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Theorem dvferm 26130

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 4302	. . . . . . 7 ⊢ (𝑈 ∈ (𝐴(,)𝐵) → (𝐴(,)𝐵) ≠ ∅)
7		ndmioo 13402	. . . . . . . 8 ⊢ (¬ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) = ∅)
8	7	necon1ai 2992	. . . . . . 7 ⊢ ((𝐴(,)𝐵) ≠ ∅ → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
9	3, 6, 8	3syl 19	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
10	9	simpld 499	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
11		ioossre 13437	. . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ ℝ
12	11, 3	sselid 3943	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ ℝ)
13	12	rexrd 11262	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ ℝ^*)
14		eliooord 13435	. . . . . . . 8 ⊢ (𝑈 ∈ (𝐴(,)𝐵) → (𝐴 < 𝑈 ∧ 𝑈 < 𝐵))
15	3, 14	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝐴 < 𝑈 ∧ 𝑈 < 𝐵))
16	15	simpld 499	. . . . . 6 ⊢ (𝜑 → 𝐴 < 𝑈)
17	10, 13, 16	xrltled 13178	. . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝑈)
18		iooss1 13410	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≤ 𝑈) → (𝑈(,)𝐵) ⊆ (𝐴(,)𝐵))
19	10, 17, 18	syl2anc 595	. . . 4 ⊢ (𝜑 → (𝑈(,)𝐵) ⊆ (𝐴(,)𝐵))
20		dvferm.r	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ (𝐴(,)𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈))
21		ssralv 4014	. . . 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 26127	. 2 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) ≤ 0)
24	9	simprd 500	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
25	15	simprd 500	. . . . . 6 ⊢ (𝜑 → 𝑈 < 𝐵)
26	13, 24, 25	xrltled 13178	. . . . 5 ⊢ (𝜑 → 𝑈 ≤ 𝐵)
27		iooss2 13411	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝑈 ≤ 𝐵) → (𝐴(,)𝑈) ⊆ (𝐴(,)𝐵))
28	24, 26, 27	syl2anc 595	. . . 4 ⊢ (𝜑 → (𝐴(,)𝑈) ⊆ (𝐴(,)𝐵))
29		ssralv 4014	. . . 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 26129	. 2 ⊢ (𝜑 → 0 ≤ ((ℝ D 𝐹)‘𝑈))
32		dvfre 26093	. . . . 5 ⊢ ((𝐹:𝑋⟶ℝ ∧ 𝑋 ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
33	1, 2, 32	syl2anc 595	. . . 4 ⊢ (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
34	33, 5	ffvelcdmd 7084	. . 3 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) ∈ ℝ)
35		0re 11213	. . 3 ⊢ 0 ∈ ℝ
36		letri3 11298	. . 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 597	. 2 ⊢ (𝜑 → (((ℝ D 𝐹)‘𝑈) = 0 ↔ (((ℝ D 𝐹)‘𝑈) ≤ 0 ∧ 0 ≤ ((ℝ D 𝐹)‘𝑈))))
38	23, 31, 37	mpbir2and 725	1 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) = 0)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1568 ∈ wcel 2150 ≠ wne 2965 ∀wral 3086 ⊆ wss 3913 ∅c0 4294 class class class wbr 5114 dom cdm 5665 ⟶wf 6536 ‘cfv 6540 (class class class)co 7414 ℝcr 11102 0cc0 11103 ℝ^*cxr 11245 < clt 11246 ≤ cle 11247 (,)cioo 13375 D cdv 26005

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2152 ax-9 2160 ax-10 2183 ax-11 2199 ax-12 2220 ax-ext 2742 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5340 ax-pr 5408 ax-un 7736 ax-cnex 11159 ax-resscn 11160 ax-1cn 11161 ax-icn 11162 ax-addcl 11163 ax-addrcl 11164 ax-mulcl 11165 ax-mulrcl 11166 ax-mulcom 11167 ax-addass 11168 ax-mulass 11169 ax-distr 11170 ax-i2m1 11171 ax-1ne0 11172 ax-1rid 11173 ax-rnegex 11174 ax-rrecex 11175 ax-cnre 11176 ax-pre-lttri 11177 ax-pre-lttrn 11178 ax-pre-ltadd 11179 ax-pre-mulgt0 11180 ax-pre-sup 11181

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2099 df-mo 2574 df-eu 2604 df-clab 2749 df-cleq 2762 df-clel 2845 df-nfc 2919 df-ne 2966 df-nel 3072 df-ral 3087 df-rex 3097 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3464 df-sbc 3753 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-iin 4964 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5560 df-eprel 5565 df-po 5573 df-so 5574 df-fr 5618 df-we 5620 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7866 df-1st 7989 df-2nd 7990 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8456 df-er 8697 df-map 8829 df-pm 8830 df-en 8947 df-dom 8948 df-sdom 8949 df-fin 8950 df-fi 9374 df-sup 9405 df-inf 9406 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11446 df-neg 11447 df-div 11875 df-nn 12237 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12508 df-z 12595 df-dec 12715 df-uz 12866 df-q 12976 df-rp 13020 df-xneg 13140 df-xadd 13141 df-xmul 13142 df-ioo 13379 df-icc 13382 df-fz 13539 df-seq 14041 df-exp 14101 df-cj 15153 df-re 15154 df-im 15155 df-sqrt 15289 df-abs 15290 df-struct 17210 df-slot 17245 df-ndx 17257 df-base 17273 df-plusg 17326 df-mulr 17327 df-starv 17328 df-tset 17332 df-ple 17333 df-ds 17335 df-unif 17336 df-rest 17478 df-topn 17479 df-topgen 17499 df-psmet 21497 df-xmet 21498 df-met 21499 df-bl 21500 df-mopn 21501 df-fbas 21502 df-fg 21503 df-cnfld 21506 df-top 23034 df-topon 23051 df-topsp 23073 df-bases 23086 df-cld 23159 df-ntr 23160 df-cls 23161 df-nei 23238 df-lp 23276 df-perf 23277 df-cn 23367 df-cnp 23368 df-haus 23455 df-fil 23986 df-fm 24078 df-flim 24079 df-flf 24080 df-xms 24460 df-ms 24461 df-cncf 25020 df-limc 26008 df-dv 26009

This theorem is referenced by: rollelem 26131 dvivthlem1 26150

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