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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fdvneggt | Structured version Visualization version GIF version |
Description: Functions with a negative derivative, i.e. monotonously decreasing functions, inverse strict ordering. (Contributed by Thierry Arnoux, 20-Dec-2021.) |
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)) |
fdvneggt.lt | ⊢ (𝜑 → 𝐴 < 𝐵) |
fdvneggt.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) < 0) |
Ref | Expression |
---|---|
fdvneggt | ⊢ (𝜑 → (𝐹‘𝐵) < (𝐹‘𝐴)) |
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 7035 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐸) → (𝐹‘𝑦) ∈ ℝ) |
6 | 5 | renegcld 11582 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐸) → -(𝐹‘𝑦) ∈ ℝ) |
7 | 6 | fmpttd 7063 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦)):𝐸⟶ℝ) |
8 | reelprrecn 11143 | . . . . . . 7 ⊢ ℝ ∈ {ℝ, ℂ} | |
9 | 8 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ}) |
10 | ax-resscn 11108 | . . . . . . 7 ⊢ ℝ ⊆ ℂ | |
11 | 10, 5 | sselid 3942 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐸) → (𝐹‘𝑦) ∈ ℂ) |
12 | fvexd 6857 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐸) → ((ℝ D 𝐹)‘𝑦) ∈ V) | |
13 | 4 | feqmptd 6910 | . . . . . . . 8 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ 𝐸 ↦ (𝐹‘𝑦))) |
14 | 13 | oveq2d 7373 | . . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑦 ∈ 𝐸 ↦ (𝐹‘𝑦)))) |
15 | fdvposlt.c | . . . . . . . . 9 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (𝐸–cn→ℝ)) | |
16 | cncff 24256 | . . . . . . . . 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 6910 | . . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹) = (𝑦 ∈ 𝐸 ↦ ((ℝ D 𝐹)‘𝑦))) |
19 | 14, 18 | eqtr3d 2778 | . . . . . 6 ⊢ (𝜑 → (ℝ D (𝑦 ∈ 𝐸 ↦ (𝐹‘𝑦))) = (𝑦 ∈ 𝐸 ↦ ((ℝ D 𝐹)‘𝑦))) |
20 | 9, 11, 12, 19 | dvmptneg 25330 | . . . . 5 ⊢ (𝜑 → (ℝ D (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦))) = (𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦))) |
21 | 17 | ffvelcdmda 7035 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐸) → ((ℝ D 𝐹)‘𝑦) ∈ ℝ) |
22 | 21 | renegcld 11582 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐸) → -((ℝ D 𝐹)‘𝑦) ∈ ℝ) |
23 | 22 | fmpttd 7063 | . . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦)):𝐸⟶ℝ) |
24 | ssid 3966 | . . . . . . . . . 10 ⊢ ℂ ⊆ ℂ | |
25 | cncfss 24262 | . . . . . . . . . 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 | sselid 3942 | . . . . . . . 8 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (𝐸–cn→ℂ)) |
28 | eqid 2736 | . . . . . . . . 9 ⊢ (𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦)) = (𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦)) | |
29 | 28 | negfcncf 24286 | . . . . . . . 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 24261 | . . . . . . 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 256 | . . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦)) ∈ (𝐸–cn→ℝ)) |
34 | 20, 33 | eqeltrd 2838 | . . . 4 ⊢ (𝜑 → (ℝ D (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦))) ∈ (𝐸–cn→ℝ)) |
35 | fdvneggt.lt | . . . 4 ⊢ (𝜑 → 𝐴 < 𝐵) | |
36 | fdvneggt.1 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) < 0) | |
37 | 17 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (ℝ D 𝐹):𝐸⟶ℝ) |
38 | ioossicc 13350 | . . . . . . . . . . 11 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) | |
39 | 38 | a1i 11 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)) |
40 | 1, 2, 3 | fct2relem 33210 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐸) |
41 | 39, 40 | sstrd 3954 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐸) |
42 | 41 | sselda 3944 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ 𝐸) |
43 | 37, 42 | ffvelcdmd 7036 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ) |
44 | 43 | lt0neg1d 11724 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (((ℝ D 𝐹)‘𝑥) < 0 ↔ 0 < -((ℝ D 𝐹)‘𝑥))) |
45 | 36, 44 | mpbid 231 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < -((ℝ D 𝐹)‘𝑥)) |
46 | 20 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (ℝ D (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦))) = (𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦))) |
47 | 46 | fveq1d 6844 | . . . . . 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 485 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥) | |
50 | 49 | fveq2d 6846 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑦 = 𝑥) → ((ℝ D 𝐹)‘𝑦) = ((ℝ D 𝐹)‘𝑥)) |
51 | 50 | negeqd 11395 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑦 = 𝑥) → -((ℝ D 𝐹)‘𝑦) = -((ℝ D 𝐹)‘𝑥)) |
52 | 43 | renegcld 11582 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → -((ℝ D 𝐹)‘𝑥) ∈ ℝ) |
53 | 48, 51, 42, 52 | fvmptd 6955 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦))‘𝑥) = -((ℝ D 𝐹)‘𝑥)) |
54 | 47, 53 | eqtrd 2776 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦)))‘𝑥) = -((ℝ D 𝐹)‘𝑥)) |
55 | 45, 54 | breqtrrd 5133 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < ((ℝ D (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦)))‘𝑥)) |
56 | 1, 2, 3, 7, 34, 35, 55 | fdvposlt 33212 | . . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦))‘𝐴) < ((𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦))‘𝐵)) |
57 | eqidd 2737 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦)) = (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦))) | |
58 | simpr 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → 𝑦 = 𝐴) | |
59 | 58 | fveq2d 6846 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → (𝐹‘𝑦) = (𝐹‘𝐴)) |
60 | 59 | negeqd 11395 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → -(𝐹‘𝑦) = -(𝐹‘𝐴)) |
61 | 4, 2 | ffvelcdmd 7036 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℝ) |
62 | 61 | renegcld 11582 | . . . 4 ⊢ (𝜑 → -(𝐹‘𝐴) ∈ ℝ) |
63 | 57, 60, 2, 62 | fvmptd 6955 | . . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦))‘𝐴) = -(𝐹‘𝐴)) |
64 | simpr 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵) | |
65 | 64 | fveq2d 6846 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → (𝐹‘𝑦) = (𝐹‘𝐵)) |
66 | 65 | negeqd 11395 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → -(𝐹‘𝑦) = -(𝐹‘𝐵)) |
67 | 4, 3 | ffvelcdmd 7036 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ℝ) |
68 | 67 | renegcld 11582 | . . . 4 ⊢ (𝜑 → -(𝐹‘𝐵) ∈ ℝ) |
69 | 57, 66, 3, 68 | fvmptd 6955 | . . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦))‘𝐵) = -(𝐹‘𝐵)) |
70 | 56, 63, 69 | 3brtr3d 5136 | . 2 ⊢ (𝜑 → -(𝐹‘𝐴) < -(𝐹‘𝐵)) |
71 | 67, 61 | ltnegd 11733 | . 2 ⊢ (𝜑 → ((𝐹‘𝐵) < (𝐹‘𝐴) ↔ -(𝐹‘𝐴) < -(𝐹‘𝐵))) |
72 | 70, 71 | mpbird 256 | 1 ⊢ (𝜑 → (𝐹‘𝐵) < (𝐹‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3445 ⊆ wss 3910 {cpr 4588 class class class wbr 5105 ↦ cmpt 5188 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 ℝcr 11050 0cc0 11051 < clt 11189 -cneg 11386 (,)cioo 13264 [,]cicc 13267 –cn→ccncf 24239 D cdv 25227 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cc 10371 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-symdif 4202 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-disj 5071 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-ofr 7618 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-oadd 8416 df-omul 8417 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-dju 9837 df-card 9875 df-acn 9878 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ioc 13269 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-limsup 15353 df-clim 15370 df-rlim 15371 df-sum 15571 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lp 22487 df-perf 22488 df-cn 22578 df-cnp 22579 df-haus 22666 df-cmp 22738 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241 df-ovol 24828 df-vol 24829 df-mbf 24983 df-itg1 24984 df-itg2 24985 df-ibl 24986 df-itg 24987 df-0p 25034 df-limc 25230 df-dv 25231 |
This theorem is referenced by: (None) |
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