Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 3921 | . . . . . . 7 ⊢ (𝑈 ∈ (𝐴(,)𝐵) → (𝐴(,)𝐵) ≠ ∅) | |
7 | ndmioo 12202 | . . . . . . . 8 ⊢ (¬ (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴(,)𝐵) = ∅) | |
8 | 7 | necon1ai 2821 | . . . . . . 7 ⊢ ((𝐴(,)𝐵) ≠ ∅ → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) |
9 | 3, 6, 8 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) |
10 | 9 | simpld 475 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
11 | eliooord 12233 | . . . . . . . 8 ⊢ (𝑈 ∈ (𝐴(,)𝐵) → (𝐴 < 𝑈 ∧ 𝑈 < 𝐵)) | |
12 | 3, 11 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝐴 < 𝑈 ∧ 𝑈 < 𝐵)) |
13 | 12 | simpld 475 | . . . . . 6 ⊢ (𝜑 → 𝐴 < 𝑈) |
14 | ioossre 12235 | . . . . . . . . 9 ⊢ (𝐴(,)𝐵) ⊆ ℝ | |
15 | 14, 3 | sseldi 3601 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ ℝ) |
16 | 15 | rexrd 10089 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ ℝ*) |
17 | xrltle 11982 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝑈 ∈ ℝ*) → (𝐴 < 𝑈 → 𝐴 ≤ 𝑈)) | |
18 | 10, 16, 17 | syl2anc 693 | . . . . . 6 ⊢ (𝜑 → (𝐴 < 𝑈 → 𝐴 ≤ 𝑈)) |
19 | 13, 18 | mpd 15 | . . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝑈) |
20 | iooss1 12210 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≤ 𝑈) → (𝑈(,)𝐵) ⊆ (𝐴(,)𝐵)) | |
21 | 10, 19, 20 | syl2anc 693 | . . . 4 ⊢ (𝜑 → (𝑈(,)𝐵) ⊆ (𝐴(,)𝐵)) |
22 | dvferm.r | . . . 4 ⊢ (𝜑 → ∀𝑦 ∈ (𝐴(,)𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈)) | |
23 | ssralv 3666 | . . . 4 ⊢ ((𝑈(,)𝐵) ⊆ (𝐴(,)𝐵) → (∀𝑦 ∈ (𝐴(,)𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈) → ∀𝑦 ∈ (𝑈(,)𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈))) | |
24 | 21, 22, 23 | sylc 65 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ (𝑈(,)𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈)) |
25 | 1, 2, 3, 4, 5, 24 | dvferm1 23748 | . 2 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) ≤ 0) |
26 | 9 | simprd 479 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
27 | 12 | simprd 479 | . . . . . 6 ⊢ (𝜑 → 𝑈 < 𝐵) |
28 | xrltle 11982 | . . . . . . 7 ⊢ ((𝑈 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑈 < 𝐵 → 𝑈 ≤ 𝐵)) | |
29 | 16, 26, 28 | syl2anc 693 | . . . . . 6 ⊢ (𝜑 → (𝑈 < 𝐵 → 𝑈 ≤ 𝐵)) |
30 | 27, 29 | mpd 15 | . . . . 5 ⊢ (𝜑 → 𝑈 ≤ 𝐵) |
31 | iooss2 12211 | . . . . 5 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑈 ≤ 𝐵) → (𝐴(,)𝑈) ⊆ (𝐴(,)𝐵)) | |
32 | 26, 30, 31 | syl2anc 693 | . . . 4 ⊢ (𝜑 → (𝐴(,)𝑈) ⊆ (𝐴(,)𝐵)) |
33 | ssralv 3666 | . . . 4 ⊢ ((𝐴(,)𝑈) ⊆ (𝐴(,)𝐵) → (∀𝑦 ∈ (𝐴(,)𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈) → ∀𝑦 ∈ (𝐴(,)𝑈)(𝐹‘𝑦) ≤ (𝐹‘𝑈))) | |
34 | 32, 22, 33 | sylc 65 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ (𝐴(,)𝑈)(𝐹‘𝑦) ≤ (𝐹‘𝑈)) |
35 | 1, 2, 3, 4, 5, 34 | dvferm2 23750 | . 2 ⊢ (𝜑 → 0 ≤ ((ℝ D 𝐹)‘𝑈)) |
36 | dvfre 23714 | . . . . 5 ⊢ ((𝐹:𝑋⟶ℝ ∧ 𝑋 ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ) | |
37 | 1, 2, 36 | syl2anc 693 | . . . 4 ⊢ (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ) |
38 | 37, 5 | ffvelrnd 6360 | . . 3 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) ∈ ℝ) |
39 | 0re 10040 | . . 3 ⊢ 0 ∈ ℝ | |
40 | letri3 10123 | . . 3 ⊢ ((((ℝ D 𝐹)‘𝑈) ∈ ℝ ∧ 0 ∈ ℝ) → (((ℝ D 𝐹)‘𝑈) = 0 ↔ (((ℝ D 𝐹)‘𝑈) ≤ 0 ∧ 0 ≤ ((ℝ D 𝐹)‘𝑈)))) | |
41 | 38, 39, 40 | sylancl 694 | . 2 ⊢ (𝜑 → (((ℝ D 𝐹)‘𝑈) = 0 ↔ (((ℝ D 𝐹)‘𝑈) ≤ 0 ∧ 0 ≤ ((ℝ D 𝐹)‘𝑈)))) |
42 | 25, 35, 41 | mpbir2and 957 | 1 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ⊆ wss 3574 ∅c0 3915 class class class wbr 4653 dom cdm 5114 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 0cc0 9936 ℝ*cxr 10073 < clt 10074 ≤ cle 10075 (,)cioo 12175 D cdv 23627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fi 8317 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-icc 12182 df-fz 12327 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-plusg 15954 df-mulr 15955 df-starv 15956 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-rest 16083 df-topn 16084 df-topgen 16104 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-cncf 22681 df-limc 23630 df-dv 23631 |
This theorem is referenced by: rollelem 23752 dvivthlem1 23771 |
Copyright terms: Public domain | W3C validator |