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Theorem dvferm 25963

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 4295	. . . . . . 7 ⊢ (𝑈 ∈ (𝐴(,)𝐵) → (𝐴(,)𝐵) ≠ ∅)
7		ndmioo 13300	. . . . . . . 8 ⊢ (¬ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) = ∅)
8	7	necon1ai 2960	. . . . . . 7 ⊢ ((𝐴(,)𝐵) ≠ ∅ → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
9	3, 6, 8	3syl 18	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
10	9	simpld 494	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
11		ioossre 13335	. . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ ℝ
12	11, 3	sselid 3933	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ ℝ)
13	12	rexrd 11194	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ ℝ^*)
14		eliooord 13333	. . . . . . . 8 ⊢ (𝑈 ∈ (𝐴(,)𝐵) → (𝐴 < 𝑈 ∧ 𝑈 < 𝐵))
15	3, 14	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐴 < 𝑈 ∧ 𝑈 < 𝐵))
16	15	simpld 494	. . . . . 6 ⊢ (𝜑 → 𝐴 < 𝑈)
17	10, 13, 16	xrltled 13076	. . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝑈)
18		iooss1 13308	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≤ 𝑈) → (𝑈(,)𝐵) ⊆ (𝐴(,)𝐵))
19	10, 17, 18	syl2anc 585	. . . 4 ⊢ (𝜑 → (𝑈(,)𝐵) ⊆ (𝐴(,)𝐵))
20		dvferm.r	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ (𝐴(,)𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈))
21		ssralv 4004	. . . 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 25960	. 2 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) ≤ 0)
24	9	simprd 495	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
25	15	simprd 495	. . . . . 6 ⊢ (𝜑 → 𝑈 < 𝐵)
26	13, 24, 25	xrltled 13076	. . . . 5 ⊢ (𝜑 → 𝑈 ≤ 𝐵)
27		iooss2 13309	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝑈 ≤ 𝐵) → (𝐴(,)𝑈) ⊆ (𝐴(,)𝐵))
28	24, 26, 27	syl2anc 585	. . . 4 ⊢ (𝜑 → (𝐴(,)𝑈) ⊆ (𝐴(,)𝐵))
29		ssralv 4004	. . . 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 25962	. 2 ⊢ (𝜑 → 0 ≤ ((ℝ D 𝐹)‘𝑈))
32		dvfre 25926	. . . . 5 ⊢ ((𝐹:𝑋⟶ℝ ∧ 𝑋 ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
33	1, 2, 32	syl2anc 585	. . . 4 ⊢ (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
34	33, 5	ffvelcdmd 7039	. . 3 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) ∈ ℝ)
35		0re 11146	. . 3 ⊢ 0 ∈ ℝ
36		letri3 11230	. . 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 587	. 2 ⊢ (𝜑 → (((ℝ D 𝐹)‘𝑈) = 0 ↔ (((ℝ D 𝐹)‘𝑈) ≤ 0 ∧ 0 ≤ ((ℝ D 𝐹)‘𝑈))))
38	23, 31, 37	mpbir2and 714	1 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) = 0)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ⊆ wss 3903 ∅c0 4287 class class class wbr 5100 dom cdm 5632 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 ℝcr 11037 0cc0 11038 ℝ^*cxr 11177 < clt 11178 ≤ cle 11179 (,)cioo 13273 D cdv 25835

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-map 8777 df-pm 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fi 9326 df-sup 9357 df-inf 9358 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13277 df-icc 13280 df-fz 13436 df-seq 13937 df-exp 13997 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-struct 17086 df-slot 17121 df-ndx 17133 df-base 17149 df-plusg 17202 df-mulr 17203 df-starv 17204 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-rest 17354 df-topn 17355 df-topgen 17375 df-psmet 21316 df-xmet 21317 df-met 21318 df-bl 21319 df-mopn 21320 df-fbas 21321 df-fg 21322 df-cnfld 21325 df-top 22853 df-topon 22870 df-topsp 22892 df-bases 22905 df-cld 22978 df-ntr 22979 df-cls 22980 df-nei 23057 df-lp 23095 df-perf 23096 df-cn 23186 df-cnp 23187 df-haus 23274 df-fil 23805 df-fm 23897 df-flim 23898 df-flf 23899 df-xms 24279 df-ms 24280 df-cncf 24842 df-limc 25838 df-dv 25839

This theorem is referenced by: rollelem 25964 dvivthlem1 25984

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