fdvnegge - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > fdvnegge		Structured version Visualization version GIF version

Theorem fdvnegge 34634

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 7074	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐸) → (𝐹‘𝑦) ∈ ℝ)
6	5	renegcld 11664	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐸) → -(𝐹‘𝑦) ∈ ℝ)
7	6	fmpttd 7105	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦)):𝐸⟶ℝ)
8		reelprrecn 11221	. . . . . . 7 ⊢ ℝ ∈ {ℝ, ℂ}
9	8	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
10		ax-resscn 11186	. . . . . . 7 ⊢ ℝ ⊆ ℂ
11	10, 5	sselid 3956	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐸) → (𝐹‘𝑦) ∈ ℂ)
12		fvexd 6891	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐸) → ((ℝ D 𝐹)‘𝑦) ∈ V)
13	4	feqmptd 6947	. . . . . . . 8 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ 𝐸 ↦ (𝐹‘𝑦)))
14	13	oveq2d 7421	. . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑦 ∈ 𝐸 ↦ (𝐹‘𝑦))))
15		fdvposlt.c	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (𝐸–cn→ℝ))
16		cncff 24837	. . . . . . . . 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 6947	. . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹) = (𝑦 ∈ 𝐸 ↦ ((ℝ D 𝐹)‘𝑦)))
19	14, 18	eqtr3d 2772	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑦 ∈ 𝐸 ↦ (𝐹‘𝑦))) = (𝑦 ∈ 𝐸 ↦ ((ℝ D 𝐹)‘𝑦)))
20	9, 11, 12, 19	dvmptneg 25922	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦))) = (𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦)))
21	17	ffvelcdmda 7074	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐸) → ((ℝ D 𝐹)‘𝑦) ∈ ℝ)
22	21	renegcld 11664	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐸) → -((ℝ D 𝐹)‘𝑦) ∈ ℝ)
23	22	fmpttd 7105	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦)):𝐸⟶ℝ)
24		ssid 3981	. . . . . . . . . 10 ⊢ ℂ ⊆ ℂ
25		cncfss 24843	. . . . . . . . . 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 3956	. . . . . . . 8 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (𝐸–cn→ℂ))
28		eqid 2735	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦)) = (𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦))
29	28	negfcncf 24868	. . . . . . . 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 24842	. . . . . . 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 13450	. . . . . . . . . . 11 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
39	38	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
40	1, 2, 3	fct2relem 34629	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐸)
41	39, 40	sstrd 3969	. . . . . . . . 9 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐸)
42	41	sselda 3958	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ 𝐸)
43	37, 42	ffvelcdmd 7075	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ)
44	43	le0neg1d 11808	. . . . . 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 6878	. . . . . 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 6880	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑦 = 𝑥) → ((ℝ D 𝐹)‘𝑦) = ((ℝ D 𝐹)‘𝑥))
51	50	negeqd 11476	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑦 = 𝑥) → -((ℝ D 𝐹)‘𝑦) = -((ℝ D 𝐹)‘𝑥))
52	43	renegcld 11664	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → -((ℝ D 𝐹)‘𝑥) ∈ ℝ)
53	48, 51, 42, 52	fvmptd 6993	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦))‘𝑥) = -((ℝ D 𝐹)‘𝑥))
54	47, 53	eqtrd 2770	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦)))‘𝑥) = -((ℝ D 𝐹)‘𝑥))
55	45, 54	breqtrrd 5147	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ≤ ((ℝ D (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦)))‘𝑥))
56	1, 2, 3, 7, 34, 35, 55	fdvposle 34633	. . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦))‘𝐴) ≤ ((𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦))‘𝐵))
57		eqidd 2736	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦)) = (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦)))
58		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → 𝑦 = 𝐴)
59	58	fveq2d 6880	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → (𝐹‘𝑦) = (𝐹‘𝐴))
60	59	negeqd 11476	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → -(𝐹‘𝑦) = -(𝐹‘𝐴))
61	4, 2	ffvelcdmd 7075	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℝ)
62	61	renegcld 11664	. . . 4 ⊢ (𝜑 → -(𝐹‘𝐴) ∈ ℝ)
63	57, 60, 2, 62	fvmptd 6993	. . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦))‘𝐴) = -(𝐹‘𝐴))
64		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
65	64	fveq2d 6880	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → (𝐹‘𝑦) = (𝐹‘𝐵))
66	65	negeqd 11476	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → -(𝐹‘𝑦) = -(𝐹‘𝐵))
67	4, 3	ffvelcdmd 7075	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ℝ)
68	67	renegcld 11664	. . . 4 ⊢ (𝜑 → -(𝐹‘𝐵) ∈ ℝ)
69	57, 66, 3, 68	fvmptd 6993	. . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦))‘𝐵) = -(𝐹‘𝐵))
70	56, 63, 69	3brtr3d 5150	. 2 ⊢ (𝜑 → -(𝐹‘𝐴) ≤ -(𝐹‘𝐵))
71	67, 61	lenegd 11816	. 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 1540 ∈ wcel 2108 Vcvv 3459 ⊆ wss 3926 {cpr 4603 class class class wbr 5119 ↦ cmpt 5201 ⟶wf 6527 ‘cfv 6531 (class class class)co 7405 ℂcc 11127 ℝcr 11128 0cc0 11129 ≤ cle 11270 -cneg 11467 (,)cioo 13362 [,]cicc 13365 –cn→ccncf 24820 D cdv 25816

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-inf2 9655 ax-cc 10449 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207 ax-addf 11208

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-symdif 4228 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-disj 5087 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7671 df-ofr 7672 df-om 7862 df-1st 7988 df-2nd 7989 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-oadd 8484 df-omul 8485 df-er 8719 df-map 8842 df-pm 8843 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9374 df-fi 9423 df-sup 9454 df-inf 9455 df-oi 9524 df-dju 9915 df-card 9953 df-acn 9956 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-z 12589 df-dec 12709 df-uz 12853 df-q 12965 df-rp 13009 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-ioo 13366 df-ioc 13367 df-ico 13368 df-icc 13369 df-fz 13525 df-fzo 13672 df-fl 13809 df-mod 13887 df-seq 14020 df-exp 14080 df-hash 14349 df-cj 15118 df-re 15119 df-im 15120 df-sqrt 15254 df-abs 15255 df-limsup 15487 df-clim 15504 df-rlim 15505 df-sum 15703 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-starv 17286 df-sca 17287 df-vsca 17288 df-ip 17289 df-tset 17290 df-ple 17291 df-ds 17293 df-unif 17294 df-hom 17295 df-cco 17296 df-rest 17436 df-topn 17437 df-0g 17455 df-gsum 17456 df-topgen 17457 df-pt 17458 df-prds 17461 df-xrs 17516 df-qtop 17521 df-imas 17522 df-xps 17524 df-mre 17598 df-mrc 17599 df-acs 17601 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-submnd 18762 df-mulg 19051 df-cntz 19300 df-cmn 19763 df-psmet 21307 df-xmet 21308 df-met 21309 df-bl 21310 df-mopn 21311 df-fbas 21312 df-fg 21313 df-cnfld 21316 df-top 22832 df-topon 22849 df-topsp 22871 df-bases 22884 df-cld 22957 df-ntr 22958 df-cls 22959 df-nei 23036 df-lp 23074 df-perf 23075 df-cn 23165 df-cnp 23166 df-haus 23253 df-cmp 23325 df-tx 23500 df-hmeo 23693 df-fil 23784 df-fm 23876 df-flim 23877 df-flf 23878 df-xms 24259 df-ms 24260 df-tms 24261 df-cncf 24822 df-ovol 25417 df-vol 25418 df-mbf 25572 df-itg1 25573 df-itg2 25574 df-ibl 25575 df-itg 25576 df-0p 25623 df-limc 25819 df-dv 25820

This theorem is referenced by: logdivsqrle 34682

W3C validator