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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 3880 | . . . . . . 7 ⊢ (𝑈 ∈ (𝐴(,)𝐵) → (𝐴(,)𝐵) ≠ ∅) | |
7 | ndmioo 12073 | . . . . . . . 8 ⊢ (¬ (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴(,)𝐵) = ∅) | |
8 | 7 | necon1ai 2809 | . . . . . . 7 ⊢ ((𝐴(,)𝐵) ≠ ∅ → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) |
9 | 3, 6, 8 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) |
10 | 9 | simpld 474 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
11 | eliooord 12104 | . . . . . . . 8 ⊢ (𝑈 ∈ (𝐴(,)𝐵) → (𝐴 < 𝑈 ∧ 𝑈 < 𝐵)) | |
12 | 3, 11 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝐴 < 𝑈 ∧ 𝑈 < 𝐵)) |
13 | 12 | simpld 474 | . . . . . 6 ⊢ (𝜑 → 𝐴 < 𝑈) |
14 | ioossre 12106 | . . . . . . . . 9 ⊢ (𝐴(,)𝐵) ⊆ ℝ | |
15 | 14, 3 | sseldi 3566 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ ℝ) |
16 | 15 | rexrd 9968 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ ℝ*) |
17 | xrltle 11858 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝑈 ∈ ℝ*) → (𝐴 < 𝑈 → 𝐴 ≤ 𝑈)) | |
18 | 10, 16, 17 | syl2anc 691 | . . . . . 6 ⊢ (𝜑 → (𝐴 < 𝑈 → 𝐴 ≤ 𝑈)) |
19 | 13, 18 | mpd 15 | . . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝑈) |
20 | iooss1 12081 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≤ 𝑈) → (𝑈(,)𝐵) ⊆ (𝐴(,)𝐵)) | |
21 | 10, 19, 20 | syl2anc 691 | . . . 4 ⊢ (𝜑 → (𝑈(,)𝐵) ⊆ (𝐴(,)𝐵)) |
22 | dvferm.r | . . . 4 ⊢ (𝜑 → ∀𝑦 ∈ (𝐴(,)𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈)) | |
23 | ssralv 3629 | . . . 4 ⊢ ((𝑈(,)𝐵) ⊆ (𝐴(,)𝐵) → (∀𝑦 ∈ (𝐴(,)𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈) → ∀𝑦 ∈ (𝑈(,)𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈))) | |
24 | 21, 22, 23 | sylc 63 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ (𝑈(,)𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈)) |
25 | 1, 2, 3, 4, 5, 24 | dvferm1 23552 | . 2 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) ≤ 0) |
26 | 9 | simprd 478 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
27 | 12 | simprd 478 | . . . . . 6 ⊢ (𝜑 → 𝑈 < 𝐵) |
28 | xrltle 11858 | . . . . . . 7 ⊢ ((𝑈 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑈 < 𝐵 → 𝑈 ≤ 𝐵)) | |
29 | 16, 26, 28 | syl2anc 691 | . . . . . 6 ⊢ (𝜑 → (𝑈 < 𝐵 → 𝑈 ≤ 𝐵)) |
30 | 27, 29 | mpd 15 | . . . . 5 ⊢ (𝜑 → 𝑈 ≤ 𝐵) |
31 | iooss2 12082 | . . . . 5 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑈 ≤ 𝐵) → (𝐴(,)𝑈) ⊆ (𝐴(,)𝐵)) | |
32 | 26, 30, 31 | syl2anc 691 | . . . 4 ⊢ (𝜑 → (𝐴(,)𝑈) ⊆ (𝐴(,)𝐵)) |
33 | ssralv 3629 | . . . 4 ⊢ ((𝐴(,)𝑈) ⊆ (𝐴(,)𝐵) → (∀𝑦 ∈ (𝐴(,)𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈) → ∀𝑦 ∈ (𝐴(,)𝑈)(𝐹‘𝑦) ≤ (𝐹‘𝑈))) | |
34 | 32, 22, 33 | sylc 63 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ (𝐴(,)𝑈)(𝐹‘𝑦) ≤ (𝐹‘𝑈)) |
35 | 1, 2, 3, 4, 5, 34 | dvferm2 23554 | . 2 ⊢ (𝜑 → 0 ≤ ((ℝ D 𝐹)‘𝑈)) |
36 | dvfre 23520 | . . . . 5 ⊢ ((𝐹:𝑋⟶ℝ ∧ 𝑋 ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ) | |
37 | 1, 2, 36 | syl2anc 691 | . . . 4 ⊢ (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ) |
38 | 37, 5 | ffvelrnd 6268 | . . 3 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) ∈ ℝ) |
39 | 0re 9919 | . . 3 ⊢ 0 ∈ ℝ | |
40 | letri3 10002 | . . 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 693 | . 2 ⊢ (𝜑 → (((ℝ D 𝐹)‘𝑈) = 0 ↔ (((ℝ D 𝐹)‘𝑈) ≤ 0 ∧ 0 ≤ ((ℝ D 𝐹)‘𝑈)))) |
42 | 25, 35, 41 | mpbir2and 959 | 1 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ⊆ wss 3540 ∅c0 3874 class class class wbr 4583 dom cdm 5038 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 (,)cioo 12046 D cdv 23433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fi 8200 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-icc 12053 df-fz 12198 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-plusg 15781 df-mulr 15782 df-starv 15783 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-rest 15906 df-topn 15907 df-topgen 15927 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-xms 21935 df-ms 21936 df-cncf 22489 df-limc 23436 df-dv 23437 |
This theorem is referenced by: rollelem 23556 dvivthlem1 23575 |
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