fdvnegge - Mathbox for Thierry Arnoux

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Theorem fdvnegge 31873

Description: Functions with a nonpositive derivative, i.e., decreasing functions, preserve 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))
fdvnegge.le	⊢ (𝜑 → 𝐴 ≤ 𝐵)
fdvnegge.1	⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ≤ 0)

**Assertion**
Ref	Expression
fdvnegge	⊢ (𝜑 → (𝐹‘𝐵) ≤ (𝐹‘𝐴))

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

Proof of Theorem fdvnegge Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fdvposlt.d	. . . 4 ⊢ 𝐸 = (𝐶(,)𝐷)
2		fdvposlt.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐸)
3		fdvposlt.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐸)
4		fdvposlt.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐸⟶ℝ)
5	4	ffvelrnda 6850	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐸) → (𝐹‘𝑦) ∈ ℝ)
6	5	renegcld 11066	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐸) → -(𝐹‘𝑦) ∈ ℝ)
7	6	fmpttd 6878	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦)):𝐸⟶ℝ)
8		reelprrecn 10628	. . . . . . 7 ⊢ ℝ ∈ {ℝ, ℂ}
9	8	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
10		ax-resscn 10593	. . . . . . 7 ⊢ ℝ ⊆ ℂ
11	10, 5	sseldi 3964	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐸) → (𝐹‘𝑦) ∈ ℂ)
12		fvexd 6684	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐸) → ((ℝ D 𝐹)‘𝑦) ∈ V)
13	4	feqmptd 6732	. . . . . . . 8 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ 𝐸 ↦ (𝐹‘𝑦)))
14	13	oveq2d 7171	. . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑦 ∈ 𝐸 ↦ (𝐹‘𝑦))))
15		fdvposlt.c	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (𝐸–cn→ℝ))
16		cncff 23500	. . . . . . . . 9 ⊢ ((ℝ D 𝐹) ∈ (𝐸–cn→ℝ) → (ℝ D 𝐹):𝐸⟶ℝ)
17	15, 16	syl 17	. . . . . . . 8 ⊢ (𝜑 → (ℝ D 𝐹):𝐸⟶ℝ)
18	17	feqmptd 6732	. . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹) = (𝑦 ∈ 𝐸 ↦ ((ℝ D 𝐹)‘𝑦)))
19	14, 18	eqtr3d 2858	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑦 ∈ 𝐸 ↦ (𝐹‘𝑦))) = (𝑦 ∈ 𝐸 ↦ ((ℝ D 𝐹)‘𝑦)))
20	9, 11, 12, 19	dvmptneg 24562	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦))) = (𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦)))
21	17	ffvelrnda 6850	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐸) → ((ℝ D 𝐹)‘𝑦) ∈ ℝ)
22	21	renegcld 11066	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐸) → -((ℝ D 𝐹)‘𝑦) ∈ ℝ)
23	22	fmpttd 6878	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦)):𝐸⟶ℝ)
24		ssid 3988	. . . . . . . . . 10 ⊢ ℂ ⊆ ℂ
25		cncfss 23506	. . . . . . . . . 10 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝐸–cn→ℝ) ⊆ (𝐸–cn→ℂ))
26	10, 24, 25	mp2an 690	. . . . . . . . 9 ⊢ (𝐸–cn→ℝ) ⊆ (𝐸–cn→ℂ)
27	26, 15	sseldi 3964	. . . . . . . 8 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (𝐸–cn→ℂ))
28		eqid 2821	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦)) = (𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦))
29	28	negfcncf 23526	. . . . . . . 8 ⊢ ((ℝ D 𝐹) ∈ (𝐸–cn→ℂ) → (𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦)) ∈ (𝐸–cn→ℂ))
30	27, 29	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦)) ∈ (𝐸–cn→ℂ))
31		cncffvrn 23505	. . . . . . 7 ⊢ ((ℝ ⊆ ℂ ∧ (𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦)) ∈ (𝐸–cn→ℂ)) → ((𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦)) ∈ (𝐸–cn→ℝ) ↔ (𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦)):𝐸⟶ℝ))
32	10, 30, 31	sylancr 589	. . . . . 6 ⊢ (𝜑 → ((𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦)) ∈ (𝐸–cn→ℝ) ↔ (𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦)):𝐸⟶ℝ))
33	23, 32	mpbird 259	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦)) ∈ (𝐸–cn→ℝ))
34	20, 33	eqeltrd 2913	. . . 4 ⊢ (𝜑 → (ℝ D (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦))) ∈ (𝐸–cn→ℝ))
35		fdvnegge.le	. . . 4 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
36		fdvnegge.1	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ≤ 0)
37	17	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (ℝ D 𝐹):𝐸⟶ℝ)
38		ioossicc 12821	. . . . . . . . . . 11 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
39	38	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
40	1, 2, 3	fct2relem 31868	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐸)
41	39, 40	sstrd 3976	. . . . . . . . 9 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐸)
42	41	sselda 3966	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ 𝐸)
43	37, 42	ffvelrnd 6851	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ)
44	43	le0neg1d 11210	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D 𝐹)‘𝑥) ≤ 0 ↔ 0 ≤ -((ℝ D 𝐹)‘𝑥)))
45	36, 44	mpbid 234	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ≤ -((ℝ D 𝐹)‘𝑥))
46	20	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (ℝ D (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦))) = (𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦)))
47	46	fveq1d 6671	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦)))‘𝑥) = ((𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦))‘𝑥))
48	28	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦)) = (𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦)))
49		simpr 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥)
50	49	fveq2d 6673	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑦 = 𝑥) → ((ℝ D 𝐹)‘𝑦) = ((ℝ D 𝐹)‘𝑥))
51	50	negeqd 10879	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑦 = 𝑥) → -((ℝ D 𝐹)‘𝑦) = -((ℝ D 𝐹)‘𝑥))
52	43	renegcld 11066	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → -((ℝ D 𝐹)‘𝑥) ∈ ℝ)
53	48, 51, 42, 52	fvmptd 6774	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦))‘𝑥) = -((ℝ D 𝐹)‘𝑥))
54	47, 53	eqtrd 2856	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦)))‘𝑥) = -((ℝ D 𝐹)‘𝑥))
55	45, 54	breqtrrd 5093	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ≤ ((ℝ D (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦)))‘𝑥))
56	1, 2, 3, 7, 34, 35, 55	fdvposle 31872	. . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦))‘𝐴) ≤ ((𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦))‘𝐵))
57		eqidd 2822	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦)) = (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦)))
58		simpr 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → 𝑦 = 𝐴)
59	58	fveq2d 6673	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → (𝐹‘𝑦) = (𝐹‘𝐴))
60	59	negeqd 10879	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → -(𝐹‘𝑦) = -(𝐹‘𝐴))
61	4, 2	ffvelrnd 6851	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℝ)
62	61	renegcld 11066	. . . 4 ⊢ (𝜑 → -(𝐹‘𝐴) ∈ ℝ)
63	57, 60, 2, 62	fvmptd 6774	. . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦))‘𝐴) = -(𝐹‘𝐴))
64		simpr 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
65	64	fveq2d 6673	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → (𝐹‘𝑦) = (𝐹‘𝐵))
66	65	negeqd 10879	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → -(𝐹‘𝑦) = -(𝐹‘𝐵))
67	4, 3	ffvelrnd 6851	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ℝ)
68	67	renegcld 11066	. . . 4 ⊢ (𝜑 → -(𝐹‘𝐵) ∈ ℝ)
69	57, 66, 3, 68	fvmptd 6774	. . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦))‘𝐵) = -(𝐹‘𝐵))
70	56, 63, 69	3brtr3d 5096	. 2 ⊢ (𝜑 → -(𝐹‘𝐴) ≤ -(𝐹‘𝐵))
71	67, 61	lenegd 11218	. 2 ⊢ (𝜑 → ((𝐹‘𝐵) ≤ (𝐹‘𝐴) ↔ -(𝐹‘𝐴) ≤ -(𝐹‘𝐵)))
72	70, 71	mpbird 259	1 ⊢ (𝜑 → (𝐹‘𝐵) ≤ (𝐹‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ⊆ wss 3935 {cpr 4568 class class class wbr 5065 ↦ cmpt 5145 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 ℂcc 10534 ℝcr 10535 0cc0 10536 ≤ cle 10675 -cneg 10870 (,)cioo 12737 [,]cicc 12740 –cn→ccncf 23483 D cdv 24460

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 ax-cc 9856 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 ax-addf 10615 ax-mulf 10616

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-symdif 4218 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-disj 5031 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-ofr 7409 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-omul 8106 df-er 8288 df-map 8407 df-pm 8408 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-fi 8874 df-sup 8905 df-inf 8906 df-oi 8973 df-dju 9329 df-card 9367 df-acn 9370 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-q 12348 df-rp 12389 df-xneg 12506 df-xadd 12507 df-xmul 12508 df-ioo 12741 df-ioc 12742 df-ico 12743 df-icc 12744 df-fz 12892 df-fzo 13033 df-fl 13161 df-mod 13237 df-seq 13369 df-exp 13429 df-hash 13690 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-limsup 14827 df-clim 14844 df-rlim 14845 df-sum 15042 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-starv 16579 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ds 16586 df-unif 16587 df-hom 16588 df-cco 16589 df-rest 16695 df-topn 16696 df-0g 16714 df-gsum 16715 df-topgen 16716 df-pt 16717 df-prds 16720 df-xrs 16774 df-qtop 16779 df-imas 16780 df-xps 16782 df-mre 16856 df-mrc 16857 df-acs 16859 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-submnd 17956 df-mulg 18224 df-cntz 18446 df-cmn 18907 df-psmet 20536 df-xmet 20537 df-met 20538 df-bl 20539 df-mopn 20540 df-fbas 20541 df-fg 20542 df-cnfld 20545 df-top 21501 df-topon 21518 df-topsp 21540 df-bases 21553 df-cld 21626 df-ntr 21627 df-cls 21628 df-nei 21705 df-lp 21743 df-perf 21744 df-cn 21834 df-cnp 21835 df-haus 21922 df-cmp 21994 df-tx 22169 df-hmeo 22362 df-fil 22453 df-fm 22545 df-flim 22546 df-flf 22547 df-xms 22929 df-ms 22930 df-tms 22931 df-cncf 23485 df-ovol 24064 df-vol 24065 df-mbf 24219 df-itg1 24220 df-itg2 24221 df-ibl 24222 df-itg 24223 df-0p 24270 df-limc 24463 df-dv 24464

This theorem is referenced by: logdivsqrle 31921

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