fdvposlt - Mathbox for Thierry Arnoux

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Theorem fdvposlt 35107

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 13460	. . . . . . . . 9 ⊢ (𝐶(,)𝐷) ⊆ ℝ
4	2, 3	eqsstri 3977	. . . . . . . 8 ⊢ 𝐸 ⊆ ℝ
5		fdvposlt.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝐸)
6	4, 5	sselid 3929	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
7		fdvposlt.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ 𝐸)
8	4, 7	sselid 3929	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
9	6, 8	posdifd 11825	. . . . . 6 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)))
10	1, 9	mpbid 235	. . . . 5 ⊢ (𝜑 → 0 < (𝐵 − 𝐴))
11	6, 8, 1	ltled 11382	. . . . . 6 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
12		volioo 25797	. . . . . 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 5133	. . . 4 ⊢ (𝜑 → 0 < (vol‘(𝐴(,)𝐵)))
15		ioossicc 13486	. . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
16	15	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
17		ioombl 25793	. . . . . 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 25121	. . . . . . . 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 35105	. . . . . . 7 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐸)
24	23	sselda 3931	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ 𝐸)
25	22, 24	ffvelcdmd 7078	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ)
26		ax-resscn 11181	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
27		ssid 3953	. . . . . . . 8 ⊢ ℂ ⊆ ℂ
28		cncfss 25127	. . . . . . . 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 6947	. . . . . . . 8 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) = (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥)))
31		rescncf 25125	. . . . . . . . 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 2861	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
34	29, 33	sselid 3929	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
35		cniccibl 26068	. . . . . 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 26032	. . . 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 3931	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝐴[,]𝐵))
40	39, 24	syldan 603	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ 𝐸)
41	38, 40	ffvelcdmd 7078	. . . . 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 13044	. . . . 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 26071	. . 3 ⊢ (𝜑 → 0 < ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) d𝑥)
46		fdvposlt.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝐸⟶ℝ)
47		fss 6719	. . . . 5 ⊢ ((𝐹:𝐸⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:𝐸⟶ℂ)
48	46, 26, 47	sylancl 598	. . . 4 ⊢ (𝜑 → 𝐹:𝐸⟶ℂ)
49		cncfss 25127	. . . . . 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 3929	. . . 4 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (𝐸–cn→ℂ))
52	2, 5, 7, 11, 48, 51	ftc2re 35106	. . 3 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) d𝑥 = ((𝐹‘𝐵) − (𝐹‘𝐴)))
53	45, 52	breqtrd 5131	. 2 ⊢ (𝜑 → 0 < ((𝐹‘𝐵) − (𝐹‘𝐴)))
54	46, 5	ffvelcdmd 7078	. . 3 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℝ)
55	46, 7	ffvelcdmd 7078	. . 3 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ℝ)
56	54, 55	posdifd 11825	. 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 2145 ⊆ wss 3899 class class class wbr 5103 ↦ cmpt 5186 dom cdm 5655 ↾ cres 5657 ⟶wf 6529 ‘cfv 6533 (class class class)co 7413 ℂcc 11122 ℝcr 11123 0cc0 11124 < clt 11267 ≤ cle 11268 − cmin 11465 ℝ⁺crp 13042 (,)cioo 13398 [,]cicc 13401 –cn→ccncf 25104 volcvol 25691 𝐿¹cibl 25845 ∫citg 25846 D cdv 26090

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-inf2 9620 ax-cc 10437 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 ax-addf 11203

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-symdif 4199 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-disj 5071 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7678 df-ofr 7679 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-oadd 8459 df-omul 8460 df-er 8696 df-map 8828 df-pm 8829 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-fi 9381 df-sup 9412 df-inf 9413 df-oi 9482 df-dju 9906 df-card 9944 df-acn 9947 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-div 11896 df-nn 12258 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12529 df-z 12616 df-dec 12737 df-uz 12888 df-q 12998 df-rp 13043 df-xneg 13163 df-xadd 13164 df-xmul 13165 df-ioo 13402 df-ioc 13403 df-ico 13404 df-icc 13405 df-fz 13562 df-fzo 13710 df-fl 13853 df-mod 13931 df-seq 14066 df-exp 14126 df-hash 14395 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-limsup 15558 df-clim 15575 df-rlim 15576 df-sum 15774 df-struct 17239 df-sets 17256 df-slot 17274 df-ndx 17286 df-base 17302 df-ress 17323 df-plusg 17355 df-mulr 17356 df-starv 17357 df-sca 17358 df-vsca 17359 df-ip 17360 df-tset 17361 df-ple 17362 df-ds 17364 df-unif 17365 df-hom 17366 df-cco 17367 df-rest 17507 df-topn 17508 df-0g 17526 df-gsum 17527 df-topgen 17528 df-pt 17529 df-prds 17532 df-xrs 17588 df-qtop 17593 df-imas 17594 df-xps 17596 df-mre 17670 df-mrc 17671 df-acs 17673 df-mgm 18730 df-sgrp 18821 df-mnd 18837 df-submnd 18892 df-mulg 19191 df-cntz 19444 df-cmn 19909 df-psmet 21577 df-xmet 21578 df-met 21579 df-bl 21580 df-mopn 21581 df-fbas 21582 df-fg 21583 df-cnfld 21586 df-top 23119 df-topon 23136 df-topsp 23158 df-bases 23171 df-cld 23244 df-ntr 23245 df-cls 23246 df-nei 23323 df-lp 23361 df-perf 23362 df-cn 23452 df-cnp 23453 df-haus 23540 df-cmp 23612 df-tx 23788 df-hmeo 23981 df-fil 24072 df-fm 24164 df-flim 24165 df-flf 24166 df-xms 24546 df-ms 24547 df-tms 24548 df-cncf 25106 df-ovol 25692 df-vol 25693 df-mbf 25847 df-itg1 25848 df-itg2 25849 df-ibl 25850 df-itg 25851 df-0p 25898 df-limc 26093 df-dv 26094

This theorem is used by: fdvneggt 35108

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