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Theorem dvferm 26363

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 4290	. . . . . . 7 ⊢ (𝑈 ∈ (𝐴(,)𝐵) → (𝐴(,)𝐵) ≠ ∅)
7		ndmioo 13543	. . . . . . . 8 ⊢ (¬ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) = ∅)
8	7	necon1ai 2986	. . . . . . 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 13578	. . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ ℝ
12	11, 3	sselid 3911	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ ℝ)
13	12	rexrd 11398	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ ℝ^*)
14		eliooord 13576	. . . . . . . 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 13319	. . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝑈)
18		iooss1 13551	. . . . 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 3982	. . . 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 26360	. 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 13319	. . . . 5 ⊢ (𝜑 → 𝑈 ≤ 𝐵)
27		iooss2 13552	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝑈 ≤ 𝐵) → (𝐴(,)𝑈) ⊆ (𝐴(,)𝐵))
28	24, 26, 27	syl2anc 596	. . . 4 ⊢ (𝜑 → (𝐴(,)𝑈) ⊆ (𝐴(,)𝐵))
29		ssralv 3982	. . . 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 26362	. 2 ⊢ (𝜑 → 0 ≤ ((ℝ D 𝐹)‘𝑈))
32		dvfre 26326	. . . . 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 7117	. . 3 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) ∈ ℝ)
35		0re 11349	. . 3 ⊢ 0 ∈ ℝ
36		letri3 11434	. . 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 1570 ∈ wcel 2147 ≠ wne 2959 ∀wral 3080 ⊆ wss 3900 ∅c0 4281 class class class wbr 5108 dom cdm 5658 ⟶wf 6562 ‘cfv 6566 (class class class)co 7451 ℝcr 11238 0cc0 11239 ℝ^*cxr 11381 < clt 11382 ≤ cle 11383 (,)cioo 13516 D cdv 26238

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-1st 8035 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-er 8755 df-map 8887 df-pm 8888 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-fi 9440 df-sup 9471 df-inf 9472 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-z 12734 df-dec 12855 df-uz 13006 df-q 13116 df-rp 13161 df-xneg 13281 df-xadd 13282 df-xmul 13283 df-ioo 13520 df-icc 13523 df-fz 13680 df-seq 14185 df-exp 14245 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-struct 17369 df-slot 17404 df-ndx 17416 df-base 17432 df-plusg 17486 df-mulr 17487 df-starv 17488 df-tset 17492 df-ple 17493 df-ds 17495 df-unif 17496 df-euf 17499 df-rest 17643 df-topn 17644 df-topgen 17664 df-psmet 21721 df-xmet 21722 df-met 21723 df-bl 21724 df-mopn 21725 df-fbas 21726 df-fg 21727 df-cnfld 21730 df-top 23267 df-topon 23284 df-topsp 23306 df-bases 23319 df-cld 23392 df-ntr 23393 df-cls 23394 df-nei 23471 df-lp 23509 df-perf 23510 df-cn 23600 df-cnp 23601 df-haus 23688 df-fil 24220 df-fm 24312 df-flim 24313 df-flf 24314 df-xms 24694 df-ms 24695 df-cncf 25254 df-limc 26241 df-dv 26242

This theorem is used by: rollelem 26364 dvivthlem1 26383

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