dvferm - Metamath Proof Explorer

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Theorem dvferm 26320

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.)

**Hypotheses**
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))

**Assertion**
Ref	Expression
dvferm	⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) = 0)

Distinct variable groups: 𝑦,𝐴 𝑦,𝐵 𝑦,𝐹 𝑦,𝑈 𝑦,𝑋 𝜑,𝑦

**Proof of Theorem dvferm**
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 4289	. . . . . . 7 ⊢ (𝑈 ∈ (𝐴(,)𝐵) → (𝐴(,)𝐵) ≠ ∅)
7		ndmioo 13513	. . . . . . . 8 ⊢ (¬ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) = ∅)
8	7	necon1ai 2985	. . . . . . 7 ⊢ ((𝐴(,)𝐵) ≠ ∅ → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
9	3, 6, 8	3syl 19	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
10	9	simpld 500	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
11		ioossre 13548	. . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ ℝ
12	11, 3	sselid 3910	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ ℝ)
13	12	rexrd 11369	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ ℝ^*)
14		eliooord 13546	. . . . . . . 8 ⊢ (𝑈 ∈ (𝐴(,)𝐵) → (𝐴 < 𝑈 ∧ 𝑈 < 𝐵))
15	3, 14	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝐴 < 𝑈 ∧ 𝑈 < 𝐵))
16	15	simpld 500	. . . . . 6 ⊢ (𝜑 → 𝐴 < 𝑈)
17	10, 13, 16	xrltled 13289	. . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝑈)
18		iooss1 13521	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≤ 𝑈) → (𝑈(,)𝐵) ⊆ (𝐴(,)𝐵))
19	10, 17, 18	syl2anc 596	. . . 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 66	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ (𝑈(,)𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈))
23	1, 2, 3, 4, 5, 22	dvferm1 26317	. 2 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) ≤ 0)
24	9	simprd 501	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
25	15	simprd 501	. . . . . 6 ⊢ (𝜑 → 𝑈 < 𝐵)
26	13, 24, 25	xrltled 13289	. . . . 5 ⊢ (𝜑 → 𝑈 ≤ 𝐵)
27		iooss2 13522	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝑈 ≤ 𝐵) → (𝐴(,)𝑈) ⊆ (𝐴(,)𝐵))
28	24, 26, 27	syl2anc 596	. . . 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 66	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ (𝐴(,)𝑈)(𝐹‘𝑦) ≤ (𝐹‘𝑈))
31	1, 2, 3, 4, 5, 30	dvferm2 26319	. 2 ⊢ (𝜑 → 0 ≤ ((ℝ D 𝐹)‘𝑈))
32		dvfre 26283	. . . . 5 ⊢ ((𝐹:𝑋⟶ℝ ∧ 𝑋 ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
33	1, 2, 32	syl2anc 596	. . . 4 ⊢ (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
34	33, 5	ffvelcdmd 7093	. . 3 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) ∈ ℝ)
35		0re 11320	. . 3 ⊢ 0 ∈ ℝ
36		letri3 11405	. . 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 598	. 2 ⊢ (𝜑 → (((ℝ D 𝐹)‘𝑈) = 0 ↔ (((ℝ D 𝐹)‘𝑈) ≤ 0 ∧ 0 ≤ ((ℝ D 𝐹)‘𝑈))))
38	23, 31, 37	mpbir2and 726	1 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) = 0)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1569 ∈ wcel 2146 ≠ wne 2958 ∀wral 3079 ⊆ wss 3899 ∅c0 4280 class class class wbr 5105 dom cdm 5654 ⟶wf 6542 ‘cfv 6546 (class class class)co 7427 ℝcr 11209 0cc0 11210 ℝ^*cxr 11352 < clt 11353 ≤ cle 11354 (,)cioo 13486 D cdv 26195

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5234 ax-sep 5251 ax-nul 5262 ax-pow 5330 ax-pr 5394 ax-un 7758 ax-cnex 11266 ax-resscn 11267 ax-1cn 11268 ax-icn 11269 ax-addcl 11270 ax-addrcl 11271 ax-mulcl 11272 ax-mulrcl 11273 ax-mulcom 11274 ax-addass 11275 ax-mulass 11276 ax-distr 11277 ax-i2m1 11278 ax-1ne0 11279 ax-1rid 11280 ax-rnegex 11281 ax-rrecex 11282 ax-cnre 11283 ax-pre-lttri 11284 ax-pre-lttrn 11285 ax-pre-ltadd 11286 ax-pre-mulgt0 11287 ax-pre-sup 11288

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3455 df-sbc 3742 df-csb 3851 df-ss 3900 df-pss 4016 df-dif 4049 df-in 4093 df-un 4153 df-nul 4281 df-if 4485 df-pw 4561 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4838 df-uni 4870 df-int 4910 df-iun 4955 df-iin 4956 df-br 5106 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5549 df-eprel 5554 df-po 5562 df-so 5563 df-fr 5607 df-we 5609 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6310 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6500 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7384 df-ov 7430 df-oprab 7431 df-mpo 7432 df-om 7885 df-1st 8008 df-2nd 8009 df-frecs 8310 df-wrecs 8341 df-recs 8390 df-rdg 8429 df-1o 8487 df-er 8728 df-map 8860 df-pm 8861 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fi 9413 df-sup 9444 df-inf 9445 df-pnf 11355 df-mnf 11356 df-xr 11357 df-ltxr 11358 df-le 11359 df-sub 11553 df-neg 11554 df-div 11984 df-nn 12346 df-2 12415 df-3 12416 df-4 12417 df-5 12418 df-6 12419 df-7 12420 df-8 12421 df-9 12422 df-n0 12617 df-z 12704 df-dec 12825 df-uz 12976 df-q 13086 df-rp 13131 df-xneg 13251 df-xadd 13252 df-xmul 13253 df-ioo 13490 df-icc 13493 df-fz 13650 df-seq 14155 df-exp 14215 df-cj 15278 df-re 15279 df-im 15280 df-sqrt 15414 df-abs 15415 df-struct 17337 df-slot 17372 df-ndx 17384 df-base 17400 df-plusg 17453 df-mulr 17454 df-starv 17455 df-tset 17459 df-ple 17460 df-ds 17462 df-unif 17463 df-rest 17605 df-topn 17606 df-topgen 17626 df-psmet 21682 df-xmet 21683 df-met 21684 df-bl 21685 df-mopn 21686 df-fbas 21687 df-fg 21688 df-cnfld 21691 df-top 23224 df-topon 23241 df-topsp 23263 df-bases 23276 df-cld 23349 df-ntr 23350 df-cls 23351 df-nei 23428 df-lp 23466 df-perf 23467 df-cn 23557 df-cnp 23558 df-haus 23645 df-fil 24177 df-fm 24269 df-flim 24270 df-flf 24271 df-xms 24651 df-ms 24652 df-cncf 25211 df-limc 26198 df-dv 26199

This theorem is used by: rollelem 26321 dvivthlem1 26340

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