fdvnegge - Mathbox for Thierry Arnoux

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Theorem fdvnegge 34708

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	ffvelcdmda 7027	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐸) → (𝐹‘𝑦) ∈ ℝ)
6	5	renegcld 11562	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐸) → -(𝐹‘𝑦) ∈ ℝ)
7	6	fmpttd 7058	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦)):𝐸⟶ℝ)
8		reelprrecn 11116	. . . . . . 7 ⊢ ℝ ∈ {ℝ, ℂ}
9	8	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
10		ax-resscn 11081	. . . . . . 7 ⊢ ℝ ⊆ ℂ
11	10, 5	sselid 3929	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐸) → (𝐹‘𝑦) ∈ ℂ)
12		fvexd 6847	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐸) → ((ℝ D 𝐹)‘𝑦) ∈ V)
13	4	feqmptd 6900	. . . . . . . 8 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ 𝐸 ↦ (𝐹‘𝑦)))
14	13	oveq2d 7372	. . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑦 ∈ 𝐸 ↦ (𝐹‘𝑦))))
15		fdvposlt.c	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (𝐸–cn→ℝ))
16		cncff 24840	. . . . . . . . 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 6900	. . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹) = (𝑦 ∈ 𝐸 ↦ ((ℝ D 𝐹)‘𝑦)))
19	14, 18	eqtr3d 2771	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑦 ∈ 𝐸 ↦ (𝐹‘𝑦))) = (𝑦 ∈ 𝐸 ↦ ((ℝ D 𝐹)‘𝑦)))
20	9, 11, 12, 19	dvmptneg 25924	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦))) = (𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦)))
21	17	ffvelcdmda 7027	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐸) → ((ℝ D 𝐹)‘𝑦) ∈ ℝ)
22	21	renegcld 11562	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐸) → -((ℝ D 𝐹)‘𝑦) ∈ ℝ)
23	22	fmpttd 7058	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦)):𝐸⟶ℝ)
24		ssid 3954	. . . . . . . . . 10 ⊢ ℂ ⊆ ℂ
25		cncfss 24846	. . . . . . . . . 10 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝐸–cn→ℝ) ⊆ (𝐸–cn→ℂ))
26	10, 24, 25	mp2an 692	. . . . . . . . 9 ⊢ (𝐸–cn→ℝ) ⊆ (𝐸–cn→ℂ)
27	26, 15	sselid 3929	. . . . . . . 8 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (𝐸–cn→ℂ))
28		eqid 2734	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦)) = (𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦))
29	28	negfcncf 24871	. . . . . . . 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		cncfcdm 24845	. . . . . . 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 587	. . . . . 6 ⊢ (𝜑 → ((𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦)) ∈ (𝐸–cn→ℝ) ↔ (𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦)):𝐸⟶ℝ))
33	23, 32	mpbird 257	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦)) ∈ (𝐸–cn→ℝ))
34	20, 33	eqeltrd 2834	. . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (ℝ D 𝐹):𝐸⟶ℝ)
38		ioossicc 13347	. . . . . . . . . . 11 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
39	38	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
40	1, 2, 3	fct2relem 34703	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐸)
41	39, 40	sstrd 3942	. . . . . . . . 9 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐸)
42	41	sselda 3931	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ 𝐸)
43	37, 42	ffvelcdmd 7028	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ)
44	43	le0neg1d 11706	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D 𝐹)‘𝑥) ≤ 0 ↔ 0 ≤ -((ℝ D 𝐹)‘𝑥)))
45	36, 44	mpbid 232	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ≤ -((ℝ D 𝐹)‘𝑥))
46	20	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (ℝ D (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦))) = (𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦)))
47	46	fveq1d 6834	. . . . . 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 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥)
50	49	fveq2d 6836	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑦 = 𝑥) → ((ℝ D 𝐹)‘𝑦) = ((ℝ D 𝐹)‘𝑥))
51	50	negeqd 11372	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑦 = 𝑥) → -((ℝ D 𝐹)‘𝑦) = -((ℝ D 𝐹)‘𝑥))
52	43	renegcld 11562	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → -((ℝ D 𝐹)‘𝑥) ∈ ℝ)
53	48, 51, 42, 52	fvmptd 6946	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦))‘𝑥) = -((ℝ D 𝐹)‘𝑥))
54	47, 53	eqtrd 2769	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦)))‘𝑥) = -((ℝ D 𝐹)‘𝑥))
55	45, 54	breqtrrd 5124	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ≤ ((ℝ D (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦)))‘𝑥))
56	1, 2, 3, 7, 34, 35, 55	fdvposle 34707	. . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦))‘𝐴) ≤ ((𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦))‘𝐵))
57		eqidd 2735	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦)) = (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦)))
58		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → 𝑦 = 𝐴)
59	58	fveq2d 6836	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → (𝐹‘𝑦) = (𝐹‘𝐴))
60	59	negeqd 11372	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → -(𝐹‘𝑦) = -(𝐹‘𝐴))
61	4, 2	ffvelcdmd 7028	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℝ)
62	61	renegcld 11562	. . . 4 ⊢ (𝜑 → -(𝐹‘𝐴) ∈ ℝ)
63	57, 60, 2, 62	fvmptd 6946	. . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦))‘𝐴) = -(𝐹‘𝐴))
64		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
65	64	fveq2d 6836	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → (𝐹‘𝑦) = (𝐹‘𝐵))
66	65	negeqd 11372	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → -(𝐹‘𝑦) = -(𝐹‘𝐵))
67	4, 3	ffvelcdmd 7028	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ℝ)
68	67	renegcld 11562	. . . 4 ⊢ (𝜑 → -(𝐹‘𝐵) ∈ ℝ)
69	57, 66, 3, 68	fvmptd 6946	. . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦))‘𝐵) = -(𝐹‘𝐵))
70	56, 63, 69	3brtr3d 5127	. 2 ⊢ (𝜑 → -(𝐹‘𝐴) ≤ -(𝐹‘𝐵))
71	67, 61	lenegd 11714	. 2 ⊢ (𝜑 → ((𝐹‘𝐵) ≤ (𝐹‘𝐴) ↔ -(𝐹‘𝐴) ≤ -(𝐹‘𝐵)))
72	70, 71	mpbird 257	1 ⊢ (𝜑 → (𝐹‘𝐵) ≤ (𝐹‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 Vcvv 3438 ⊆ wss 3899 {cpr 4580 class class class wbr 5096 ↦ cmpt 5177 ⟶wf 6486 ‘cfv 6490 (class class class)co 7356 ℂcc 11022 ℝcr 11023 0cc0 11024 ≤ cle 11165 -cneg 11363 (,)cioo 13259 [,]cicc 13262 –cn→ccncf 24823 D cdv 25818

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-inf2 9548 ax-cc 10343 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102 ax-addf 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-symdif 4203 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-iin 4947 df-disj 5064 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-ofr 7621 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-omul 8400 df-er 8633 df-map 8763 df-pm 8764 df-ixp 8834 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-fsupp 9263 df-fi 9312 df-sup 9343 df-inf 9344 df-oi 9413 df-dju 9811 df-card 9849 df-acn 9852 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-z 12487 df-dec 12606 df-uz 12750 df-q 12860 df-rp 12904 df-xneg 13024 df-xadd 13025 df-xmul 13026 df-ioo 13263 df-ioc 13264 df-ico 13265 df-icc 13266 df-fz 13422 df-fzo 13569 df-fl 13710 df-mod 13788 df-seq 13923 df-exp 13983 df-hash 14252 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157 df-limsup 15392 df-clim 15409 df-rlim 15410 df-sum 15608 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-mulr 17189 df-starv 17190 df-sca 17191 df-vsca 17192 df-ip 17193 df-tset 17194 df-ple 17195 df-ds 17197 df-unif 17198 df-hom 17199 df-cco 17200 df-rest 17340 df-topn 17341 df-0g 17359 df-gsum 17360 df-topgen 17361 df-pt 17362 df-prds 17365 df-xrs 17421 df-qtop 17426 df-imas 17427 df-xps 17429 df-mre 17503 df-mrc 17504 df-acs 17506 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18707 df-mulg 18996 df-cntz 19244 df-cmn 19709 df-psmet 21299 df-xmet 21300 df-met 21301 df-bl 21302 df-mopn 21303 df-fbas 21304 df-fg 21305 df-cnfld 21308 df-top 22836 df-topon 22853 df-topsp 22875 df-bases 22888 df-cld 22961 df-ntr 22962 df-cls 22963 df-nei 23040 df-lp 23078 df-perf 23079 df-cn 23169 df-cnp 23170 df-haus 23257 df-cmp 23329 df-tx 23504 df-hmeo 23697 df-fil 23788 df-fm 23880 df-flim 23881 df-flf 23882 df-xms 24262 df-ms 24263 df-tms 24264 df-cncf 24825 df-ovol 25419 df-vol 25420 df-mbf 25574 df-itg1 25575 df-itg2 25576 df-ibl 25577 df-itg 25578 df-0p 25625 df-limc 25821 df-dv 25822

This theorem is referenced by: logdivsqrle 34756

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