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 | ffvelrnda 6955 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐸) → (𝐹‘𝑦) ∈ ℝ) |
6 | 5 | renegcld 11385 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐸) → -(𝐹‘𝑦) ∈ ℝ) |
7 | 6 | fmpttd 6983 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦)):𝐸⟶ℝ) |
8 | reelprrecn 10947 | . . . . . . 7 ⊢ ℝ ∈ {ℝ, ℂ} | |
9 | 8 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ}) |
10 | ax-resscn 10912 | . . . . . . 7 ⊢ ℝ ⊆ ℂ | |
11 | 10, 5 | sselid 3923 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐸) → (𝐹‘𝑦) ∈ ℂ) |
12 | fvexd 6783 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐸) → ((ℝ D 𝐹)‘𝑦) ∈ V) | |
13 | 4 | feqmptd 6831 | . . . . . . . 8 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ 𝐸 ↦ (𝐹‘𝑦))) |
14 | 13 | oveq2d 7284 | . . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑦 ∈ 𝐸 ↦ (𝐹‘𝑦)))) |
15 | fdvposlt.c | . . . . . . . . 9 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (𝐸–cn→ℝ)) | |
16 | cncff 24037 | . . . . . . . . 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 6831 | . . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹) = (𝑦 ∈ 𝐸 ↦ ((ℝ D 𝐹)‘𝑦))) |
19 | 14, 18 | eqtr3d 2781 | . . . . . 6 ⊢ (𝜑 → (ℝ D (𝑦 ∈ 𝐸 ↦ (𝐹‘𝑦))) = (𝑦 ∈ 𝐸 ↦ ((ℝ D 𝐹)‘𝑦))) |
20 | 9, 11, 12, 19 | dvmptneg 25111 | . . . . 5 ⊢ (𝜑 → (ℝ D (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦))) = (𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦))) |
21 | 17 | ffvelrnda 6955 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐸) → ((ℝ D 𝐹)‘𝑦) ∈ ℝ) |
22 | 21 | renegcld 11385 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐸) → -((ℝ D 𝐹)‘𝑦) ∈ ℝ) |
23 | 22 | fmpttd 6983 | . . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦)):𝐸⟶ℝ) |
24 | ssid 3947 | . . . . . . . . . 10 ⊢ ℂ ⊆ ℂ | |
25 | cncfss 24043 | . . . . . . . . . 10 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝐸–cn→ℝ) ⊆ (𝐸–cn→ℂ)) | |
26 | 10, 24, 25 | mp2an 688 | . . . . . . . . 9 ⊢ (𝐸–cn→ℝ) ⊆ (𝐸–cn→ℂ) |
27 | 26, 15 | sselid 3923 | . . . . . . . 8 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (𝐸–cn→ℂ)) |
28 | eqid 2739 | . . . . . . . . 9 ⊢ (𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦)) = (𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦)) | |
29 | 28 | negfcncf 24067 | . . . . . . . 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 24042 | . . . . . . 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 586 | . . . . . 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 2840 | . . . 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 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (ℝ D 𝐹):𝐸⟶ℝ) |
38 | ioossicc 13147 | . . . . . . . . . . 11 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) | |
39 | 38 | a1i 11 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)) |
40 | 1, 2, 3 | fct2relem 32556 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐸) |
41 | 39, 40 | sstrd 3935 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐸) |
42 | 41 | sselda 3925 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ 𝐸) |
43 | 37, 42 | ffvelrnd 6956 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ) |
44 | 43 | lt0neg1d 11527 | . . . . . 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 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (ℝ D (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦))) = (𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦))) |
47 | 46 | fveq1d 6770 | . . . . . 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 6772 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑦 = 𝑥) → ((ℝ D 𝐹)‘𝑦) = ((ℝ D 𝐹)‘𝑥)) |
51 | 50 | negeqd 11198 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) ∧ 𝑦 = 𝑥) → -((ℝ D 𝐹)‘𝑦) = -((ℝ D 𝐹)‘𝑥)) |
52 | 43 | renegcld 11385 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → -((ℝ D 𝐹)‘𝑥) ∈ ℝ) |
53 | 48, 51, 42, 52 | fvmptd 6876 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((𝑦 ∈ 𝐸 ↦ -((ℝ D 𝐹)‘𝑦))‘𝑥) = -((ℝ D 𝐹)‘𝑥)) |
54 | 47, 53 | eqtrd 2779 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦)))‘𝑥) = -((ℝ D 𝐹)‘𝑥)) |
55 | 45, 54 | breqtrrd 5106 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < ((ℝ D (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦)))‘𝑥)) |
56 | 1, 2, 3, 7, 34, 35, 55 | fdvposlt 32558 | . . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦))‘𝐴) < ((𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦))‘𝐵)) |
57 | eqidd 2740 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦)) = (𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦))) | |
58 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → 𝑦 = 𝐴) | |
59 | 58 | fveq2d 6772 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → (𝐹‘𝑦) = (𝐹‘𝐴)) |
60 | 59 | negeqd 11198 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → -(𝐹‘𝑦) = -(𝐹‘𝐴)) |
61 | 4, 2 | ffvelrnd 6956 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℝ) |
62 | 61 | renegcld 11385 | . . . 4 ⊢ (𝜑 → -(𝐹‘𝐴) ∈ ℝ) |
63 | 57, 60, 2, 62 | fvmptd 6876 | . . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦))‘𝐴) = -(𝐹‘𝐴)) |
64 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵) | |
65 | 64 | fveq2d 6772 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → (𝐹‘𝑦) = (𝐹‘𝐵)) |
66 | 65 | negeqd 11198 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → -(𝐹‘𝑦) = -(𝐹‘𝐵)) |
67 | 4, 3 | ffvelrnd 6956 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ℝ) |
68 | 67 | renegcld 11385 | . . . 4 ⊢ (𝜑 → -(𝐹‘𝐵) ∈ ℝ) |
69 | 57, 66, 3, 68 | fvmptd 6876 | . . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐸 ↦ -(𝐹‘𝑦))‘𝐵) = -(𝐹‘𝐵)) |
70 | 56, 63, 69 | 3brtr3d 5109 | . 2 ⊢ (𝜑 → -(𝐹‘𝐴) < -(𝐹‘𝐵)) |
71 | 67, 61 | ltnegd 11536 | . 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 395 = wceq 1541 ∈ wcel 2109 Vcvv 3430 ⊆ wss 3891 {cpr 4568 class class class wbr 5078 ↦ cmpt 5161 ⟶wf 6426 ‘cfv 6430 (class class class)co 7268 ℂcc 10853 ℝcr 10854 0cc0 10855 < clt 10993 -cneg 11189 (,)cioo 13061 [,]cicc 13064 –cn→ccncf 24020 D cdv 25008 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-inf2 9360 ax-cc 10175 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-pre-sup 10933 ax-addf 10934 ax-mulf 10935 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-symdif 4181 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-iin 4932 df-disj 5044 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-se 5544 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-isom 6439 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-of 7524 df-ofr 7525 df-om 7701 df-1st 7817 df-2nd 7818 df-supp 7962 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-2o 8282 df-oadd 8285 df-omul 8286 df-er 8472 df-map 8591 df-pm 8592 df-ixp 8660 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-fsupp 9090 df-fi 9131 df-sup 9162 df-inf 9163 df-oi 9230 df-dju 9643 df-card 9681 df-acn 9684 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-div 11616 df-nn 11957 df-2 12019 df-3 12020 df-4 12021 df-5 12022 df-6 12023 df-7 12024 df-8 12025 df-9 12026 df-n0 12217 df-z 12303 df-dec 12420 df-uz 12565 df-q 12671 df-rp 12713 df-xneg 12830 df-xadd 12831 df-xmul 12832 df-ioo 13065 df-ioc 13066 df-ico 13067 df-icc 13068 df-fz 13222 df-fzo 13365 df-fl 13493 df-mod 13571 df-seq 13703 df-exp 13764 df-hash 14026 df-cj 14791 df-re 14792 df-im 14793 df-sqrt 14927 df-abs 14928 df-limsup 15161 df-clim 15178 df-rlim 15179 df-sum 15379 df-struct 16829 df-sets 16846 df-slot 16864 df-ndx 16876 df-base 16894 df-ress 16923 df-plusg 16956 df-mulr 16957 df-starv 16958 df-sca 16959 df-vsca 16960 df-ip 16961 df-tset 16962 df-ple 16963 df-ds 16965 df-unif 16966 df-hom 16967 df-cco 16968 df-rest 17114 df-topn 17115 df-0g 17133 df-gsum 17134 df-topgen 17135 df-pt 17136 df-prds 17139 df-xrs 17194 df-qtop 17199 df-imas 17200 df-xps 17202 df-mre 17276 df-mrc 17277 df-acs 17279 df-mgm 18307 df-sgrp 18356 df-mnd 18367 df-submnd 18412 df-mulg 18682 df-cntz 18904 df-cmn 19369 df-psmet 20570 df-xmet 20571 df-met 20572 df-bl 20573 df-mopn 20574 df-fbas 20575 df-fg 20576 df-cnfld 20579 df-top 22024 df-topon 22041 df-topsp 22063 df-bases 22077 df-cld 22151 df-ntr 22152 df-cls 22153 df-nei 22230 df-lp 22268 df-perf 22269 df-cn 22359 df-cnp 22360 df-haus 22447 df-cmp 22519 df-tx 22694 df-hmeo 22887 df-fil 22978 df-fm 23070 df-flim 23071 df-flf 23072 df-xms 23454 df-ms 23455 df-tms 23456 df-cncf 24022 df-ovol 24609 df-vol 24610 df-mbf 24764 df-itg1 24765 df-itg2 24766 df-ibl 24767 df-itg 24768 df-0p 24815 df-limc 25011 df-dv 25012 |
This theorem is referenced by: (None) |
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