 51	ftc2re 34562	. . 3 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) d𝑥 = ((𝐹‘𝐵) − (𝐹‘𝐴)))
53	45, 52	breqtrd 5128	. 2 ⊢ (𝜑 → 0 < ((𝐹‘𝐵) − (𝐹‘𝐴)))
54	46, 5	ffvelcdmd 7039	. . 3 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℝ)
55	46, 7	ffvelcdmd 7039	. . 3 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ℝ)
56	54, 55	posdifd 11741	. 2 ⊢ (𝜑 → ((𝐹‘𝐴) < (𝐹‘𝐵) ↔ 0 < ((𝐹‘𝐵) − (𝐹‘𝐴))))
57	53, 56	mpbird 257	1 ⊢ (𝜑 → (𝐹‘𝐴) < (𝐹‘𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ⊆ wss 3911 class class class wbr 5102 ↦ cmpt 5183 dom cdm 5631 ↾ cres 5633 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 ℂcc 11042 ℝcr 11043 0cc0 11044 < clt 11184 ≤ cle 11185 − cmin 11381 ℝ⁺crp 12927 (,)cioo 13282 [,]cicc 13285 –cn→ccncf 24745 volcvol 25340 𝐿¹cibl 25494 ∫citg 25495 D cdv 25740

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9570 ax-cc 10364 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-symdif 4212 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-disj 5070 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-ofr 7634 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-oadd 8415 df-omul 8416 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-dju 9830 df-card 9868 df-acn 9871 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-ioc 13287 df-ico 13288 df-icc 13289 df-fz 13445 df-fzo 13592 df-fl 13730 df-mod 13808 df-seq 13943 df-exp 14003 df-hash 14272 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-limsup 15413 df-clim 15430 df-rlim 15431 df-sum 15629 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-mulg 18976 df-cntz 19225 df-cmn 19688 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-cnfld 21241 df-top 22757 df-topon 22774 df-topsp 22796 df-bases 22809 df-cld 22882 df-ntr 22883 df-cls 22884 df-nei 22961 df-lp 22999 df-perf 23000 df-cn 23090 df-cnp 23091 df-haus 23178 df-cmp 23250 df-tx 23425 df-hmeo 23618 df-fil 23709 df-fm 23801 df-flim 23802 df-flf 23803 df-xms 24184 df-ms 24185 df-tms 24186 df-cncf 24747 df-ovol 25341 df-vol 25342 df-mbf 25496 df-itg1 25497 df-itg2 25498 df-ibl 25499 df-itg 25500 df-0p 25547 df-limc 25743 df-dv 25744

This theorem is referenced by: fdvneggt 34564

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