fdvposlt - Mathbox for Thierry Arnoux

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Theorem fdvposlt 35051

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 13450	. . . . . . . . 9 ⊢ (𝐶(,)𝐷) ⊆ ℝ
4	2, 3	eqsstri 3984	. . . . . . . 8 ⊢ 𝐸 ⊆ ℝ
5		fdvposlt.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝐸)
6	4, 5	sselid 3936	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
7		fdvposlt.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ 𝐸)
8	4, 7	sselid 3936	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
9	6, 8	posdifd 11816	. . . . . 6 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)))
10	1, 9	mpbid 235	. . . . 5 ⊢ (𝜑 → 0 < (𝐵 − 𝐴))
11	6, 8, 1	ltled 11373	. . . . . 6 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
12		volioo 25779	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol‘(𝐴(,)𝐵)) = (𝐵 − 𝐴))
13	6, 8, 11, 12	syl3anc 1398	. . . . 5 ⊢ (𝜑 → (vol‘(𝐴(,)𝐵)) = (𝐵 − 𝐴))
14	10, 13	breqtrrd 5141	. . . 4 ⊢ (𝜑 → 0 < (vol‘(𝐴(,)𝐵)))
15		ioossicc 13476	. . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
16	15	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
17		ioombl 25775	. . . . . 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 25103	. . . . . . . 8 ⊢ ((ℝ D 𝐹) ∈ (𝐸–cn→ℝ) → (ℝ D 𝐹):𝐸⟶ℝ)
21	19, 20	syl 18	. . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹):𝐸⟶ℝ)
22	21	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (ℝ D 𝐹):𝐸⟶ℝ)
23	2, 5, 7	fct2relem 35049	. . . . . . 7 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐸)
24	23	sselda 3938	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ 𝐸)
25	22, 24	ffvelcdmd 7084	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ)
26		ax-resscn 11172	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
27		ssid 3960	. . . . . . . 8 ⊢ ℂ ⊆ ℂ
28		cncfss 25109	. . . . . . . 8 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ))
29	26, 27, 28	mp2an 705	. . . . . . 7 ⊢ ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ)
30	21, 23	feqresmpt 6954	. . . . . . . 8 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) = (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥)))
31		rescncf 25107	. . . . . . . . 9 ⊢ ((𝐴[,]𝐵) ⊆ 𝐸 → ((ℝ D 𝐹) ∈ (𝐸–cn→ℝ) → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ)))
32	23, 19, 31	sylc 66	. . . . . . . 8 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
33	30, 32	eqeltrrd 2866	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
34	29, 33	sselid 3936	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
35		cniccibl 26051	. . . . . 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 1398	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ 𝐿¹)
37	16, 18, 25, 36	iblss 26015	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ 𝐿¹)
38	21	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (ℝ D 𝐹):𝐸⟶ℝ)
39	16	sselda 3938	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝐴[,]𝐵))
40	39, 24	syldan 603	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ 𝐸)
41	38, 40	ffvelcdmd 7084	. . . . 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 13034	. . . . 5 ⊢ (((ℝ D 𝐹)‘𝑥) ∈ ℝ⁺ ↔ (((ℝ D 𝐹)‘𝑥) ∈ ℝ ∧ 0 < ((ℝ D 𝐹)‘𝑥)))
44	41, 42, 43	sylanbrc 595	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ⁺)
45	14, 37, 44	itggt0 26054	. . 3 ⊢ (𝜑 → 0 < ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) d𝑥)
46		fdvposlt.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝐸⟶ℝ)
47		fss 6726	. . . . 5 ⊢ ((𝐹:𝐸⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:𝐸⟶ℂ)
48	46, 26, 47	sylancl 598	. . . 4 ⊢ (𝜑 → 𝐹:𝐸⟶ℂ)
49		cncfss 25109	. . . . . 6 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝐸–cn→ℝ) ⊆ (𝐸–cn→ℂ))
50	26, 27, 49	mp2an 705	. . . . 5 ⊢ (𝐸–cn→ℝ) ⊆ (𝐸–cn→ℂ)
51	50, 19	sselid 3936	. . . 4 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (𝐸–cn→ℂ))
52	2, 5, 7, 11, 48, 51	ftc2re 35050	. . 3 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) d𝑥 = ((𝐹‘𝐵) − (𝐹‘𝐴)))
53	45, 52	breqtrd 5139	. 2 ⊢ (𝜑 → 0 < ((𝐹‘𝐵) − (𝐹‘𝐴)))
54	46, 5	ffvelcdmd 7084	. . 3 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℝ)
55	46, 7	ffvelcdmd 7084	. . 3 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ℝ)
56	54, 55	posdifd 11816	. 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

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ⊆ wss 3906 class class class wbr 5111 ↦ cmpt 5194 dom cdm 5663 ↾ cres 5665 ⟶wf 6536 ‘cfv 6540 (class class class)co 7419 ℂcc 11113 ℝcr 11114 0cc0 11115 < clt 11258 ≤ cle 11259 − cmin 11456 ℝ⁺crp 13032 (,)cioo 13388 [,]cicc 13391 –cn→ccncf 25086 volcvol 25673 𝐿¹cibl 25827 ∫citg 25828 D cdv 26073

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-inf2 9617 ax-cc 10434 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192 ax-pre-sup 11193 ax-addf 11194

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-symdif 4206 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-disj 5079 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7684 df-ofr 7685 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8163 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-oadd 8463 df-omul 8464 df-er 8700 df-map 8832 df-pm 8833 df-ixp 8902 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-fsupp 9329 df-fi 9378 df-sup 9409 df-inf 9410 df-oi 9479 df-dju 9903 df-card 9941 df-acn 9944 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-div 11887 df-nn 12249 df-2 12318 df-3 12319 df-4 12320 df-5 12321 df-6 12322 df-7 12323 df-8 12324 df-9 12325 df-n0 12520 df-z 12607 df-dec 12728 df-uz 12879 df-q 12989 df-rp 13033 df-xneg 13153 df-xadd 13154 df-xmul 13155 df-ioo 13392 df-ioc 13393 df-ico 13394 df-icc 13395 df-fz 13552 df-fzo 13700 df-fl 13843 df-mod 13921 df-seq 14056 df-exp 14116 df-hash 14385 df-cj 15174 df-re 15175 df-im 15176 df-sqrt 15310 df-abs 15311 df-limsup 15546 df-clim 15563 df-rlim 15564 df-sum 15762 df-struct 17229 df-sets 17246 df-slot 17264 df-ndx 17276 df-base 17292 df-ress 17313 df-plusg 17345 df-mulr 17346 df-starv 17347 df-sca 17348 df-vsca 17349 df-ip 17350 df-tset 17351 df-ple 17352 df-ds 17354 df-unif 17355 df-hom 17356 df-cco 17357 df-rest 17497 df-topn 17498 df-0g 17516 df-gsum 17517 df-topgen 17518 df-pt 17519 df-prds 17522 df-xrs 17578 df-qtop 17583 df-imas 17584 df-xps 17586 df-mre 17660 df-mrc 17661 df-acs 17663 df-mgm 18720 df-sgrp 18809 df-mnd 18825 df-submnd 18879 df-mulg 19178 df-cntz 19431 df-cmn 19896 df-psmet 21564 df-xmet 21565 df-met 21566 df-bl 21567 df-mopn 21568 df-fbas 21569 df-fg 21570 df-cnfld 21573 df-top 23101 df-topon 23118 df-topsp 23140 df-bases 23153 df-cld 23226 df-ntr 23227 df-cls 23228 df-nei 23305 df-lp 23343 df-perf 23344 df-cn 23434 df-cnp 23435 df-haus 23522 df-cmp 23594 df-tx 23770 df-hmeo 23963 df-fil 24054 df-fm 24146 df-flim 24147 df-flf 24148 df-xms 24528 df-ms 24529 df-tms 24530 df-cncf 25088 df-ovol 25674 df-vol 25675 df-mbf 25829 df-itg1 25830 df-itg2 25831 df-ibl 25832 df-itg 25833 df-0p 25880 df-limc 26076 df-dv 26077

This theorem is used by: fdvneggt 35052

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