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Mirrors > Home > MPE Home > Th. List > dvferm | Structured version Visualization version GIF version |
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.) |
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)) |
Ref | Expression |
---|---|
dvferm | ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) = 0) |
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 4347 | . . . . . . 7 ⊢ (𝑈 ∈ (𝐴(,)𝐵) → (𝐴(,)𝐵) ≠ ∅) | |
7 | ndmioo 13411 | . . . . . . . 8 ⊢ (¬ (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴(,)𝐵) = ∅) | |
8 | 7 | necon1ai 2966 | . . . . . . 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 13445 | . . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ ℝ | |
12 | 11, 3 | sselid 3993 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ ℝ) |
13 | 12 | rexrd 11309 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ ℝ*) |
14 | eliooord 13443 | . . . . . . . 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 13189 | . . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝑈) |
18 | iooss1 13419 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≤ 𝑈) → (𝑈(,)𝐵) ⊆ (𝐴(,)𝐵)) | |
19 | 10, 17, 18 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝑈(,)𝐵) ⊆ (𝐴(,)𝐵)) |
20 | dvferm.r | . . . 4 ⊢ (𝜑 → ∀𝑦 ∈ (𝐴(,)𝐵)(𝐹‘𝑦) ≤ (𝐹‘𝑈)) | |
21 | ssralv 4064 | . . . 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 26038 | . 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 13189 | . . . . 5 ⊢ (𝜑 → 𝑈 ≤ 𝐵) |
27 | iooss2 13420 | . . . . 5 ⊢ ((𝐵 ∈ ℝ* ∧ 𝑈 ≤ 𝐵) → (𝐴(,)𝑈) ⊆ (𝐴(,)𝐵)) | |
28 | 24, 26, 27 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝐴(,)𝑈) ⊆ (𝐴(,)𝐵)) |
29 | ssralv 4064 | . . . 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 26040 | . 2 ⊢ (𝜑 → 0 ≤ ((ℝ D 𝐹)‘𝑈)) |
32 | dvfre 26004 | . . . . 5 ⊢ ((𝐹:𝑋⟶ℝ ∧ 𝑋 ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ) | |
33 | 1, 2, 32 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ) |
34 | 33, 5 | ffvelcdmd 7105 | . . 3 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) ∈ ℝ) |
35 | 0re 11261 | . . 3 ⊢ 0 ∈ ℝ | |
36 | letri3 11344 | . . 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 586 | . 2 ⊢ (𝜑 → (((ℝ D 𝐹)‘𝑈) = 0 ↔ (((ℝ D 𝐹)‘𝑈) ≤ 0 ∧ 0 ≤ ((ℝ D 𝐹)‘𝑈)))) |
38 | 23, 31, 37 | mpbir2and 713 | 1 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑈) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ⊆ wss 3963 ∅c0 4339 class class class wbr 5148 dom cdm 5689 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 0cc0 11153 ℝ*cxr 11292 < clt 11293 ≤ cle 11294 (,)cioo 13384 D cdv 25913 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fi 9449 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-icc 13391 df-fz 13545 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-struct 17181 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-mulr 17312 df-starv 17313 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-rest 17469 df-topn 17470 df-topgen 17490 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-cncf 24918 df-limc 25916 df-dv 25917 |
This theorem is referenced by: rollelem 26042 dvivthlem1 26062 |
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