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Theorem dvferm 25973

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 4269	. . . . . . 7 ⊢ (𝑈 ∈ (𝐴(,)𝐵) → (𝐴(,)𝐵) ≠ ∅)
7		ndmioo 13316	. . . . . . . 8 ⊢ (¬ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) = ∅)
8	7	necon1ai 2961	. . . . . . 7 ⊢ ((𝐴(,)𝐵) ≠ ∅ → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
9	3, 6, 8	3syl 18	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
10	9	simpld 495	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
11		ioossre 13351	. . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ ℝ
12	11, 3	sselid 3913	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ ℝ)
13	12	rexrd 11186	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ ℝ^*)
14		eliooord 13349	. . . . . . . 8 ⊢ (𝑈 ∈ (𝐴(,)𝐵) → (𝐴 < 𝑈 ∧ 𝑈 < 𝐵))
15	3, 14	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐴 < 𝑈 ∧ 𝑈 < 𝐵))
16	15	simpld 495	. . . . . 6 ⊢ (𝜑 → 𝐴 < 𝑈)
17	10, 13, 16	xrltled 13092	. . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝑈)
18		iooss1 13324	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≤ 𝑈) → (𝑈(,)𝐵) ⊆ (𝐴(,)𝐵))
19	10, 17, 18	syl2anc 590	. . . 4 ⊢ (𝜑 → (𝑈(,)𝐵) ⊆ (𝐴(,)𝐵))
20		dvferm.r	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ (𝐴(,)𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈))
21		ssralv 3983	. . . 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 25970	. 2 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) ≤ 0)
24	9	simprd 496	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
25	15	simprd 496	. . . . . 6 ⊢ (𝜑 → 𝑈 < 𝐵)
26	13, 24, 25	xrltled 13092	. . . . 5 ⊢ (𝜑 → 𝑈 ≤ 𝐵)
27		iooss2 13325	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝑈 ≤ 𝐵) → (𝐴(,)𝑈) ⊆ (𝐴(,)𝐵))
28	24, 26, 27	syl2anc 590	. . . 4 ⊢ (𝜑 → (𝐴(,)𝑈) ⊆ (𝐴(,)𝐵))
29		ssralv 3983	. . . 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 25972	. 2 ⊢ (𝜑 → 0 ≤ ((ℝ D 𝐹)‘𝑈))
32		dvfre 25936	. . . . 5 ⊢ ((𝐹:𝑋⟶ℝ ∧ 𝑋 ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
33	1, 2, 32	syl2anc 590	. . . 4 ⊢ (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
34	33, 5	ffvelcdmd 7026	. . 3 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) ∈ ℝ)
35		0re 11137	. . 3 ⊢ 0 ∈ ℝ
36		letri3 11222	. . 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 592	. 2 ⊢ (𝜑 → (((ℝ D 𝐹)‘𝑈) = 0 ↔ (((ℝ D 𝐹)‘𝑈) ≤ 0 ∧ 0 ≤ ((ℝ D 𝐹)‘𝑈))))
38	23, 31, 37	mpbir2and 719	1 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) = 0)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 ∀wral 3053 ⊆ wss 3883 ∅c0 4261 class class class wbr 5072 dom cdm 5618 ⟶wf 6481 ‘cfv 6485 (class class class)co 7356 ℝcr 11028 0cc0 11029 ℝ^*cxr 11169 < clt 11170 ≤ cle 11171 (,)cioo 13289 D cdv 25848

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8633 df-map 8765 df-pm 8766 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fi 9314 df-sup 9345 df-inf 9346 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-icc 13296 df-fz 13453 df-seq 13955 df-exp 14015 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-struct 17108 df-slot 17143 df-ndx 17155 df-base 17171 df-plusg 17224 df-mulr 17225 df-starv 17226 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-rest 17376 df-topn 17377 df-topgen 17397 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-fbas 21344 df-fg 21345 df-cnfld 21348 df-top 22877 df-topon 22894 df-topsp 22916 df-bases 22929 df-cld 23002 df-ntr 23003 df-cls 23004 df-nei 23081 df-lp 23119 df-perf 23120 df-cn 23210 df-cnp 23211 df-haus 23298 df-fil 23829 df-fm 23921 df-flim 23922 df-flf 23923 df-xms 24303 df-ms 24304 df-cncf 24863 df-limc 25851 df-dv 25852

This theorem is referenced by: rollelem 25974 dvivthlem1 25993

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