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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 4250 | . . . . . . 7 ⊢ (𝑈 ∈ (𝐴(,)𝐵) → (𝐴(,)𝐵) ≠ ∅) | |
7 | ndmioo 12753 | . . . . . . . 8 ⊢ (¬ (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴(,)𝐵) = ∅) | |
8 | 7 | necon1ai 3014 | . . . . . . 7 ⊢ ((𝐴(,)𝐵) ≠ ∅ → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) |
9 | 3, 6, 8 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) |
10 | 9 | simpld 498 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
11 | ioossre 12786 | . . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ ℝ | |
12 | 11, 3 | sseldi 3913 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ ℝ) |
13 | 12 | rexrd 10680 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ ℝ*) |
14 | eliooord 12784 | . . . . . . . 8 ⊢ (𝑈 ∈ (𝐴(,)𝐵) → (𝐴 < 𝑈 ∧ 𝑈 < 𝐵)) | |
15 | 3, 14 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝐴 < 𝑈 ∧ 𝑈 < 𝐵)) |
16 | 15 | simpld 498 | . . . . . 6 ⊢ (𝜑 → 𝐴 < 𝑈) |
17 | 10, 13, 16 | xrltled 12531 | . . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝑈) |
18 | iooss1 12761 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≤ 𝑈) → (𝑈(,)𝐵) ⊆ (𝐴(,)𝐵)) | |
19 | 10, 17, 18 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (𝑈(,)𝐵) ⊆ (𝐴(,)𝐵)) |
20 | dvferm.r | . . . 4 ⊢ (𝜑 → ∀𝑦 ∈ (𝐴(,)𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈)) | |
21 | ssralv 3981 | . . . 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 24588 | . 2 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) ≤ 0) |
24 | 9 | simprd 499 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
25 | 15 | simprd 499 | . . . . . 6 ⊢ (𝜑 → 𝑈 < 𝐵) |
26 | 13, 24, 25 | xrltled 12531 | . . . . 5 ⊢ (𝜑 → 𝑈 ≤ 𝐵) |
27 | iooss2 12762 | . . . . 5 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑈 ≤ 𝐵) → (𝐴(,)𝑈) ⊆ (𝐴(,)𝐵)) | |
28 | 24, 26, 27 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (𝐴(,)𝑈) ⊆ (𝐴(,)𝐵)) |
29 | ssralv 3981 | . . . 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 24590 | . 2 ⊢ (𝜑 → 0 ≤ ((ℝ D 𝐹)‘𝑈)) |
32 | dvfre 24554 | . . . . 5 ⊢ ((𝐹:𝑋⟶ℝ ∧ 𝑋 ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ) | |
33 | 1, 2, 32 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ) |
34 | 33, 5 | ffvelrnd 6829 | . . 3 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) ∈ ℝ) |
35 | 0re 10632 | . . 3 ⊢ 0 ∈ ℝ | |
36 | letri3 10715 | . . 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 589 | . 2 ⊢ (𝜑 → (((ℝ D 𝐹)‘𝑈) = 0 ↔ (((ℝ D 𝐹)‘𝑈) ≤ 0 ∧ 0 ≤ ((ℝ D 𝐹)‘𝑈)))) |
38 | 23, 31, 37 | mpbir2and 712 | 1 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ⊆ wss 3881 ∅c0 4243 class class class wbr 5030 dom cdm 5519 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 0cc0 10526 ℝ*cxr 10663 < clt 10664 ≤ cle 10665 (,)cioo 12726 D cdv 24466 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fi 8859 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-icc 12733 df-fz 12886 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-plusg 16570 df-mulr 16571 df-starv 16572 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-rest 16688 df-topn 16689 df-topgen 16709 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-perf 21742 df-cn 21832 df-cnp 21833 df-haus 21920 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-xms 22927 df-ms 22928 df-cncf 23483 df-limc 24469 df-dv 24470 |
This theorem is referenced by: rollelem 24592 dvivthlem1 24611 |
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