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Mirrors > Home > MPE Home > Th. List > dvferm | Structured version Visualization version GIF version |
Description: Fermat's theorem on stationary points. A point 𝑈 which is a local maximum has derivative equal to zero. (Contributed by Mario Carneiro, 1-Sep-2014.) |
Ref | Expression |
---|---|
dvferm.a | ⊢ (𝜑 → 𝐹:𝑋⟶ℝ) |
dvferm.b | ⊢ (𝜑 → 𝑋 ⊆ ℝ) |
dvferm.u | ⊢ (𝜑 → 𝑈 ∈ (𝐴(,)𝐵)) |
dvferm.s | ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝑋) |
dvferm.d | ⊢ (𝜑 → 𝑈 ∈ dom (ℝ D 𝐹)) |
dvferm.r | ⊢ (𝜑 → ∀𝑦 ∈ (𝐴(,)𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈)) |
Ref | Expression |
---|---|
dvferm | ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvferm.a | . . 3 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ) | |
2 | dvferm.b | . . 3 ⊢ (𝜑 → 𝑋 ⊆ ℝ) | |
3 | dvferm.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ (𝐴(,)𝐵)) | |
4 | dvferm.s | . . 3 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝑋) | |
5 | dvferm.d | . . 3 ⊢ (𝜑 → 𝑈 ∈ dom (ℝ D 𝐹)) | |
6 | ne0i 4120 | . . . . . . 7 ⊢ (𝑈 ∈ (𝐴(,)𝐵) → (𝐴(,)𝐵) ≠ ∅) | |
7 | ndmioo 12448 | . . . . . . . 8 ⊢ (¬ (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴(,)𝐵) = ∅) | |
8 | 7 | necon1ai 2997 | . . . . . . 7 ⊢ ((𝐴(,)𝐵) ≠ ∅ → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) |
9 | 3, 6, 8 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) |
10 | 9 | simpld 489 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
11 | ioossre 12481 | . . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ ℝ | |
12 | 11, 3 | sseldi 3795 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ ℝ) |
13 | 12 | rexrd 10377 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ ℝ*) |
14 | eliooord 12479 | . . . . . . . 8 ⊢ (𝑈 ∈ (𝐴(,)𝐵) → (𝐴 < 𝑈 ∧ 𝑈 < 𝐵)) | |
15 | 3, 14 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝐴 < 𝑈 ∧ 𝑈 < 𝐵)) |
16 | 15 | simpld 489 | . . . . . 6 ⊢ (𝜑 → 𝐴 < 𝑈) |
17 | 10, 13, 16 | xrltled 12227 | . . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝑈) |
18 | iooss1 12456 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≤ 𝑈) → (𝑈(,)𝐵) ⊆ (𝐴(,)𝐵)) | |
19 | 10, 17, 18 | syl2anc 580 | . . . 4 ⊢ (𝜑 → (𝑈(,)𝐵) ⊆ (𝐴(,)𝐵)) |
20 | dvferm.r | . . . 4 ⊢ (𝜑 → ∀𝑦 ∈ (𝐴(,)𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈)) | |
21 | ssralv 3861 | . . . 4 ⊢ ((𝑈(,)𝐵) ⊆ (𝐴(,)𝐵) → (∀𝑦 ∈ (𝐴(,)𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈) → ∀𝑦 ∈ (𝑈(,)𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈))) | |
22 | 19, 20, 21 | sylc 65 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ (𝑈(,)𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈)) |
23 | 1, 2, 3, 4, 5, 22 | dvferm1 24086 | . 2 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) ≤ 0) |
24 | 9 | simprd 490 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
25 | 15 | simprd 490 | . . . . . 6 ⊢ (𝜑 → 𝑈 < 𝐵) |
26 | 13, 24, 25 | xrltled 12227 | . . . . 5 ⊢ (𝜑 → 𝑈 ≤ 𝐵) |
27 | iooss2 12457 | . . . . 5 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑈 ≤ 𝐵) → (𝐴(,)𝑈) ⊆ (𝐴(,)𝐵)) | |
28 | 24, 26, 27 | syl2anc 580 | . . . 4 ⊢ (𝜑 → (𝐴(,)𝑈) ⊆ (𝐴(,)𝐵)) |
29 | ssralv 3861 | . . . 4 ⊢ ((𝐴(,)𝑈) ⊆ (𝐴(,)𝐵) → (∀𝑦 ∈ (𝐴(,)𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈) → ∀𝑦 ∈ (𝐴(,)𝑈)(𝐹‘𝑦) ≤ (𝐹‘𝑈))) | |
30 | 28, 20, 29 | sylc 65 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ (𝐴(,)𝑈)(𝐹‘𝑦) ≤ (𝐹‘𝑈)) |
31 | 1, 2, 3, 4, 5, 30 | dvferm2 24088 | . 2 ⊢ (𝜑 → 0 ≤ ((ℝ D 𝐹)‘𝑈)) |
32 | dvfre 24052 | . . . . 5 ⊢ ((𝐹:𝑋⟶ℝ ∧ 𝑋 ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ) | |
33 | 1, 2, 32 | syl2anc 580 | . . . 4 ⊢ (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ) |
34 | 33, 5 | ffvelrnd 6585 | . . 3 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) ∈ ℝ) |
35 | 0re 10329 | . . 3 ⊢ 0 ∈ ℝ | |
36 | letri3 10412 | . . 3 ⊢ ((((ℝ D 𝐹)‘𝑈) ∈ ℝ ∧ 0 ∈ ℝ) → (((ℝ D 𝐹)‘𝑈) = 0 ↔ (((ℝ D 𝐹)‘𝑈) ≤ 0 ∧ 0 ≤ ((ℝ D 𝐹)‘𝑈)))) | |
37 | 34, 35, 36 | sylancl 581 | . 2 ⊢ (𝜑 → (((ℝ D 𝐹)‘𝑈) = 0 ↔ (((ℝ D 𝐹)‘𝑈) ≤ 0 ∧ 0 ≤ ((ℝ D 𝐹)‘𝑈)))) |
38 | 23, 31, 37 | mpbir2and 705 | 1 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ≠ wne 2970 ∀wral 3088 ⊆ wss 3768 ∅c0 4114 class class class wbr 4842 dom cdm 5311 ⟶wf 6096 ‘cfv 6100 (class class class)co 6877 ℝcr 10222 0cc0 10223 ℝ*cxr 10361 < clt 10362 ≤ cle 10363 (,)cioo 12421 D cdv 23965 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2776 ax-rep 4963 ax-sep 4974 ax-nul 4982 ax-pow 5034 ax-pr 5096 ax-un 7182 ax-cnex 10279 ax-resscn 10280 ax-1cn 10281 ax-icn 10282 ax-addcl 10283 ax-addrcl 10284 ax-mulcl 10285 ax-mulrcl 10286 ax-mulcom 10287 ax-addass 10288 ax-mulass 10289 ax-distr 10290 ax-i2m1 10291 ax-1ne0 10292 ax-1rid 10293 ax-rnegex 10294 ax-rrecex 10295 ax-cnre 10296 ax-pre-lttri 10297 ax-pre-lttrn 10298 ax-pre-ltadd 10299 ax-pre-mulgt0 10300 ax-pre-sup 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2785 df-cleq 2791 df-clel 2794 df-nfc 2929 df-ne 2971 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3386 df-sbc 3633 df-csb 3728 df-dif 3771 df-un 3773 df-in 3775 df-ss 3782 df-pss 3784 df-nul 4115 df-if 4277 df-pw 4350 df-sn 4368 df-pr 4370 df-tp 4372 df-op 4374 df-uni 4628 df-int 4667 df-iun 4711 df-iin 4712 df-br 4843 df-opab 4905 df-mpt 4922 df-tr 4945 df-id 5219 df-eprel 5224 df-po 5232 df-so 5233 df-fr 5270 df-we 5272 df-xp 5317 df-rel 5318 df-cnv 5319 df-co 5320 df-dm 5321 df-rn 5322 df-res 5323 df-ima 5324 df-pred 5897 df-ord 5943 df-on 5944 df-lim 5945 df-suc 5946 df-iota 6063 df-fun 6102 df-fn 6103 df-f 6104 df-f1 6105 df-fo 6106 df-f1o 6107 df-fv 6108 df-riota 6838 df-ov 6880 df-oprab 6881 df-mpt2 6882 df-om 7299 df-1st 7400 df-2nd 7401 df-wrecs 7644 df-recs 7706 df-rdg 7744 df-1o 7798 df-oadd 7802 df-er 7981 df-map 8096 df-pm 8097 df-en 8195 df-dom 8196 df-sdom 8197 df-fin 8198 df-fi 8558 df-sup 8589 df-inf 8590 df-pnf 10364 df-mnf 10365 df-xr 10366 df-ltxr 10367 df-le 10368 df-sub 10557 df-neg 10558 df-div 10976 df-nn 11312 df-2 11373 df-3 11374 df-4 11375 df-5 11376 df-6 11377 df-7 11378 df-8 11379 df-9 11380 df-n0 11578 df-z 11664 df-dec 11781 df-uz 11928 df-q 12031 df-rp 12072 df-xneg 12190 df-xadd 12191 df-xmul 12192 df-ioo 12425 df-icc 12428 df-fz 12578 df-seq 13053 df-exp 13112 df-cj 14177 df-re 14178 df-im 14179 df-sqrt 14313 df-abs 14314 df-struct 16183 df-ndx 16184 df-slot 16185 df-base 16187 df-plusg 16277 df-mulr 16278 df-starv 16279 df-tset 16283 df-ple 16284 df-ds 16286 df-unif 16287 df-rest 16395 df-topn 16396 df-topgen 16416 df-psmet 20057 df-xmet 20058 df-met 20059 df-bl 20060 df-mopn 20061 df-fbas 20062 df-fg 20063 df-cnfld 20066 df-top 21024 df-topon 21041 df-topsp 21063 df-bases 21076 df-cld 21149 df-ntr 21150 df-cls 21151 df-nei 21228 df-lp 21266 df-perf 21267 df-cn 21357 df-cnp 21358 df-haus 21445 df-fil 21975 df-fm 22067 df-flim 22068 df-flf 22069 df-xms 22450 df-ms 22451 df-cncf 23006 df-limc 23968 df-dv 23969 |
This theorem is referenced by: rollelem 24090 dvivthlem1 24109 |
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